mm-4649 === Subject: Question about Apostol Text Two questions about Tom Apostol's 2 vol set on calculus. 1) I am considering going to grad school for applied math (starting at the MS level) and I am wondering if Apostol is a good book to review (it's been about 8 years since I got my BS). I've noticed that it appears to be the source book that Rensselaer Poly applied math bases its prelims on. (Obviously, there is no algebra or topology, so I'm looking at Apostol as a way to review the essential facts about real analysis, matrices, vector analysis and diff eqs) 2) When I read the various reviews of Aopstol on Amazon.com, I get the impression that the editions vary significantly. I read one review that said Apostol treated Fourier analysis, yet in the editions available in my library, I did not see any such material covered. Matt === Subject: Re: Question about Apostol Text posting-account=JpxxPAgAAAAgwzQIYqn4j6syK-YhOmcF Gecko/20070309 Firefox/2.0.0.3,gzip(gfe),gzip(gfe) Two questions about Tom Apostol's 2 vol set on calculus. 1) I am considering going to grad school for applied math (starting at > the MS level) and I am wondering if Apostol is a good book to review > (it's been about 8 years since I got my BS). I've noticed that it > appears to be the source book that Rensselaer Poly applied math bases > its prelims on. (Obviously, there is no algebra or topology, so I'm > looking at Apostol as a way to review the essential facts about real > analysis, matrices, vector analysis and diff eqs) 2) When I read the various reviews of Aopstol on Amazon.com, I get the > impression that the editions vary significantly. I read one review > that said Apostol treated Fourier analysis, yet in the editions > available in my library, I did not see any such material covered. > Matt I have Apostol's Calculus, both volumes, not in front on me, but I don't recall seeing anything on Fourier Analysis. There is another book by Apostol, Mathematical Analysis, and this is where he goes in depth into Fourier Analysis (after covering Lebesgue Integrals). Those who contrast Apostol's Calculus and Rudin's Principles of Analysis are comparing apples and oranges. Apostol's Calculus is just - well - Calculus, very rigorous, fairly comprehensive, in fact both volumes include large sections on Linear Algebra and Differential Equations - and yet it's still Calculus rather than Analysis. I would say this is probably the best Calculus book you can get (I still use it as a reference). Im not sure what you mean by 'Various editions exist'. The latest edition was, if I remember correctly, the 2nd edition. You can get inexpensive 'international edition (softcover)' from eBay. Apostol's Calculus is commonly used in 'Honor Calculus' (Calculus with Theory, Rigorous Calculus or whatever the course name is), like at MIT. Im not sure if it's good for reviewing of Calculus, though. The level of details may be overwhelming. I would recommend to get both volumes anyway, just have it as a reference. If you just want to review some Calculus theorem, without rigor or sufficient depth, something like Stewart will be sufficient. Ridin's Principles of Analysis is not a Calculus book and completely unsuitable for reviewing the Calculus. (Yes, I took two semesters of Analysis, with Baby Rudin, so I'm quite familiar with that book). On a subject of Fourier Analysis, for rigorous exposure you need to be familiar with Lebesgue Integrals first. That's what Apostol does in his 'Mathematical Analysis. Incidently, this is another excellent book by Apostol, but once again - not a Calculus book. Highly recommend, though, as a good Introduction to Analysis and not as terse as Rudin's. To summarize, if you want to review Calculus, get some Calculus book, like Stewart. Apostol's Calculus is great but probably not the best book for reviewing. On the other hand, Apostol's Calculus also contains Linear Algebra and Differential Equations, you can read those sections for review as well. I'm not sure how much of analysis you would need for Applied math and whether you need to have some knowledge of it (on a level of, say, the first 7-8 Chapters of Rudin's Principles of Analysis), or having solid Calculus, Linear Algebra and ODE is all that's required. === Subject: Re: Question about Apostol Text posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > I have Apostol's Calculus, both volumes, not in front on me, but I > don't recall seeing anything on Fourier Analysis. There is another > book by Apostol, Mathematical Analysis, and this is where he goes in > depth into Fourier Analysis (after covering Lebesgue Integrals). > OK, that is where the Fourier analysis is coming from: my mistake in getting the two texts confused. So that answers my question about where Apostol covers Fourier analysis. Matt === Subject: Re: Question about Apostol Text Rudin's 'Principles Of Mathematical Analysis' is easier to carry and better preparation for grad school. === Subject: Re: Question about Apostol Text <25705036.1217602850801.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Rudin's 'Principles Of Mathematical Analysis' is easier to carry and better preparation for grad school. When you say better preparation in what sense do you mean this? Do you think Rudin is better in that it is more rigorous than Apostol? I had Rudin as an undergrad for a 2 semester course in analysis and got A's in both classes, but the reasons why I have shied away from it are: i) does not seem to be as comprehensive as Apostol (as far as the variety of topics it covers). ii) I am also not totally convinced that Rudin covers the material in sufficiently greater depth than Apostol does (with the possible exception of the last few chapters of Rudin that deals with measure and differential forms). BUT, I could be wrong here, so I'm interested in what others think in this regard. iii) Since my interest is in applied math, I was not crazy about a book dealing with the basic theory of calculus that has no figures, motivations, or applications. Matt === Subject: Re: Question about Apostol Text In most applied math programs you probably still wouldn't see many numbers. So in that sense Rudin might be better preparation. Also, if you get used to Rudin's style, he has texts on functional analysis and other topics that are useful. I suppose taste plays a part. === Subject: Re: An exact simplification challenge - 70 (MeijerG) posting-account=htkO2QoAAABig9a_npSTZGp0tWMcSMXY Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] Here's what I entered in Maple: s:=convert(c,StandardFunctions); e:=convert(s,Ei); simplify(e); Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this can be simplified further. One could convert it into 1-argument Ei calls (see FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer expression. -- Thomas Richard Maple Support Scientific Computers GmbH http://www.scientific.de === Subject: Re: An exact simplification challenge - 70 (MeijerG) posting-account=uD9kfgoAAABaqjCF8ol-EFTFI3g2PjhE Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] Here's what I entered in Maple: s:=convert(c,StandardFunctions); > e:=convert(s,Ei); > simplify(e); Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this > can be simplified further. > One could convert it into 1-argument Ei calls (see > FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer > expression. > Or, in terms of Si,Ci: 1/2*Pi^(1/2)*(Pi^2-4*Pi*Si(1)+4*Ci(1)^2+4*Si(1)^2) Mate === Subject: Re: An exact simplification challenge - 70 (MeijerG) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] > Here's what I entered in Maple: > s:=convert(c,StandardFunctions); > e:=convert(s,Ei); > simplify(e); > Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this > can be simplified further. > One could convert it into 1-argument Ei calls (see > FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer > expression. Or, in terms of Si,Ci: 1/2*Pi^(1/2)*(Pi^2-4*Pi*Si(1)+4*Ci(1)^2+4*Si(1)^2) > Mate For me it is a bit senseless to ask for simplifications for arbitrary MeijerG, since that class is certainly too large. Already for the hypergeometric type there would be enough, where Maple and other CAS will fail to 'simplify' missing special identities. For example convert 3F1 to MeijerG and try to return ... (which gives identities ...) === Subject: Re: An exact simplification challenge - 70 (MeijerG) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] > Here's what I entered in Maple: > s:=convert(c,StandardFunctions); > e:=convert(s,Ei); > simplify(e); > Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this > can be simplified further. > One could convert it into 1-argument Ei calls (see > FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer > expression. > Or, in terms of Si,Ci: > 1/2*Pi^(1/2)*(Pi^2-4*Pi*Si(1)+4*Ci(1)^2+4*Si(1)^2) > Mate > For me it is a bit senseless to ask for simplifications for > arbitrary MeijerG, since that class is certainly too large. > Already for the hypergeometric type there would be enough, > where Maple and other CAS will fail to 'simplify' missing > special identities. > For example convert 3F1 to MeijerG and try to return ... > (which gives identities ...) > AV> For me it is a bit senseless to ask for simplifications > AV> for arbitrary MeijerG, since that class is certainly too > AV> large. For arbitrary MeijerG, yes. Still, what we post (and have in store) is not arbitrary. Computer algebra systems will deliver to the customers more > powers/flexibility with a better implementation of approaches > to MeijerG (simplification). For example, http://www.planetquantum.com I agree with Mate (essentially: for internal use as mighty tool), that does not say 'do not care for MeijerG'. Well, seeing no systematics (except the one already said) your choices are arbitrary - it makes not that much difference from which chapters in what books or pages they are taken. Even it would be nice if CAS know the vast variety of all the identities - we know: they don't. So why to repeat it. They even do not know all 'useful' identities or transformations for hypergeometric functions. So why repeat it. Of course occasionally one can guess which in your 'tasks' has may be considered as variable and look up books etc. But it is a mess to type those formulae (besides the notational burdens). For example your task 69 reduces to a duplication formula, but no reasonable person would like to type that in hoping to come out with a simple result. BTW: Kelly Roach (=your link) uses exp before the usual MeijerG (at least in older papers). AFAIK in former days he did some work for/with Maple. Certainly a strong Mathematician - and those may 'always' beat existing CAS, giving enhancements. === Subject: Re: An exact simplification challenge - 70 (MeijerG) posting-account=uD9kfgoAAABaqjCF8ol-EFTFI3g2PjhE Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] > Here's what I entered in Maple: > s:=convert(c,StandardFunctions); > e:=convert(s,Ei); > simplify(e); > Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this > can be simplified further. > One could convert it into 1-argument Ei calls (see > FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer > expression. > Or, in terms of Si,Ci: > 1/2*Pi^(1/2)*(Pi^2-4*Pi*Si(1)+4*Ci(1)^2+4*Si(1)^2) > Mate > For me it is a bit senseless to ask for simplifications for > arbitrary MeijerG, since that class is certainly too large. > Already for the hypergeometric type there would be enough, > where Maple and other CAS will fail to 'simplify' missing > special identities. > For example convert 3F1 to MeijerG and try to return ... > (which gives identities ...) AV> For me it is a bit senseless to ask for simplifications > AV> for arbitrary MeijerG, since that class is certainly too > AV> large. For arbitrary MeijerG, yes. Still, what we post (and have in store) is not arbitrary. > MeijerG should be used mainly (if not exclusively) for internal tasks and not for challenges. There are so many _beautiful_ math problems ... === Subject: Re: -- approach to solving a second order differential equation (nonseries? <26983706.1217593241699.JavaMail.jakarta@nitrogen.mathforum.org>, > How might I go about solving the following second > order > differential equation xy'' + y' + xy = 0 > Is there a solution other than a series solution > that will help me solve this equation ? > http://en.wikipedia.org/wiki/Bessel_function Best wishes > Torsten. Bessel function J_0 is one solution. === Subject: Re: zeta stuff Do you know of an expression for the sum side of Product[1/(1-(1/Composite[n]^2))]= Sum = 12/Pi^2 ? n=2,3,4,... === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Note that I refer to a extra-set-theoretical reality here. If you like, > I consider a proof to be a linguistic entity . And your /sequences/ are > just mathematical means to analyze this entities. This would justify > the identification of a one-line proof with the formula making up > this proof.) > [...] I'm using 'proof' here in such a 'technical sense' and not in its > other non-technical sense as 'an argument or discourse that provides > convincing grounds for believing a proposition or an argument or > discourse that provides convincing grounds for believing that there > exists a formal mathematical proof of a certain formula'. >I was in fact using proof vs axiom mostly in the non-technical >sense. What I meant to say was exactly that stating a result as an >axiom cannot provide convincing grounds in the face of an argument >that states the opposite result from the basic properties of the >natural numbers plus induction. Possibly so. That has no relevance here, since you have not > given any such argument: The things you say are almost > always incomprehesibly garbled, and when we try to extract > what you really mean we find an error . Pathetic as usual. In spite of your reiterated bull and personal insults, your math is now provably wrong even by an absolute beginner like me! Go to hell, retard, the game is over. -LV >-LV > Same with me. The realistic view of a proof (as a certain linguistic entity) > does not contradict the notion of proof in this formal sense (due to Frege). > Anyway, I enjoyed our little exchange of ideas/thoughts. > B. > -- > For every line of Cantor's list it is true that this line does not > æcontain the diagonal number. æNevertheless the diagonal number may > æbe in the infinite list. (WM, sci.logic) David C. Ullrich Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result > Note that I refer to a extra-set-theoretical reality here. If you > like, > I consider a proof to be a _linguistic entity_. And your /sequences/ > are > just mathematical means to analyze this entities. This would > _justify_ > the identification of a one-line proof with the formula making up > this proof.) > [...] I'm using 'proof' here in such a 'technical sense' and not in > its > other non-technical sense as 'an argument or discourse that provides > convincing grounds for believing a proposition or an argument or > discourse that provides convincing grounds for believing that there > exists a formal mathematical proof of a certain formula'. >I was in fact using proof vs axiom mostly in the non-technical >sense. What I meant to say was exactly that stating a result as an >axiom cannot provide convincing grounds in the face of an argument >that states the opposite result from the basic properties of the >natural numbers plus induction. > Possibly so. That has no relevance here, since you have not > _given_ any such argument: The things you say are almost > always incomprehesibly garbled, and when we try to extract > what you really mean we find an _error_. > Pathetic as usual. In spite of your reiterated bull and personal insults, You don't realize how funny it is for _you_ to be complaining about personal insults? The other day I said something about the math, not about you, and your reply was OFF. Here I say something about the math, not about you, and you say Go to hell, retard. Exactly what personal insults are you talking about? >your math > is now provably wrong even by an absolute beginner like me! Really? I haven't seen you say _anything_ in reply to the point I made several times. Well, nothing about the math - saying OFF doesn't really count as proving what I said is wrong. By the way, when you call it _my_ math that's very funny. It's the same as the math of every mathematician on the planet. > Go to hell, retard, the game is over. Keep it up - this is the best way to convince people you're right. > -LV >-LV > Same with me. The realistic view of a proof (as a certain linguistic > entity) > does not contradict the notion of proof in this formal sense (due to > Frege). > Anyway, I enjoyed our little exchange of ideas/thoughts. > B. > -- > For every line of Cantor's list it is true that this line does not > æcontain the diagonal number. æNevertheless the diagonal number may > æbe in the infinite list. (WM, sci.logic) > David C. Ullrich > Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) -- David C. Ullrich === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > your math > is now provably wrong even by an absolute beginner like me! So far, the only suggestion of something from you that might be a proof is to take as an axiom some principle or another that contradicts mere intuitionistic logic combined with the axiom schema of separation. That is fine onto itself; we freely admit that ZFC is contradicted by any such principle. However, that leaves us with not an inkling as to how you would derive the mathematical theorems for calculus, or even for arithmetic, or even for any interesting mathematics at all. MoeBlee === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > your math > is now provably wrong even by an absolute beginner like me! So far, the only suggestion of something from you that might be a > proof is to take as an axiom some principle or another that > contradicts mere intuitionistic logic combined with the axiom schema > of separation. That is fine onto itself; we freely admit that ZFC is > contradicted by any such principle. However, that leaves us with not > an inkling as to how you would derive the mathematical theorems for > calculus, or even for arithmetic, or even for any interesting > mathematics at all. MoeBlee Yes, but you can understand that that job cannot be done in this context. There is just too much noise. -LV === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result > your math > is now provably wrong even by an absolute beginner like me! > So far, the only suggestion of something from you that might be a > proof is to take as an axiom some principle or another that > contradicts mere intuitionistic logic combined with the axiom schema > of separation. That is fine onto itself; we freely admit that ZFC is > contradicted by any such principle. However, that leaves us with not > an inkling as to how you would derive the mathematical theorems for > calculus, or even for arithmetic, or even for any interesting > mathematics at all. > MoeBlee Yes, but you can understand that that job cannot be done in this > context. There is just too much noise. Not a very good excuse. I have no problem posting correct mathematics here in spite of the noise. All you have to do is post a correct proof of what you assert - there's no way that the other posts can prevent you from doing that. Of course, the fact that what you're trying to prove is _false_ is going to make it hard. > -LV -- David C. Ullrich === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > your math > is now provably wrong even by an absolute beginner like me! > So far, the only suggestion of something from you that might be a > proof is to take as an axiom some principle or another that > contradicts mere intuitionistic logic combined with the axiom schema > of separation. That is fine onto itself; we freely admit that ZFC is > contradicted by any such principle. However, that leaves us with not > an inkling as to how you would derive the mathematical theorems for > calculus, or even for arithmetic, or even for any interesting > mathematics at all. > Yes, but you can understand that that job cannot be done in this > context. There is just too much noise. That classical mathematics goes on its merry way is no impediment for you to devise whatever theory you like. Indeed, you could learn from classical mathematics (and many other alternatives to classical mathematics that are part of the literature of the subject) how theories are put together, even if then your twist is to put together a quite different theory. MoeBlee === Subject: Alternate proof of Th7.13 in Rudin's PMA? posting-account=dUaYtQoAAABjEW06yhX-7UQnMbSgN7mB rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Hi folks, I'm self-studying my way through baby Rudin. I've reached theorem 7.13 and have come up with a proof. However, it doesn't use one of the givens so I suspect that it's incorrect. I was hoping someone here could find the flaw in my reasoning. (Please let me know if this is the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and a) {f_n} is a sequence of continuous functions on K, b) {f_n} converges pointwise to a continuous function f on K, c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, we can assign to each point x an integer M(x) such that |f_{n>=M(x)} (x) - f(x)| < epsilon. We have to show that an M can be found that satisfies this condition for all x in K. Due to the continuity of the f_n and of f, we can define a neighbourhood N(x) around each point x such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). These neighbourhoods form an open cover of K. Since K is compact, a finite number of the N(x) covers K, centered at x_i = {x_1,x_2,...x_P}. We can then set M = max M(x_i). Then |f_{n>=M}(x)-f(x)} < epsilon for all x in K, and uniform convergence is proved. As you can see, I haven't used (c) anywhere. Where's my mistake? Sina === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? >I'm self-studying my way through baby Rudin. I've reached theorem 7.13 >and have come up with a proof. However, it doesn't use one of the >givens so I suspect that it's incorrect. I was hoping someone here >could find the flaw in my reasoning. (Please let me know if this is >the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and >a) {f_n} is a sequence of continuous functions on K, >b) {f_n} converges pointwise to a continuous function f on K, >c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... >Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, >we can assign to each point x an integer M(x) such that |f_{n>=M(x)} >(x) - f(x)| < epsilon. We have to show that an M can be found that >satisfies this condition for all x in K. Due to the continuity of the >f_n and of f, we can define a neighbourhood N(x) around each point x >such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). These >neighbourhoods form an open cover of K. Since K is compact, a finite >number of the N(x) covers K, centered at x_i = {x_1,x_2,...x_P}. We >can then set M = max M(x_i). Then |f_{n>=M}(x)-f(x)} < epsilon for >all x in K, and uniform convergence is proved. As you can see, I haven't used (c) anywhere. Where's my mistake? Consider the following counterexample without c. For x in [0,1] and n >= 1, let f_n(x) = 4nx(1-x)/(1+(n-1)x)^2. Note that for all x in [0,1], lim f (x) = 0 n->oo n Therefore, the f_n satisfy a and b, yet 1 f ( --- ) = 1 n n+1 Thus, the f_n do not converge uniformly. Rob Johnson take out the trash before replying === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? > Hi folks, I'm self-studying my way through baby Rudin. I've reached theorem 7.13 > and have come up with a proof. However, it doesn't use one of the > givens so I suspect that it's incorrect. I was hoping someone here > could find the flaw in my reasoning. (Please let me know if this is > the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and > a) {f_n} is a sequence of continuous functions on K, > b) {f_n} converges pointwise to a continuous function f on K, > c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... > Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, > we can assign to each point x an integer M(x) such that |f_{n>=M(x)} > (x) - f(x)| < epsilon. We have to show that an M can be found that > satisfies this condition for all x in K. Due to the continuity of the > f_n and of f, we can define a neighbourhood N(x) around each point x > such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). The mistake is here. For each n >= M(x), there is a neighborhood U_n(x) that works for f_n - f, but the intersection of all these U_n(x)'s is what you need, and there's no reason for the intersection to be open. > These > neighbourhoods form an open cover of K. Since K is compact, a finite > number of the N(x) covers K, centered at x_i = {x_1,x_2,...x_P}. We > can then set M = max M(x_i). Then |f_{n>=M}(x)-f(x)} < epsilon for > all x in K, and uniform convergence is proved. As you can see, I haven't used (c) anywhere. Where's my mistake? > Sina === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? , Hi folks, I'm self-studying my way through baby Rudin. I've reached theorem 7.13 > and have come up with a proof. However, it doesn't use one of the > givens so I suspect that it's incorrect. I was hoping someone here > could find the flaw in my reasoning. (Please let me know if this is > the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and > a) {f_n} is a sequence of continuous functions on K, > b) {f_n} converges pointwise to a continuous function f on K, > c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... > Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, > we can assign to each point x an integer M(x) such that |f_{n>=M(x)} > (x) - f(x)| < epsilon. We have to show that an M can be found that > satisfies this condition for all x in K. Due to the continuity of the > f_n and of f, we can define a neighbourhood N(x) around each point x > such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). The mistake is here. For each n >= M(x), there is a neighborhood > U_n(x) that works for f_n - f, but the intersection of all these > U_n(x)'s is what you need, and there's no reason for the intersection > to be open. Is this acceptable? M_n = max|f - f_n| >= 0 exists, because f and f_n are continuous on a compact set. M_n -> 0, because f_n -> f everywhere on K. M_n >= M_{n+1} by c). For eps > 0 exists N such that for all n > N, we have M_n < eps. This last statement is equivalent to the desired result. -- Michael Press === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? > , > Hi folks, I'm self-studying my way through baby Rudin. I've reached theorem 7.13 > and have come up with a proof. However, it doesn't use one of the > givens so I suspect that it's incorrect. I was hoping someone here > could find the flaw in my reasoning. (Please let me know if this is > the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and > a) {f_n} is a sequence of continuous functions on K, > b) {f_n} converges pointwise to a continuous function f on K, > c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... > Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, > we can assign to each point x an integer M(x) such that |f_{n>=M(x)} > (x) - f(x)| < epsilon. We have to show that an M can be found that > satisfies this condition for all x in K. Due to the continuity of the > f_n and of f, we can define a neighbourhood N(x) around each point x > such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). The mistake is here. For each n >= M(x), there is a neighborhood > U_n(x) that works for f_n - f, but the intersection of all these > U_n(x)'s is what you need, and there's no reason for the intersection > to be open. Is this acceptable? M_n = max|f - f_n| >= 0 exists, because > f and f_n are continuous on a compact set. M_n -> 0, because f_n -> f everywhere on K. Two problems with this. First, you could set M_n = sup|f - f_n|, which exists without any continuity assumptions. Second, M_n -> 0 is equivalent to f_n -> f uniformly (exercise: prove this). So you are obtaining uniform convergence without using (c), whoops. > M_n >= M_{n+1} by c). For eps > 0 exists N such that for all n > N, we have M_n < eps. > This last statement is equivalent to the desired result. === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? >, > Hi folks, > I'm self-studying my way through baby Rudin. I've reached theorem 7.13 > and have come up with a proof. However, it doesn't use one of the > givens so I suspect that it's incorrect. I was hoping someone here > could find the flaw in my reasoning. (Please let me know if this is > the wrong group to be posting questions like this). > Here's the theorem: Suppose K is compact, and > a) {f_n} is a sequence of continuous functions on K, > b) {f_n} converges pointwise to a continuous function f on K, > c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... > Then f_n => f uniformly on K. > My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, > we can assign to each point x an integer M(x) such that |f_{n>=M(x)} > (x) - f(x)| < epsilon. We have to show that an M can be found that > satisfies this condition for all x in K. Due to the continuity of the > f_n and of f, we can define a neighbourhood N(x) around each point x > such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). > The mistake is here. For each n >= M(x), there is a neighborhood > U_n(x) that works for f_n - f, but the intersection of all these > U_n(x)'s is what you need, and there's no reason for the intersection > to be open. Is this acceptable? M_n = max|f - f_n| >= 0 exists, because >f and f_n are continuous on a compact set. M_n -> 0, because f_n -> f everywhere on K. M_n >= M_{n+1} by c). For eps > 0 exists N such that for all n > N, we have M_n < eps. >This last statement is equivalent to the desired result. Just because f_n -> f everywhere does not mean that M_n -> 0. See the counterexample that I gave earlier. That is, 4nx(1-x) f (x) = ------------ n (1+(n-1)x)^2 As before, for all x in [0,1], lim f (x) = 0 n->oo n Yet, the hump just moves toward 0 1 f ( --- ) = 1 n n+1 So M_n = 1 for all n. Another counterexample is n n f (x) = 4 x (1-x ) n Here we get that the hump moves toward 1 -1/n f ( 2 ) = 1 and again, M_n = 1 for all n. Rob Johnson take out the trash before replying === Subject: Re: Graph Theory: Cutting a Graph into Two in an Artificial Chemistry I don't have time to do this properly, but the general question for general graphs has the feel of NP-Complete. However, your situation will have many many more constraints on the graph. For example, you won't have 6000 vertices, all mutually connected. It feels to me like the graphs you are interested in will have only limited sorts of minimal cycles, and each atom will have small degree. In that case there are going to be clever branch-and-bound searches that should work. I don't, however, know of anything off the shelf. === Subject: Re: Graph Theory: Cutting a Graph into Two in an Artificial Chemistry <6fer50Fbb9lnU1@mid.individual.net> posting-account=oTDIagkAAACTxHurtPutBWvNQS8ZCNO9 Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Jul 31, 3:03 pm, Chris Gordon-Smith It's unclear what you mean by cut - there is more > than one definition, and for each definition, someone > thinks it's obviously the right one. Let me put the question a different way. Here is a brute force way to find > what I am looking for:- 1) I start with a graph that has a single connected component (such that > any node can be reached from any other node by traversing a series of > edges) 2) I then divide the graph's nodes into two mutually exclusive groups and > remove all of the edges between nodes in different groups, leaving the > other edges in place. If I do this and the result is exactly two connected > components, then I count that as a valid way that my original graph > (molecule) could split. If the result is more than two components, then I > discard the result as invalid. I would count a single node on its own as a > connected component. 3) I repeat (2) until I have tried all of the possible ways of dividing the > nodes of the original graph into two mutually exclusive groups 4) The set of all 'valid' splits from (2) is the answer I want. Is there an efficient way to get this set of 'valid' splits? > Since no one else has responded, no solution, but some thoughts... First, it sounds like you don't want to just count these splits, you want to obtain a list of pairs of disjoint sub-graphs which cover the original set of nodes. Second, your problem will become quite a bit more difficult if you want only a list of non-isomorphic such pairs. For eample, for ethane, one could argue that there are only two different such pairings: (CH3, CH3) and (CH3CH2, H); even though your algorithm above would naively count 7 distinct such pairings. And determining whether two graphs are isomorphic is (generally) not so easy. At any rate, there are relatively quick algorithms for determining whether two nodes a and b are members of the same component of a graph (for a starting point see: http://en.wikipedia.org/wiki/Disjoint-set_data_structure ). This suggests that you can try removing edges E_i = (n_a, n_b) one at a time, then check for connectivity of n_a and n_b. The trick would be to bound your searches so that once you have determined that some set of edges (E_1, E_2, .., E_k) causes a split, you don't re-examine sets of edge removals which contain that subset a second time; e.g., by considering the directed graph of all subsets of edges, and then traversing /that/ tree, stopping at any depth that produces a split. This approach can probably be modified so that if you know that some set of edges are are essentially the same (e.g., the C-H bonds in ethane), then you don't revisit removing a single C-H edge 6 times. It would probably be useful to know the size and complexity of the graphs you are working with. Trivially, if your graphs are trees, then the number of such splits is the number of edges. So if they are sufficiently small and generally tree-like except for a few cycles, it may be that this will affect the average expected complexity (as opposed to the worst-case complexity); and perhaps even a brute force search will be sufficient to proceed. Anyway, post if you find a nice algorithm for this... === Subject: Re: Graph Theory: Cutting a Graph into Two in an Artificial Chemistry <25198735.1217459003945.JavaMail.jakarta@nitrogen.mathforum.org> <6fer50Fbb9lnU1@mid.individual.net > It's unclear what you mean by cut - there is more than one > definition, and for each definition, someone thinks it's obviously > the right one. > Let me put the question a different way. Here is a brute force way to > find what I am looking for:- > 1) I start with a graph that has a single connected component (such > that any node can be reached from any other node by traversing a series > of edges) > 2) I then divide the graph's nodes into two mutually exclusive groups > and remove all of the edges between nodes in different groups, leaving > the other edges in place. If I do this and the result is exactly two > connected components, then I count that as a valid way that my original > graph (molecule) could split. If the result is more than two > components, then I discard the result as invalid. I would count a > single node on its own as a connected component. > 3) I repeat (2) until I have tried all of the possible ways of > dividing the nodes of the original graph into two mutually exclusive > groups > 4) The set of all 'valid' splits from (2) is the answer I want. > Is there an efficient way to get this set of 'valid' splits? > Since no one else has responded, no solution, but some thoughts... First, it sounds like you don't want to just count these splits, you > want to obtain a list of pairs of disjoint sub-graphs which cover the > original set of nodes. Second, your problem will become quite a bit more difficult if you want > only a list of non-isomorphic such pairs. For eample, for ethane, one > could argue that there are only two different such pairings: (CH3, > CH3) and (CH3CH2, H); even though your algorithm above would naively > count 7 distinct such pairings. And determining whether two graphs are > isomorphic is (generally) not so easy. It may be OK to count all of the isomorphic cases, because each results from the breaking of a different bond and I want my model to reflect the likelihood of each outcome. Other things being equal, if one bond is broken at random then the likeliest outcome is CH3CH2. > At any rate, there are relatively quick algorithms for determining > whether two nodes a and b are members of the same component of a graph > (for a starting point see: http://en.wikipedia.org/wiki/Disjoint-set_data_structure > may well end up using this. > ). This suggests that you can try removing edges E_i = (n_a, n_b) one at a > time, then check for connectivity of n_a and n_b. The trick would be to > bound your searches so that once you have determined that some set of > edges (E_1, E_2, .., E_k) causes a split, you don't re-examine sets of > edge removals which contain that subset a second time; e.g., by > considering the directed graph of all subsets of edges, and then > traversing /that/ tree, stopping at any depth that produces a split. This approach can probably be modified so that if you know that some set > of edges are are essentially the same (e.g., the C-H bonds in ethane), > then you don't revisit removing a single C-H edge 6 times. It would probably be useful to know the size and complexity of the > graphs you are working with. Trivially, if your graphs are trees, then > the number of such splits is the number of edges. So if they are > sufficiently small and generally tree-like except for a few cycles, it > may be that this will affect the average expected complexity (as opposed > to the worst-case complexity); and perhaps even a brute force search > will be sufficient to proceed. I don't want to limit the graphs to being trees. Also, the graphs could get quite large. In real chemistry molecules can have hundreds or even thousands of atoms. I don't know whether this would make the searches too large. If so, one possibility might be to limit the search so that I find a 'representative' set of splits rather than all of the possible splits. Chris www.simsoup.info Anyway, post if you find a nice algorithm for this... > === Subject: Information that can free you financially posting-account=ca3sVwoAAABbBgOuHm48TMnvyz3CC7Ze Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) Improve Your Financial Situation NOW! No matter what your personal financial goals may be ($4K to $20K a month or more), we have the training, resources, and amazing team support to help you achieve these goals, quickly and easily! http://www.mygoldplan.com/ebusaf/ === Subject: Re: help with deducing general eqn of limacon?! posting-account=x9DlGAoAAAAvZTRvRYG8JjmPJCeyCza7 SV1),gzip(gfe),gzip(gfe) æFormulation is easier with respect to the invariant origin. æTo derive further from center of rolling circle, you want to add > æextra (p,q) or ( c cos(gam, c sin(gam) ), where gam is angle of > tracer ptæending crank to the horizontal. æLets assume the rolling circle has radius n times that of fixed > æcircle, so has angle subtended at its center is n times less, as > arc length is same. gam = t + n t - pi. æhttp://i36.tinypic.com/5p0qq1.jpg æ(picture modified from from epicycloids wikipedia where n < 1 ) æSo the first and second coordinate increments with respect to > originæas fixed circle on cranks or vector lengths (c + c n) and cn > ærespectively to generate epicycloid are: æc (1+ n) { cos(t), sin(t) } æ+ c { cos(gam), sin(gam) } æIn Mathematica notation æParametricPlot[ (n+1)c {Cos[t],Sin[t]} - n c{Cos[(1/n+1)t],Sin[(1/n > æ+1)t] }, {t,0, æ2 n Pi }, AspectRatio->Automatic] æConversion into polar coordinates: radius = Sqrt[1 + 2*n*(1 + n) - > æ2*n*(1 + n)*Cos[t/n]] and æpolar angle = ArcTan[ U Cos[t] - Cos[U], > æU Sin[t] - Sin[U] ] ; U = (1 + 1/n)*t Hi Narasimham, I am very grateful for your help, but how can I derive the simpler formula r = b + a cos .9b for the limacon?? Sorry, but I'm not getting this! Michael === Subject: Re: the adjoint Any teacher who, when asked What does > this mean?, answers by parroting the > book definition, is a CRUMMY TEACHER. > Is that what you think about my previous writings? Perhaps it would do you good if you were to learn some book definitions yourself. It also helps your intuition, you know. Ignored from now on. Sebastiaan. === Subject: rumpled surfaces? posting-account=twLK8gkAAADX2eq14ORonqtyAuVoEcNf Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) In familiar Euclidean plane geometry, a set of angles can be placed edge-to-edge, with their vertices on a common point. When the sum of the angle measures equals exactly 360 degrees, the plane region about the point is filled. If the sum is positive and less than 360 degrees, the angles can still be placed edge-to-edge, and form a vertex of a solid figure. For example, the three right angles at the corner of a cube sum to 270 degrees. If the sum is greater than 360 degrees, the surface becomes rumpled (for lack of the correct term). Try fitting together ten 60- degree angles made of paper, and you'll see what I mean. The surface described in the paragraph above is rumpled about one point. Is it possible for a surface to be rumpled about every point? If so, what special geometries or topologies are required? What is the correct terminology for this concept, so that I can look it up for myself? Ted Shoemaker === Subject: Re: rumpled surfaces? > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? > Yes. A saddle shape has everywhere negative curvature and is hence is rumpled everywhere. > What is the correct terminology for this concept, so that I can look > it up for myself? > Google negative curvature surface, or hyperbolic geometry. Many/most of the links assume the surface has negative curvature everywhere, and not just at specific points. Ted Shoemaker === Subject: Re: rumpled surfaces? posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? What is the correct terminology for this concept, so that I can look > it up for myself? > Ted Shoemaker It has to do with double curvature of the originating smooth surface, whether normal curvature in two principal perpendicular directions is on the same side or on opposite sides. If a smooth surface of positive Gauss curvature(sphere, ellipsoid,paraboloid etc., like a football) is now discretized by triangulation, say more than 4 triangles meet at such a vertex, the sum of angles is less than 180 degrees.It is a synclastic surface, belongs to elliptic geometry. If a smooth surface of negative Gauss curvature(pseudosphere,hyperbolic paraboloid, catenoid etc.,like a Pringles potato chip) is now discretized by triangulation, say more than 4 triangles meet at a vertex, the sum of angles is more than 180 degrees.It is an anticlastic surface, belongs to hyperbolic geometry, all the points are saddle points. The relationship between rotation around a contour and the solid angle ( called integral curvature in steridians ) subtended by normals is simply stated in Gauss-Bonnet theorem.It can be stated for smooth as well as discretized/ triangulated/tesselated surfaces.. Hope it helps. Narasimham === Subject: Re: rumpled surfaces? posting-account=twLK8gkAAADX2eq14ORonqtyAuVoEcNf Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) triangle is less than 180 degrees. Yes, but that's not what I'm looking for. Just to be sure, I did look up Gaussian curvature and a few related topics; I couldn't see that they were the topic I wanted. Either I'm not understanding you, or you're not understanding me. Let's try again. CASE 1 Take a sheet of paper, and cut it into angles (I didn't say triangles) of various measures. Assemble the angles so that their vertices share a common point, and the angles are adjacent but not overlapping each other. If you do this on a plane, you can exactly fill the space around a point with 360 degrees of angles. If you do this on a sphere, you STILL get 360 degrees. If you doubt that, look at the north pole on a globe, and start counting longitude lines. CASE 2 Assemble your angles as before, except use a total of less than 360 degrees. Tape the adjacent edges together. Now tape the two outside edges together. You will have formed part of a cone or a prism. Noneuclidean geometry not required. (Not yet, anyway.) CASE 3 Assemble your angles again, this time making the angle measure total more than 360 degrees. Tape the adjacent edges together. Tape the two edges at the extremes together. You now have, for lack of the right term, a wrinkled skirt, with a single vertex. THE QUESTIONS, RESTATED 1. Is it possible to have a surface such that every point is surrounded by exactly 360 degrees of surface? (YES. Examples are the plane, sphere, etc.) 2. Is it possible to have a surface such that every point is the vertex of a wrinkled skirt? 3. Is it possible to have a surface such that every point is surrounded by less than 360 degrees of surface? 4. What branch of math deals with this? 5. What is the correct terminology for me to look up? Ted Shoemaker > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. > If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. > If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. > The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? > What is the correct terminology for this concept, so that I can look > it up for myself? > Ted Shoemaker It has to do with double curvature of the originating smooth surface, > whether normal curvature in two principal perpendicular directions is > on the same side or on opposite sides. If a smooth surface of positive Gauss curvature(sphere, > ellipsoid,paraboloid etc., like a football) is now discretized by > triangulation, say more than 4 triangles meet at such a vertex, the > sum of angles is less than 180 degrees.It is a synclastic surface, > belongs to elliptic geometry. If a smooth surface of negative Gauss > curvature(pseudosphere,hyperbolic paraboloid, catenoid etc.,like a > Pringles potato chip) is now discretized by triangulation, say more > than 4 triangles meet at a vertex, the sum of angles is more than 180 > degrees.It is an anticlastic surface, belongs to hyperbolic geometry, > all the points are saddle points. The relationship between rotation around a contour and the solid angle > ( called integral curvature in steridians ) subtended by normals is > simply stated in Gauss-Bonnet theorem.It can be stated for smooth as > well as discretized/ triangulated/tesselated surfaces.. Hope it helps. Narasimham === Subject: Re: rumpled surfaces? posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > triangle is less than 180 degrees. Yes, but that's not what I'm > looking for. Just to be sure, I did look up Gaussian curvature and a > few related topics; I couldn't see that they were the topic I wanted. Either I'm not understanding you, or you're not understanding me. > Let's try again. CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. If you doubt that, look > at the north pole on a globe, and start counting longitude lines. CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. Noneuclidean geometry not required. (Not yet, anyway.) CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? > 4. What branch of math deals with this? > 5. What is the correct terminology for me to look up? > Ted Shoemaker > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. > If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. > If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. > The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? > What is the correct terminology for this concept, so that I can look > it up for myself? > Ted Shoemaker > It has to do with double curvature of the originating smooth surface, > whether normal curvature in two principal perpendicular directions is > on the same side or on opposite sides. > If a smooth surface of positive Gauss curvature(sphere, > ellipsoid,paraboloid etc., like a football) is now discretized by > triangulation, say more than 4 triangles meet at such a vertex, the > sum of angles is less than 180 degrees.It is a synclastic surface, > belongs to elliptic geometry. > If a smooth surface of negative Gauss > curvature(pseudosphere,hyperbolic paraboloid, catenoid etc.,like a > Pringles potato chip) is now discretized by triangulation, say more > than 4 triangles meet at a vertex, the sum of angles is more than 180 > degrees.It is an anticlastic surface, belongs to hyperbolic geometry, > all the points are saddle points. > The relationship between rotation around a contour and the solid angle > ( called integral curvature in steridians ) subtended by normals is > simply stated in Gauss-Bonnet theorem.It can be stated for smooth as > well as discretized/ triangulated/tesselated surfaces.. Hope it helps. > Narasimham For a smooth surface the angle sum between tangents at any point is locally always 360 degrees. For polyhedrons all the three cases are possible.So at a polyhedral vertex,three cases: On the plane, or any point of developable surface (exclude cone/ pyramid vertex singularity etc) .. sum = 360, K = 0. On elliptic convex or concave surfaces of discretized elliptic geometry, cone vertex, pyramid vertex, .. sum < 360, K > 0. On warped surfaces discretized saddle points of hyperbolic geometry,.. sum > 360, K < 0. Mathematica imaging may be helpful in seeing polyhedral vertices. But there is a hole, to avoid crowded convergent lines: << RealTime3D` Pringle =u { Cos[v] , Sin[v], Sin[2 v]/2} PR=ParametricPlot3D[Pringle,{u, 0, 1},{v,0,2 Pi},PlotPoints->{6,30} ] cone = u { Cos[v] , Sin[v], 1} CO=ParametricPlot3D[cone,{u, 0, 1},{v,0,2 Pi},PlotPoints->{6,30} ] disc = u { Cos[v] , Sin[v], 0} DI=ParametricPlot3D[disc,{u, 0, 1},{v,0,2 Pi},PlotPoints->{6,30} ] Show[PR,CO,DI] The depiction of surfaces using geodesic polar coordinates in differential geometry I think would be very useful in your further studies. Narasimham === Subject: Re: rumpled surfaces? posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) A smooth surface can be discretized and a discretized surface(rumpled or many faceted like the exterior of a diamond)can be smoothed. After discretization of a smooth surface lot of edges(E), vertices (V) and faces (F) develop, and their numbers obey the Euler law: V + F = E + 2. The sum of angles at a spherical vertex is less than 360 degrees. (Sorry, not 180 degrees as I typed wrongly).Please look at a model and faces e.g., of an icosahedron or dodecahedron or tetrahedron from among the set of ideal Platonic solids. The total sums up respectively to 60 X 5 = 300, 108 X 3 = 324 and 60 X 3 = 180 all less than 360 degrees. > triangle is less than 180 degrees. No no, by spherical triangle is meant a curvilinear triangle whose sides are geodesic arcs of great circles, the sum of three internal triangles is more than 180 degrees: Neither I referred to it nor is your query about it. You are asking about situation at any vertex of a polyhedron. Compared to a smooth differentiable surface, for a polyhedron normal curvature has sudden jumps at the edges. As the number of edges goes to infinity and area of faces created by a certain subdivision of curves passing through the vertices tends to zero, the polyhedron can be made to tend to a smooth surface, but the two are quite different. > Yes, but that's not what I'm > looking for. Just to be sure, I did look up Gaussian curvature and a > few related topics; I couldn't see that they were the topic I wanted. After some more study you will see that discretization of positive and negative Gaussian curvature surfaces is your topic. > Either I'm not understanding you, or you're not understanding me. > Let's try again. May be the former, but I shall help in explanations. > CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. > If you doubt that, look at the north pole on a globe, and start counting longitude lines. No, you are not distinguishing between a polyhedron and a smooth sphere. The sum of angles is < 360 degrees in the former, if number of faces -> Infinity, then the sum -> 360 degrees . > CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. you meant a pyramid. > CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. This warped surface that you call a wrinkled skirt can be described as discretized hyperbolic paraboloid with two hilly humps and two descending valleys for the two legs of a horse rider sitting on the horse saddle. Or even a monkey saddle , where there are three humps to go in between them are monkey's two legs and a tail. The monkey sits on it, bottom contacting a central (singular, but don't bother for now) point. > THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) plane OK, but not sphere. You can include cylinder, cone. > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? Yes. Look at the surface of a helicoid for instance.It has a simple parametrization. x= u cos(v), y= u sin(v), z = v. Two lines cut at any point, angle sum is 360 for this smooth surface. But do not make the same error as you did at the sphere north pole, because when you join the vertices forming a polytope, sum of angles at any vertex is more than 360 degrees. > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? Yes, e.g., at any vertex of Platonic solid, or any convex surface like ellipsoid suitably triangulated (by Veronoi, Delaunay etc. on convex hulls)and when the vertices are joined not by lines on the surface but by straight lines through the air. > 4. What branch of math deals with this? Differential geometry. Beware of going too deep into topology at this stage. > 5. What is the correct terminology for me to look up? Gauss curvature,Gauss-Bonnet theorem linking differential geometry and topology, parametrization of surfaces,triangulation,discretization. I suggest you build models of Platonic solids and also some from hyperbolic geometry like hyperbolic paraboloid or a catenoid or a helicoid using cardboard cuttings that may provide insight not just for what is happening at each point but how the entire surface is building up.I also suggest going through the book by David Hilbert and Cohn Vossen: Geometry and Imagination. Narasimham > Ted Shoemaker > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. > If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. > If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. > The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? > What is the correct terminology for this concept, so that I can look > it up for myself? > Ted Shoemaker > It has to do with double curvature of the originating smooth surface, > whether normal curvature in two principal perpendicular directions is > on the same side or on opposite sides. > If a smooth surface of positive Gauss curvature(sphere, > ellipsoid,paraboloid etc., like a football) is now discretized by > triangulation, say more than 4 triangles meet at such a vertex, the > sum of angles is less than 180 degrees.It is a synclastic surface, > belongs to elliptic geometry. > If a smooth surface of negative Gauss > curvature(pseudosphere,hyperbolic paraboloid, catenoid etc.,like a > Pringles potato chip) is now discretized by triangulation, say more > than 4 triangles meet at a vertex, the sum of angles is more than 180 > degrees.It is an anticlastic surface, belongs to hyperbolic geometry, > all the points are saddle points. > The relationship between rotation around a contour and the solid angle > ( called integral curvature in steridians ) subtended by normals is > simply stated in Gauss-Bonnet theorem.It can be stated for smooth as > well as discretized/ triangulated/tesselated surfaces.. Hope it helps. > Narasimham === Subject: Re: rumpled surfaces? May be the former, but I shall help in explanations. > CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. > If you doubt that, look at the north pole on a globe, and start counting > longitude lines. No, you are not distinguishing between a polyhedron and a smooth > sphere. The sum of angles is < 360 degrees in the former, if number > of faces -> Infinity, then the sum -> 360 degrees . > The sum of the angles around any point on a sphere is 360. What he says is completely true. He isn't asking about polyhedra in his question, and nor do I think its relevant to his problem. > CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. you meant a pyramid. No, I think he means a cone. His assembly process will result in a circle (if all the strips are the same length) pit a pie shaped degment removed - roll this up as he describes, and a cone is formed. > CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. This warped surface that you call a wrinkled skirt can be described as > discretized hyperbolic paraboloid with two hilly humps and two > descending valleys for the two legs of a horse rider sitting on the > horse saddle. Or even a monkey saddle , where there are three humps > to go in between them are monkey's two legs and a tail. The monkey > sits on it, bottom contacting a central (singular, but don't bother > for now) point. > Simpler than that. His wrinled shirt is a surface where the sum of the internal angles of a triangle is less than 360. Any hyperbolic surface does that. > THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) plane OK, but not sphere. You can include cylinder, cone. > plane OK, sphere OK, endless cylinder OK, cone not OK. I score you 2/4, which is no better than chance. > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? Yes. Look at the surface of a helicoid for instance.It has a simple > parametrization. x= u cos(v), y= u sin(v), z = v. Two lines cut at any > point, angle sum is 360 for this smooth surface. But do not make the > same error as you did at the sphere north pole, because when you join > the vertices forming a polytope, sum of angles at any vertex is more > than 360 degrees. > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? Yes, e.g., at any vertex of Platonic solid, or any convex surface like > ellipsoid suitably triangulated (by Veronoi, Delaunay etc. on convex > hulls)and when the vertices are joined not by lines on the surface but > by straight lines through the air. > He said every point. On Platonic surfaces, the ONLY points where the sum of the angles is less 360 are the finite number of points at the vertices. > 4. What branch of math deals with this? Differential geometry. Beware of going too deep into topology at this > stage. > 5. What is the correct terminology for me to look up? Gauss curvature,Gauss-Bonnet theorem linking differential geometry and > topology, parametrization of surfaces,triangulation,discretization. I suggest you build models of Platonic solids and also some from > hyperbolic geometry like hyperbolic paraboloid or a catenoid or a > helicoid using cardboard cuttings that may provide insight not just > for what is happening at each point but how the entire surface is > building up.I also suggest going through the book by David Hilbert and > Cohn Vossen: Geometry and Imagination. > I suggested he just Google hyperbolic geometry, which seems to be all he is really talking about. === Subject: Re: rumpled surfaces? <4895a49d$0$1025$afc38c87@news.optusnet.com.au> posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > May be the former, but I shall help in explanations. > CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. > If you doubt that, look at the north pole on a globe, and start counting > longitude lines. > No, you are not distinguishing between a polyhedron and a smooth > sphere. The sum of angles is < 360 degrees in the former, if number > of faces -> Infinity, then the sum -> 360 degrees . The sum of the angles around any point on a sphere is 360. What he says is > completely true. Indeed it is, no disagreement about the tangent plane situation where angle sum is 360 degrees. > He isn't asking about polyhedra in his question, and nor do I think its > relevant to his problem. Did you read OP's first post where he talks about the example 3 squares, 3 edges of a cube? If you assemble straight edges at a point, the point must necessarily be a polyhedron vertex. (The total can be more,equal or less than 360 degrees). A flat development for a curved surface of non-zero K is impossible by Gauss Egregium theorem and a faithful assembly in the tangent plane at the vertex where the straight adjacent edges are brought together all along contacting is also consequently impossible. > CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. > you meant a pyramid. No, I think he means a cone. His assembly process will result in a circle > (if all the strips are the same length) pit a pie shaped degment removed - > roll this up as he describes, and a cone is formed. Yes he meant a cone no doubt about it, next I suggested he meant a pyramid instead of a prism. > CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. > This warped surface that you call a wrinkled skirt can be described as > discretized hyperbolic paraboloid with two hilly humps and two > descending valleys for the two legs of a horse rider sitting on the > horse saddle. Or even a monkey saddle , where there are three humps > to go in between them are monkey's two legs and a tail. The monkey > sits on it, bottom contacting a central (singular, but don't bother > for now) point. Simpler than that. His wrinled shirt is a surface where the sum of the > internal angles of a triangle is less than 360. Any hyperbolic surface does > that. You perhaps mean sum of internal angles of a quadrilateral of 4 geodesic arcs is less than 360 degrees. I imagined a ballerina's mini skirt shape in spin dance. Does not matter, Gauss curvature K should be negative in any example given. > THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) > plane OK, but not sphere. You can include cylinder, cone. plane OK, sphere OK, endless cylinder OK, cone not OK. I mentioned cone the cylinder meaning all points of a cone except at the vertex which is a point of singularity.So cone OK except at cone vertex that should be mentioned which I hope OP would now begin to understand. Every point of a zero K surface is developable and is surrounded by exactly 360 degrees. There are other cases of zero K surfaces like the developable helicoid. > I score you 2/4, which is no better than chance. :) ! > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? > Yes. Look at the surface of a helicoid for instance.It has a simple > parametrization. x= u cos(v), y= u sin(v), z = v. Two lines cut at any > point, angle sum is 360 for this smooth surface. But do not make the > same error as you did at the sphere north pole, because when you join > the vertices forming a polytope, sum of angles at any vertex is more > than 360 degrees. > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? > Yes, e.g., at any vertex of Platonic solid, or any convex surface like > ellipsoid suitably triangulated (by Veronoi, Delaunay etc. on convex > hulls)and when the vertices are joined not by lines on the surface but > by straight lines through the air. He said every point. On Platonic surfaces, the ONLY points where the sum > of the angles is less 360 are the finite number of points at the vertices. Specifically mentioning it first for a vertex only, I proceeded to convex surface as I think that going from known to unknown would be more instructive for a student... without splitting hairs too much. > 4. What branch of math deals with this? > Differential geometry. Beware of going too deep into topology at this > stage. > 5. What is the correct terminology for me to look up? > Gauss curvature,Gauss-Bonnet theorem linking differential geometry and > topology, parametrization of surfaces,triangulation,discretization. > I suggest you build models of Platonic solids and also some from > hyperbolic geometry like hyperbolic paraboloid or a catenoid or a > helicoid using cardboard cuttings that may provide insight not just > for what is happening at each point but how the entire surface is > building up.I also suggest going through the book by David Hilbert and > Cohn Vossen: Geometry and Imagination. I suggested he just Google hyperbolic geometry, which seems to be all he > is really talking about. Differential and Riemannian geometries are more comprehensive for his guidance for non-Euclidean geometries. They include elliptic,hyperbolic and flat parabolic geometries as special cases.Pure hyperbolic geometry will not help him when angle sum is less than 360 degrees. There is also a classical book by Felix Klein on Non-Euclidean geometry (Vorlesungen ueber..) in German, IIRC another by HSM Coxeter etc. Narasimham === Subject: Re: rumpled surfaces? > triangle is less than 180 degrees. Yes, but that's not what I'm > looking for. Just to be sure, I did look up Gaussian curvature and a > few related topics; I couldn't see that they were the topic I wanted. Either I'm not understanding you, or you're not understanding me. > Let's try again. CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. If you doubt that, look > at the north pole on a globe, and start counting longitude lines. CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. Noneuclidean geometry not required. (Not yet, anyway.) CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) Yes. Examples are the plane, sphere etc. Indeed, any surface that has a curvature defined at every point has this characteristic. As you zoom in on the point, the surface looks locally more and more flat (eg a lake looks flat, even though its surface is actually part of a sphere the size of the earth), so the sum of angles is always 360. > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? Yes, a hyperbolic surface. I note that you have introduced a new and undefined term - vertex. I don't actually care what a vertex is in this example, other than it is a point, as every point on a hyperbolic plane has this characteristic you have described. So I don't care which points are also verices. > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? No, or at least not one which is everywhere differentiable (but see note below). > 4. What branch of math deals with this? Non-euclidean geometry, for starters. Topology for desert. > 5. What is the correct terminology for me to look up? > Non-euclidean geometry. Hyperbolic surfaces. Ted Shoemaker All everywhere differentiable surfaces are locally Euclidean (but see note below), so the sum of the angles around a point is always 360 degrees. Your cone fails because this is not true at the point of the cone, and this is the only place it fails. Zoom in as much as you like, but the area around the point never looks flat. Your use of the word vertex would seem to highlight your confusion. I am assuming that a vertex is a point on the surface which is not differentiable. This is a different issue to whether a triangle on the surface has interior angles adding to 180. As long as the triangle on a cone does not include the vertex, its interior angle will add to 180 degrees; a cone has zero curvature except at the vertex, where the curvature is undefined (or infinite, if you prefer). Note: This is well outside my field of expertise, and I am only guessing that the requirement is that the surface is everywhere differentiable; I think this is a necessary condition but may not be sufficient. There may be pathological functions that are everywhere differentiable but still have points with undefined curvature. === Subject: Re: rumpled surfaces? > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? What is the correct terminology for this concept, so that I can look > it up for myself? I think you're talking about a surface of negative Gaussian curvature. Look up: Gaussian curvature (e.g. ) -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: ? general norm soln for a model with errors > Hi: Most commonly seen approximated soln to a system of linear eqns A*x = b is the least-squares soln. One can find the solutions either when the system matrix A without and with uncertainty. But that assumes 2-norm being used. How does one solve the same problem when the matrix A is with uncertainty by using more general > norm(s)? > By using more general optimization algorithms. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: ? general norm soln for a model with errors posting-account=H-IscAoAAABkDNrURGSxo9jPN3MJ3a8A 1.0.3705; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) On Aug 1, 2:55æpm, Robert Israel > Hi: > æMost commonly seen approximated soln to a system of linear eqns > A*x = b is the least-squares soln. One can find the solutions either > when the system matrix A without and with uncertainty. > æBut that assumes 2-norm being used. How does one solve the same > problem when the matrix A is with uncertainty by using more general > norm(s)? By using more general optimization algorithms. > -- > Robert Israel æ æ æ æ æ æ æisr...@math.MyUniversitysInitials.ca > Department of Mathematics æ æ æ æhttp://www.math.ubc.ca/~israel > University of British Columbia æ æ æ æ æ æVancouver, BC, Canada I think I should make it more precise: can one have some more explicit formula to the soln of this kind of problem? I tried pure numerical approach for the case simply using the 2-norm and this pure numerical approach suffers convergent problem. When the 2-norm is used, there is explicit soln to this model-with-error problem: solving SVD of the extended system matrix and then take the (n+1) right singular vector, assuming full-ranked system. So I can compare numerical soln with explicit soln; numerical soln does not always provide a convergent soln. For exmaple, using MatLab built-in optimization routine with the following objective function. function y = funcTLS(x, AExtend) xExtend = [-1.0; x]; y = AExtend*xExtend; The above does not always give comparable soln with that of explicit form; depending on IC used. So let's narrow down my question: are there explicit soln for model- with-error problem using a norm that is not 2-norm or are there more reliable numerical approach to solve this? === Subject: Re: ? general norm soln for a model with errors posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > On Aug 1, 2:55 pm, Robert Israel > Hi: > Most commonly seen approximated soln to a system of linear eqns > A*x = b is the least-squares soln. One can find the solutions either > when the system matrix A without and with uncertainty. > But that assumes 2-norm being used. How does one solve the same > problem when the matrix A is with uncertainty by using more general > norm(s)? > By using more general optimization algorithms. > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada I think I should make it more precise: can one have some more explicit > formula to the soln of this kind of problem? I tried pure numerical approach for the case simply using the 2-norm > and this pure numerical approach suffers convergent problem. When the 2-norm is > used, there is explicit soln to this model-with-error problem: solving SVD > of the extended system matrix and then take the (n+1) right singular vector, > assuming full-ranked system. So I can compare numerical soln with explicit > soln; numerical soln does not always provide a convergent soln. For exmaple, using MatLab built-in optimization routine with the > following objective function. function y = funcTLS(x, AExtend) xExtend = [-1.0; x]; > y = AExtend*xExtend; The above does not always give comparable soln with that of explicit > form; depending on IC used. So let's narrow down my question: are there explicit soln for model- > with-error problem using a norm that is not 2-norm or are there more reliable > numerical approach to solve this? Are you saying that you have a system of equations A*x = b in which the elements of A and b are not known exactly? If so, you might benefit by looking at the literature on chance-constrained programming (now rather old) or the much more recent (and reportedly much more effective) approach of robust optimization. Reportedly, this technique is starting to have important impacts in areas such as bridge and building design, machine design, financial planning and the like. Of course, your problem does not look immediately like an optimization problem, but you could consider it as the problem of maximizing 0*x, subject to A*x = b, and so turn it into an optimization. Good luck. R.G. Vickson === Subject: Binary Sequences posting-account=dGiPYgkAAABSJ3xUlNLViQdT0h489hR6 AppleWebKit/523.10.3 (KHTML, like Gecko) Version/3.0.4 Safari/523.10,gzip(gfe),gzip(gfe) I have recently posted these three related sequences to the Encyclopedia Of Integer Sequences: (I don't know if they have appeared yet.) %S A143220 1,1,0,1,0,1,1,0,0,0,1,1,1,1,0,1,0,1,0,0,1,1,1,0 %N A143220 a(0)=1. For n >=1, a(n) = 1 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 0 otherwise. %e A143220 The binary representation of 21 is 10101. This occurs in the concatenation of terms a(0) through a(20) like so: 1101011000111(10101)001. So a(21) = 1. %Y A143220 A118268,A143221,A143222 %O A143220 0 %K A143220 ,base,more,nonn, %S A143221 1,0,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0,0 %N A143221 a(0)=1. For n >=1, a(n) = 0 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 1 otherwise. %e A143221 The binary representation of 21 is 10101. This occurs in the concatenation of terms a(0) through a(20) like so: 10010111(10101)00010010. So a(21) = 0. %Y A143221 A143220,A143222 %O A143221 0 %K A143221 ,base,more,nonn, %S A143222 0,1,1,0,1,0,0,1,1,0,0,1,0,0,1,1,1,1,0,0,0,1 %N A143222 a(0)=0. For n >=1, a(n) = 0 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 1 otherwise. %e A143222 The binary representation of 20 is 10100. This occurs in the concatenation of terms a(0) through a(19) like so: 01(10100)1100100111100. So a(20) = 0. %Y A143222 A143220,A143221 %O A143222 0 %K A143222 ,base,more,nonn, (Every term of the sequence where a(0)=0 and a(n)=1 if n does occur and a(n)=0 otherwise is trivially 0.) Let a = sum{k=0 to inf} A143220(k)/2^k, b = sum{k=0 to inf} A143221(k)/2^k, and c = sum{k=0 to inf} A143222(k)/2^k. Can a, b, and/or c be connected via any mathematical relations? Are there closed expressions for a, b, or c? Maybe the sequences have closed forms for their generating functions, I wonder. (I am guessing that the GFs with closed forms, if any, would be the ordinary GFs, but I could be wrong.) -- By the way, doing a search on Google Groups for Leroy Quet now brings up ZERO hits, when in fact I have posted THOUSANDS of posts to sci.math. (And I signed each one. So my name appears in each post, and should have been seen by a search.) Does anyone know what the hell is wrong with Goggle's search? Leroy Quet === Subject: Re: Binary Sequences posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) > I have recently posted these three related sequences to the > Encyclopedia Of Integer Sequences: > (I don't know if they have appeared yet.) %S A143220 1,1,0,1,0,1,1,0,0,0,1,1,1,1,0,1,0,1,0,0,1,1,1,0 > %N A143220 a(0)=1. For n >=1, a(n) = 1 if the binary representation of > n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). > a(n) = 0 otherwise. > %e A143220 The binary representation of 21 is 10101. This occurs in > the concatenation of terms a(0) through a(20) like so: > 1101011000111(10101)001. So a(21) = 1. > %Y A143220 A118268,A143221,A143222 > %O A143220 0 > %K A143220 ,base,more,nonn, %S A143221 1,0,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0,0 > %N A143221 a(0)=1. For n >=1, a(n) = 0 if the binary representation of > n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). > a(n) = 1 otherwise. > %e A143221 The binary representation of 21 is 10101. This occurs in > the concatenation of terms a(0) through a(20) like so: > 10010111(10101)00010010. So a(21) = 0. > %Y A143221 A143220,A143222 > %O A143221 0 > %K A143221 ,base,more,nonn, %S A143222 0,1,1,0,1,0,0,1,1,0,0,1,0,0,1,1,1,1,0,0,0,1 > %N A143222 a(0)=0. For n >=1, a(n) = 0 if the binary representation of > n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). > a(n) = 1 otherwise. > %e A143222 The binary representation of 20 is 10100. This occurs in > the concatenation of terms a(0) through a(19) like so: > 01(10100)1100100111100. So a(20) = 0. > %Y A143222 A143220,A143221 > %O A143222 0 > %K A143222 ,base,more,nonn, (Every term of the sequence where a(0)=0 and a(n)=1 if n does occur > and a(n)=0 otherwise is trivially 0.) Let a = sum{k=0 to inf} A143220(k)/2^k, > b = æsum{k=0 to inf} A143221(k)/2^k, and > c = æsum{k=0 to inf} A143222(k)/2^k. Can a, b, and/or c be connected via any mathematical relations? > Are there closed expressions for a, b, or c? Maybe the sequences have closed forms for their generating functions, > I wonder. > (I am guessing that the GFs with closed forms, if any, would be the > ordinary GFs, but I could be wrong.) -- By the way, doing a search on Google Groups for Leroy Quet now > brings up ZERO hits, when in fact I have posted THOUSANDS of posts to > sci.math. (And I signed each one. So my name appears in each post, and > should have been seen by a search.) > Does anyone know what the hell is wrong with Goggle's search? I've noticed curious bevaviour also. Just a couple minutes ago and it couldn't find it. I did find it eventually by looking for different keywords. In looking for leroy quet, I got 4 hits. Looking for quet returned 6000, so I would guess they are still all there. Here's one of the summaries. I've put square brackets [] around the bolded words that mark keyword matches. But note only 2 of the 3 occurences of quet are bolded (searching for Quet gave me exactly the same result, quet bolded, Quet not bolded). There definitely appears to be something wrong with the matching of keywords to content. That sucks. Imposter? Re:Sum Of Product Involving Group: rec.puzzles A leroy [quet], not this Leroy Quet, has posted something in my name, or so it seems. A few possibilities; 1) There is another leroy [quet]. 2) Someone is trying to mess with my mind. (Who? A long while back, someone else DID post something in my name. Have they returned?) 3) Looking at the content, I am starting ... Leroy Quet === Subject: moderator analysis posting-account=r5Mu_AoAAAA4Or1qD_TiZuru2F3PYCEL Gecko/20080702 Firefox/2.0.0.16 eMusic DLM/4.0_1.0.0.1,gzip(gfe),gzip(gfe) Hi everyone, I am analyzing the retention of subjects across the 3 timepoints of a study. I would like to determine whether the level of a variable (say varA) at baseline moderates continuation in the study as measured by presence at time 1 and time2. I am not sure how to best run that analysis. I was thinking I should get the correlations between varA, PresentAtTime1, and PresentAtTime2 where PresentAtTime1 is an indicator variable. I was also thinking I should run some logistic models. I am confused about those. PresentAtTime1= logistic(varA)? PresentAtTime2= logistic(PresentAtTime1+PresentAtTime1*varA)? I would appreciate some help or some pointers. === Subject: Re: How long did it take him to get home? I currently am reading Me, Myself and Them by Kurt > Snyder, which is > a well-written, interesting first person account of > paranoid > schizophrenia. At one point he is driving home and > developes the > delusion that the CIA is somehow controlling his > route home. To thwart > semi-random route > home, instead of letting them control me. I would > only make a turn in > the direction of my house if the last digit of the > minutes on the > digital clock read 1, 3, 5, 7. Otherwise, I would > continue straight > ahead. As you can probably guess, it took me a *very* > long time to get > home. (pgs. 79-80). Under certain simplifying assumptions, this can be > modeled by a Markov > chain. Take as states ordered triples (i,j,d) where > (i,j) is a point > with integer coordinates and d is in {N,S,E,W}. Home > states are of the > form (0,0,d). State transitions are hard to spell out > in detail > (especially if u-turns are not permitted). Home > states are absorbing. > Assuming u-turns it would be like this. With > probability p = 0.6 you > would move in the same direction. For example > (1,-3,S) would lead to > (1,-4,S). With probability 1-p you would go to a > neighboring point in > the plane (and adjust d accordingly) in such a way > that the taxicab > distance is decreased. If there is a tie between two > adjacent states > in this situation make the choice randomly. Note that > decreasing the > distance home might imply going in the direction you > are already > heading, so in some states the probability of going > straight might be > 1 rather than 0.6 (in which case you make a bee-line > home). He didn't > explicitly say that in the passage quoted above, but > it seems like a > natural interpretation. I don't know enough about > Markov chains to > know for sure, but it seems clear enough that with > probability one he > *does* get home with the expected time heavily > dependent on the > initial state. It is easy to see that, for any > initial state, > infinitely many states are inaccessible These > includine those on the > coordinate axis heading in the wrong direction > (unless the initial > state is on the same axis and also heading in the > wrong direction). > Are there other inaccessible states? Has anybody studied such semi-random walks? At the > very least it > would make for some fun computer simulations. -scattered p.s. The author also describes how when he was in > college he had > delusions of grandeur: I believed that I was going > to discover some > fabulous new mathematical principle that would > transform the way we > view the universe. I told no one about these > thoughts. I started > looking for clues to this mathematical theory in math > books I found at > the library. I actually learned very little about > math though, because > I couldn't focus on any material for any significant > length of > time ... but I still thought that one day I would get > a flash of > inspiration and become famous. (pgs. 1,2) I suspect > that most of the > cranks who post here are *not* schizophrenic, but > passages like that > just quoted make you wonder. OT, but if you liked that book (haven't read it, though I think I might like to)you might want to read another first person account of descent into schizophrenia-- Mark Vonnegut, The Eden Express. It takes place in the 1960s, when Vonnegut (son of Kurt Vonnegut, from whom he inherited the condition)was young. Mark Vonnegut, after treatment, became a successful medical doctor. Tom === Subject: Re: How long did it take him to get home? >p.s. The author also describes how when he was in college he had >delusions of grandeur: I believed that I was going to discover some >fabulous new mathematical principle that would transform the way we >view the universe. I told no one about these thoughts. I started >looking for clues to this mathematical theory in math books I found at >the library. I actually learned very little about math though, because >I couldn't focus on any material for any significant length of >time ... but I still thought that one day I would get a flash of >inspiration and become famous. (pgs. 1,2) That feels awfully familiar! I think I'd find this book a good read. (I've never actually been schizophrenic, but I'm a long way off from being normal). >I suspect that most of the >cranks who post here are *not* schizophrenic, but passages like that >just quoted make you wonder. Psychology is not an empirical science (notwithstanding the undoubted existence of an empirical science that calls itself psychology), so it's hardly surprising if its categories are as fluid and ambiguous as its subject matter. -- Angus Rodgers Contains mild peril === Subject: Efficient method to find roots of equations. Hi everyone! I currently encounter a root finding problem. f_1(x1)=f_2(x2)=...=f_N(xN) g_1(x1)+g_2(x2)+...+g_N(xN)=B === Subject: Re: Efficient method to find roots of equations. posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Hi everyone! I currently encounter a root finding problem. f 1(x1)=f 2(x2)=...=f N(xN) g 1(x1)+g 2(x2)+...+g N(xN)=B I know there is and only is one vector X=[x1,x1,...,xN] satisfying these equations. There is N equations and N variables. I wonder whether the Newton[CapitalEth]Raphson can be used to find the roots of these equations? If there there is one equation ,the Newton[CapitalEth]Raphson works, but does it work for multiple equations? Yes, of course, and it is one of the underlying methods of multivariate optimization. For multiple-equation Newton-Raphson, see, eg., www.haoli.org/nr/bookf/f9-6.ps or Section 7 of http://en.wikipedia.org/wiki/Newton's method or http://trond.hjorteland.com/thesis/node28.html . R.G. Vickson === Subject: Re: Efficient method to find roots of equations. === Subject: Re: Partially Persistent Tree Implementation Face: iVBORw0KGgoAAAANSUhEUgAAADAAAAAwAQMAAABtzGvEAAAABlBMVEUAAAD///+l2Z/dAAAA oElEQVR4nK3OsRHCMAwF0O8YQufUNIQRGIAja9CxSA55AxZgFO4coMgYrEDDQZWPIlNAjwq9 033pbOBPtbXuB6PKNBn5gZkhGa86Z4x2wE67O+06WxGD/HCOGR0deY3f9Ijwwt7rNGNf6Oac l/GuZTF1wFGKiYYHKSFAkjIo1b6sCYS1sVmFhhhahKQssRjRT90ITWUk6vvK3RsPGs+M1RuR mV+hO/VvFAAAAABJRU5ErkJggg== > I'm trying to implement a partially persistent tree structure. I plan > to cache a portion of the frequently accessed nodes in the memory > while the full tree will reside on the disk and changes committed to > memory/on-disk nodes will get propogated to their on-disk/memory > correspondents. Before going into further implementation details, I > want to check the literature for anything had done in the past > subject? I've got no literature, but once I implemented a quad-tree as a memory-mappable data structure. Since the number of children in a quad-tree is fixed (to 4), there's no need for pointers or offsets, you can compute directly the offset of a node from the position in the tree. However, for random trees, you can still do it, simply using offsets instead of pointers, so the data can be mapped at any address. Well I guess the only literature you need is mmap(2). -- __Pascal Bourguignon__ http://www.informatimago.com/ Indentation! -- I will show you how to indent when I indent your skull! === Subject: solution manual posting-account=N_qD3woAAACvn11QjAmpGm8Ieo5qY02Y 2.0.50727),gzip(gfe),gzip(gfe) Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ A first course in probability - Sheldon M. Ross - 7 ed Adaptive Control - Karl J. Astrom - 2 ed Advanced Macroeconomics - Jeffrey Rohaly Advanced Microeconomic Theory - Geoffrey Jehle Advanced Modern Engineering Mathematics - Glyn James - 3 ed Algebra- Baldor An introduction to numerical analysis - E. Suli, F. Mayers Analytical Mechanics - Fowles and Cassiday - 7 ed Antenna theory - Constantine Balanis - 2 ed Applied Numerical Analysis - Curtis F. Gerald, Patrick O. Wheatley - 7 ed Applied Numerical Methods - Steven Chapra Applied Probability models with optimization applications - Sheldon M. Ross Applied strength of materials - Robert L. Mott - 4 ed Artificial Intelligence - Stuart J. Russell y Peter Norvig - 2 ed Automatic control systems - Kuo and Golnaraghi - 8 ed Basic Engineering Circuit Analysis - David Irwin - 8 ed Calculo several variables - Hallet, Gleason McCallum - 4 ed Calculus - George B. Thomas - 11 ed - Vol 1 Calculus - George B. Thomas - 11 ed - Vol 2 Calculus - George Thomas Vol.2 Calculus - James Stewart - 5 ed Calculus - Jerrold Marsden, Alan Weinstein - vol 1 Calculus - Leithold - 7 ed Calculus - Purcell - 9 ed Calculus several variables - James Stewart - 4 Ed Calculus - Wards y Penney 4 ed Calculus 1 variable - Hallet, Gleason McCallum - 4 ed Calculus 1 variable - James Stewart - 4 Ed Calculus one and several variables - Salas Hille Etgen - 8 ed Calculus several variables - Neta B Calculus.Early.Transcendentals - Edition.James.Stewart - 5 ed vol1 and 2 Chemical and Engineering Thermodynamics - Stanley Sandler - 3 ed Communication systems engineering - John G.Proakis - 2 ed Computer Networking - Kurose, W. Ross - 3 ed Computer Networks - Andrew Tanenbaum - 4 ed Control Systems Engineering - Norman Nice Design and analysis of experiments - Douglas C. Montgomery - 6 ed Design of Machinery - Robert Norton - 3 ed Device Electronics for Integrated Circuits - Richard S. Muller, Theodore I. Kamins - 3 ed Differential equations - Dennis G Zill - 7 ed Differential equations linear algebra - Jerry Farlow - 2 ed Digital Comunications - Bernard Sklar - 2 ed Digital Comunications - John G. Proakis - 4 ed Digital image processing - Rafael C. Gonzalez, Richard E. Woods - 2 ed Digital Signal Processing - John G. Proakis - 3 ed Digital signal processing - Sanjit K. Mitra Discrete time signal processing - Alan V. Oppenheim Dynamics - Bedford Fowler - 4 ed Dynamics - Bedford Fowler - 5 ed Dynamics - Hibbeler - 11 ed Economics econometric analysis - William H. Greene - 5 ed Electric Circuits - Nilsson - 7 ed Electric machinery - Fitzgerald , Kingsley, Uman - 6 ed Electric Machinery Fundamentals - Stephen Chapman - 4 ed Elementary mechanics and Thermodynamics - Jhon W. Norbury Elementary Principles of Chemical Processes - Richard Felder y Ronald Rousseau Engineering Mechanics, Statics - R. C. Hibbeler - 10 ed Engineering Circuit Analysis - William H. Hayt - 6 ed Engineering electromagnetics - Hayt - 6 ed Engineering fluid mechanics - Clayton T. Crowe - 6 ed Engineering fluid mechanics - Crowe, Elger, Robertson - 7 ed Engineering mathematics - John Bird - 4 ed Engineering Mechanics, Statics - Hibbeler - 11 ed Feedback Control Dynamic Systems - Franklin Powel Emami - 4 ed Field and Wave Electromagnetics - David K. Cheng - 2 ed Field Theory Electromagneticos - Alexander Sadiku Fluid mechanics - Frank M. White - 6 ed Fluid mechanics, Thermodynamics of turbomachinery - 5 ed Fourier and laplace transforms Fracture mechanics fundamentals and applications - T.L. Anderson - 2 ed Fundamentals of Aerodynamics - John D. Anderson - 3 ed Fundamentals of Applied Electromagnetics - Fawwaz T. Ulaby - 5 ed Fundamentals of engineering electromagnetics - David K. Cheng Fundamentals of engineering thermodynamics - Moran M.J, Shapiro H.N - 5 ed Fundamentals of fluid mechanics - Bruce R. Munson - 4 ed Fundamentals of Physics - Halliday Resnick vol 1 - 7 ed Fundamentals of Physics - Halliday Resnick vol 2 - 7 ed Fundamentals of thermodynamics - Sonntag, Bognakke, Van Wyler - 6 ed Fundamentals.of.Electric.Circuits - C.K.Alexander, M.N.O.Sadiku - 2 ed Heat transfer - Fundamentals of heat and mass transfer - Frank P. Incropera, David Dewitt heat transfer, fundamentals of heat and mass transfer - P. Incropera, D. P. Dewitt Introduction to algorithms - Thomas H. Cormen Charles E. Leiserson - 2 ed Introduction to Electric Circuits - R. C. Dorf y J. A. Svoboda - 6 ed Introduction to electrodynamics - David J. Griffiths - 3 ed Introduction to linear algebra - Gilbert Strang - 3 ed Introduction to mechanics of Fluidos - Robert Fox, Alan McDonald, y Philip J. Pritchard - 6 ed Introduction to probability- Dimitri P. Bertsekas and John N. Tsitsiklis Introduction to Queueing theory - Robert B. Cooper - 2 ed Introductions to chemical engineering Thermodynamics - J. M. Smith, H. C. Van Ness, M. M Abbott - 6 ed Introductions to chemical engineering Thermodynamics - J. M. Smith, H. C. Van Ness, M. M Abbott - 7 ed Introductory quantum optics - C. C. Gerry and P. L. Knight Linear Algebra - Jim Hefferon Linear Algebra and its Applications - David C. Lay - 3rd ed Linear circuit analysis - R. A DeCarlo, Pen Min Lin - 2 ed Materials science and engineering - W.D. Callister - 6 ed Mathematical Analysis - Apostol Mechanical engineering - Shigleys - 8 ed Mechanical Engineering Design - S Mischke, R Budynas - 7 ed Mechanics of Fluids - Bernard Massey - 8 ed Mechanics of materials - Beer Johnston and Dewolf - 3 ed Mechanics of materials - Gere - 6 ed Mechanics of materials - Hibbeler - 4 ed Mechanics of materials - Hibbeler - 4 ed Microeconomic analysis - Hal R. Varian - 3 ed Microelectonics - Millman Microelectronic Circuits - Adel S. Sedra - 4 ed Microelectronic Circuits - Kenneth C. Sedra, Kc Smith - 4 ed Modern Control Engineering - Problems B - Katsuhiko Ogata - 3 ed Modern control system - Richard Dorf y Robert Bishop - 9 ed Modern digital and analog communications systems - B. P. Lathi Organic Chemistry - Carey - 5 ed Organic Chemistry - Hornback - 2 ed Physical chemistry - Peter Atkins, Julio de Paula - 7 ed Physical chemistry - Peter W. Atkins - 7 ed Physics - Paul A. Tipler - 5 ed Physics - Sears, Zemansky, Young, Freedman vol1 - 11 ed Physics - Sears, Zemansky, Young, Freedman vol2 - 11 ed Physics by Resnick Halliday Krane vol 2 - 5 ed Physics for scientists and engineers - Raymond Serway - vol 1 - 5 ed Physics for scientists and engineers - Raymond Serway - vol 1 - 6 ed Physics for scientists and engineers - Raymond Serway - vol 2 - 5 ed Physics for scientists and engineers - Raymond Serway - vol 2 - 6 ed Physics: Principles with Applications - Douglas Giancoli - 6 ed Power System Analysis - John J. Grainger, William D. Stevenson Principles and applications of electrical engineering - Giorgio Rizzoni Principles of electronic materials and devices - S. O. Kasap - 2 ed Probability and statistics for engineers and scientists - Anthony Hayter - 3 ed Probability and statistics for engineers and scientists - Jay L. Devore - 6 ed Probability and statistics for engineers and scientists - Walpole, Myers - 8 ed Probability, Random Variables and Stochastic Processes Solutions - Athanasios Papoulis.- 4 ed Process system analysis and control - Donald R. Coughanowr Quantum Mechanics - Yung-Kuo Lim Science and engineering of materials - Donald R. Askeland - 4 ed Signals and systems - Simon Haykin - 2 ed Signals and systems - Michael J. Roberts Signals and systems - Oppenheim - Willsky - 2 ed Solid state electronic devices - B. G. Streetman, B. Sanjay Solid state physics - Charles Kittel - 8 ed Statics - Meriam Structural analysis - hibbeler - 5 ed System dynamics- Katsuhiko Ogata - 3 ed The econometrics of financial markets - Craig MacKinlay, Andrew W. Lo & John Y. Campbell Thermodynamics an engieneering approach - Yunus Cengel - 5 ed Transport Phenomena - R. Byron Bird, Warren E. Stewart - 2 ed Unit operations of chemical engineering - Warren McCabe, Juan C. Smith, Peter Harriott - 6 ed Vector Mechanics for Engineers: Dynamic - Ferdinand P. Beer - 6 ed Vector Mechanics for Engineers: Dynamics - Ferdinand P. Beer - 7 ed Vector Mechanics for Engineers: Statics - Ferdinand P. Beer - 6 ed Vector Mechanics for Engineers: Statics - Ferdinand P. Beer - 7 ed Wireless Communications - Theodore Rappaport - 2 ed Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ === Subject: Mathieu Functions and Differential Equation Solution posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Hi All, I used Mathematica to solve the following DEQ: x''[t] + (2 Pi)^2(1 + (1/10) Cos[(2 Pi/100) t]) x[t] = 0 The result uses even and odd Mathieu functions. The results are not matching what the paper I am reading shows. Can someone out there use some package and provide a solution to the DEQ to make sure I am getting the correct result (maybe there is an error in the paper - or I could be missing something)? ~A === Subject: Re: Mathieu Functions and Differential Equation Solution > Hi All, I used Mathematica to solve the following DEQ: x''[t] + (2 Pi)^2(1 + (1/10) Cos[(2 Pi/100) t]) x[t] = 0 The result uses even and odd Mathieu functions. The results are not matching what the paper I am reading shows. Can someone out there use some package and provide a solution to the > DEQ to make sure I am getting the correct result (maybe there is an > error in the paper - or I could be missing something)? Maple 12 says: x(t) = _C1*MathieuC(40000,-2000,1/100*Pi*t)+_C2*MathieuS(40000,-2000,1/100*Pi*t) -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Mathieu Functions and Differential Equation Solution posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) On Aug 1, 5:40æpm, Robert Israel > Hi All, > I used Mathematica to solve the following DEQ: > x''[t] + (2 Pi)^2(1 + (1/10) Cos[(2 Pi/100) t]) x[t] = 0 > The result uses even and odd Mathieu functions. > The results are not matching what the paper I am reading shows. > Can someone out there use some package and provide a solution to the > DEQ to make sure I am getting the correct result (maybe there is an > error in the paper - or I could be missing something)? Maple 12 says: x(t) = C1*MathieuC(40000,-2000,1/100*Pi*t)+ C2*MathieuS(40000,-2000,1/100*Pi*t) > -- > Robert Israel æ æ æ æ æ æ æisr...@math.MyUniversitysInitials.ca > Department of Mathematics æ æ æ æhttp://www.math.ubc.ca/~israel > University of British Columbia æ æ æ æ æ æVancouver, BC, Canada If you don't mind - are you able to duplicate Figures 1 through 3 in this paper? http://epsppd.epfl.ch/Roma/pdf/P2 091.pdf ~A === Subject: Re: the Science of Math-DR SALIL PANDE Importance: low >I have always found Math to be a fantastic venture. My father was a >Mathematician and it drew me into the exact sciences. As a doctor I >use math everyday and for some strange reason I find the concepts >behind our most complex theories in math and science fascinating. >Now, I don't want to be wishy-washy here, I just wanted to set up a >post so that people who care about what I care about know how I feel. >Nobody else I tell these things to cares, since my siblings were >always on the more rebellious side. >I think this is a great forum, all of you take care. as my work load >goes down a bit I look forward to being a regular contributor to this >site. >Blessings, >Dr. Salil Pande Welcome! I've never been able to talk about maths either, not even (that's the > problem) to other mathematicians. I'm trying to get into the habit of > talking about it online, in the hope that I might eventually even be > able to literally talk about it when an occasion arises to do so. Maths is indeed fascinating. You forgot your anagram, Angus... Dr. Salil Pande ~ plans a riddle Phil -- -- Microsoft voice recognition live demonstration === Subject: Re: the Science of Math-DR SALIL PANDE indeed, math is quite an interesting field. there are so many possibilities and ways to integrate concepts into our lives. i hope you join our conversations in the near future. - Lana Todorovich === Subject: Prime function(related to zeta) Anything known about the function defined by prod(1/(1 - 1/(p_k - x)^s))? obviously f(x,s) = 0 when x is positive and prime and if x = 0 then it reduces to zeta(s). I'm mainly interested in it for negative values as its better behaved. There seems to be some relationship between it and the zeta function. (I do have a relationship between it and the zeta function but its not closed form) === Subject: Re: the problem with Cantor > When you consider computable reals 0<=cr<1 > what variety do you get in the decimal expansions? > And so another nonstandard mathematician |-|erc, has > appeared out of lurkdom to refute Cantor. > I've only glanced at this thread, but it appears that > |-|erc's argument is another one based on the > assumption (that doesn't hold in ZFC) that if for every > natural number n, phi(n) holds, then we must have > phi(N) holding as well. *************************************************************** [Tonico] > I can't be sure (pretty confussing with all those non-defined terms > and stuff), but at the bottom line it seems |-|erc tries somehow to > turn over the argument used in Cantor's Diagonal proof and he says: OK, I can write down all the computable numbers and that way I get > ALL the possible real numbers, [quote: Computable reals displays EVERY > type of decimal expansion imaginable, there is nothing it misses.]. > Proof? Very simple: tell me which number would I miss doing this! Oh, > but if you can point such a number then it is computable, and thus I > would have written it...taraaaaan! Well, there seems to exist a rather huge logical flaw up there: how > can you know a priori whether you've written down all the conmputable > real numbers? > Or even better, and turning over the tortilla once again: what if > you present me the list (because it will be a list...right??) of ALL > computable real numbers, and then I use Cantor's Diagonal argument and > pinpoint a real number which is NOT in that list? Because believe me: > in any list of real numbers there will be a number I can pinpoint and > prove it is NOT in that list... Of course, it could be |-|erc didn't actually mean the above... [Herc] > Well spotted flipside of my claim - the diag argument. The list will contain every finite prefix of the anti-diagonal. When the diag argument was invented we imagine something like this. 123 > 456 > 789 The diagonal is 159 > The antidiagonal is 261 > Voila - 261 is not on the list. But, if the list is the computable reals, > 2 is on the list > 26 is on the list > 261 is on the list Every finite prefix of the antidiagonal, UP TO OO LENGTH is on the list. This gives me grave doubts of claims that the sequence is missing. > In a way the sequence is there, every finite prefix to infinite length means > you don't miss a digit. Sorry to have to tell you but you're all fools, > only when Artificial Intelligence arrives in 20 years will you be corrected. :)~ Herc ********************************************************** Yeah...that sounded like :you're wrong, you suck, I'm right I win. Wait till jesus (muhammed, buddha, your-favourite-god) arrives...just wait! Then you'll see who's the fool! Well, some fundies already mention NOW the year 2012, after many failed dates in the past, and now you've mentioned A.I. in 20 years more (which I can't fully understand what has to do with this issue...). We shall see, I supose, and no further debate is worth. Tonio You ignored my argument and focused on the little emoticon ending. No wonder you want to withdraw now, your superinfinity is looking pretty weak. You can't even form a unique sequence of digits using diagonalisation on my set can you? http://www.freewebs.com/namesort/linux.html Tell me in your own words what this site does. There's 3 buttons numbered 1 2 3. Press 1 2 3 2 3 2 3 and describe what happens. Then I'll present my proof. Herc === Subject: Re: the problem with Cantor > When you consider computable reals 0<=cr<1 > what variety do you get in the decimal expansions? > And so another nonstandard mathematician |-|erc, has > appeared out of lurkdom to refute Cantor. > I've only glanced at this thread, but it appears that > |-|erc's argument is another one based on the > assumption (that doesn't hold in ZFC) that if for every > natural number n, phi(n) holds, then we must have > phi(N) holding as well. *************************************************************** [Tonico] > I can't be sure (pretty confussing with all those non-defined terms > and stuff), but at the bottom line it seems |-|erc tries somehow to > turn over the argument used in Cantor's Diagonal proof and he says: OK, I can write down all the computable numbers and that way I get > ALL the possible real numbers, [quote: Computable reals displays EVERY > type of decimal expansion imaginable, there is nothing it misses.]. > Proof? Very simple: tell me which number would I miss doing this! Oh, > but if you can point such a number then it is computable, and thus I > would have written it...taraaaaan! Well, there seems to exist a rather huge logical flaw up there: how > can you know a priori whether you've written down all the conmputable > real numbers? Who says they must all be written down in order to know that they are all included in a set? > Or even better, and turning over the tortilla once again: what if > you present me the list (because it will be a list...right??) of ALL > computable real numbers, and then I use Cantor's Diagonal argument and > pinpoint a real number which is NOT in that list? Because believe me: > in any list of real numbers there will be a number I can pinpoint and > prove it is NOT in that list... Of course, it could be |-|erc didn't actually mean the above... > Is Huck really claiming to be able to show that there is a number in in a list and not in that same list? He seems to be challenging WM's position as King Kook. > [Herc] > Well spotted flipside of my claim - the diag argument. The list will contain every finite prefix of the anti-diagonal. When the diag argument was invented we imagine something like this. 123 > 456 > 789 The diagonal is 159 > The antidiagonal is 261 > Voila - 261 is not on the list. But, if the list is the computable reals, > 2 is on the list > 26 is on the list > 261 is on the list Every finite prefix of the antidiagonal, UP TO OO LENGTH is on the list. Up to but not including oo, which means every member of the list has a sufficiently long but finite prefix differing from an equally long prefix of the diagonal. This gives me grave doubts of claims that the sequence is missing. Doubt all you want, but until you can support your doubts with better evidence than you so far have been able to do, they will not convince anyone else. > In a way the sequence is there, every finite prefix to infinite length means > you don't miss a digit. Sorry to have to tell you but you're all fools But not so foolish as to be convinced by such foolish arguments as you have been presenting. , > only when Artificial Intelligence arrives in 20 years will you be corrected. > :)~ Herc === Subject: Re: the problem with Cantor [VIRGIN] You dumb , I can still make a new number [HERC] I don't care about your new number, give me a new sequence of digits. Herc === Subject: Re: the problem with Cantor posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; FunWebProducts; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > [VIRGIN] > You dumb , I can still make a new number [HERC] > I don't care about your new number, give me a new sequence of digits. Herc *************************************************************** I answer here you prior post since for some rather misterious and completely unreachable for me reason that post doesn't accept any answer... 1) Well, let's play along: I entered your link, pressed 1232323 and got a left list of some 60 numbers, then an apaprently randomly generated numbers, and a second list which apparently contains the randly gen. number in its main diagoal entries...so? I don't understand what is this supposed to show. 2) What is superinfinity? And why is it mine? 3) I did not ignore your argument: I did address it in my first post, misunderstood. You ignored this addressing of mine. 4) no wonder I want to withdraw now....from what? A debate? If it indeed is a debate, what path is left for a debate after you write Sorry to have to tell you but you're all fools, only when Artificial Intelligence arrives in 20 years will you be corrected.? It seems not only you've utterly made up your mind about this, but you've also decided we all are fools, you only are wise, we're wrong, you're right...and we all shall see this in 20 years more! Well, you've wrapped it all tight and dandy: YOU closed the door. Tonio === Subject: Re: the problem with Cantor posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > 1) Well, let's play along: I entered your link, pressed 1232323 and > got a left list of some 60 numbers, then an apaprently randomly > generated numbers, and a second list which apparently contains the > randly gen. number in its main diagoal entries...so? I don't > understand what is this supposed to show. I believe I know what Herc is trying to accomplish. Herc is making the following claim (quoted verbatim): Claim: Let L be a list of numbers in [0,1] such that for each d in {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal representation of L_i is d} is infinite. Let D be a random variable uniform on [0,1]. Then almost surely there is a permutation L' of L such that D is the diagonal of L'. But there are two big problems here: 1) The claim most likely _fails_ in ZFC -- even for finite lists! 2) Even if the claim does hold in ZFC, it has absolutely _no_ bearing on the validity of Cantor in ZFC. Let's deal with 1) first. Herc's link is an applet consisting of three steps: 1. Populate list -- clicking this causes a list of sixty real numbers between zero and one, apparently in ternary since only the digits 0,1,2, appear, each with sixty digits. 2. Randomise diagonal -- clicking this causes a 61st real number to appear. 3. Generate list -- clicking this causes a reordering of the first list to appear such that the real number chosen in the second step appears on the diagonal. On the face of it, this should always work. We know that there are 60! (60 factorial) permutations of the original list and only 3^60 possible diagonals -- and factorials increase much faster than exponentiation. (60! ~ 8*10^81, but 3^60 ~ 4*10^28.) So there ought to be more than enough permutations for all the possible diagonals. But I decided to try a little trick. I clicked on the first button to populate the list -- but rather than clicking on the second button, I manually typed in my own diagonal. What I did was look at the first row, which happened to be: 0.021100001022012201100101102022102002102112020210111021110120 Then I used one of the usual tricks for generating an antidiagonal (in Cantor's proof) from a diagonal to make an antirow. I replaced the digits 1 and 2 with 0 and the digit 0 with 1, to obtain the number: 0.100011110100100010011010010100010110010000101001000100001001 which I typed into the diagonal box. Then I clicked on the third button. Sure enough, only 59 numbers appeared rather than 60. And of course, the number that was present on the first list but missing from the second list was the first number. It would be _impossible_ for the first number to appear, since it differs from the diagonal in _every_ digit! And it's easy to see that the same problem would happen for _any_ list, whether finite or infinite, whether saturated or unsaturated. Of course, Herc's claim is that almost surely such a permutation exists. But what is almost surely? Does Herc mean that the set of all real numbers for which a permutation fails to exist has Lebesgue measure zero? If so, then Herc may be right. So far, we've only shown the antirows to be counterexamples -- where an antirow is defined to be a number differing from every digit from a number in a row. Suppose the first row contains the real number 1/2 -- which is 0.111... in ternary. Then an antirow would be any number consisting of only 0 and 2 in ternary -- and the set of all such numbers is the famous Cantor middle-third set, which has Lebesgue measure zero. Similarly, the set of all possible antirows for a given row also has Lebesgue measure zero. And since this is a _list_ of reals, there's exactly one real in this list for every natural, so there are only countably many such reals and their corresponding sets of antirows. So the set of all antirows for the entire list is the countable union of null sets, and therefore itself null. Of course, this isn't to say that there doesn't exist a counterexample to Herc's claim that doesn't happen to be an antirow to any row at all. But the set of the only _known_ counterexamples to Herc's claim does have Lebesgue measure zero. But even if this makes Herc's claim true, we reach problem 2) -- it has nothing to do with Cantor at all! Indeed, let's compare my proof above to Cantor's: I show it's impossible for an _antirow_ to be a _diagonal_. Cantor shows it's impossible for an _antidiagonal_ to be a _row_. If we think about this for a moment, we see that in my proof, I only claim to show that there exists numbers which can't be on the diagonal. That a counterexample happens to be an antirow is irrelevant. I only care about the diagonal, not rows or antirows. Similarly, Cantor claims that there exist real numbers which can't be in any row of the list. That a counterexample happens to be an antidiagonal is irrevelevant. Cantor cares only about _rows_, not the diagonal or the antidiagonal. It's possible for a real number to be the diagonal of one permutation but an antidiagonal of another permutation. Herc was apparently hoping that this wouldn't be the case, but we can even consider the example: 168 249 357 So an antidiagonal is 258, but if we reorder the list: 249 357 168 suddenly 258's the diagonal. Now 369's an antidiagonal, but: 357 168 249 now 369's the diagonal. Therefore, Herc's claim, even if it were true, would neither prove nor disprove _anything_ about Cantor. > 2) What is superinfinity? And why is it mine? By superinfinity, most likely Herc (pejoratively) means uncountable infinity. And it's yours (Tonio's), as much as it belongs to every standard mathematician who believes in its existence (i.e., who adheres to a set theory, such as ZFC, which proves the existence of uncountable sets), as opposed to Herc, who doesn't believe in the existence of uncountable sets. === Subject: Re: the problem with Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > But there are two big problems here: > 1) The claim most likely fails in ZFC -- even for finite lists! > 2) Even if the claim does hold in ZFC, it has absolutely no > bearing on the validity of Cantor in ZFC. Yathink? > But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! Really? Fascinating. You know, no one has been able to see that before. Just to think that all this time we all thought Herc had it absolutely correct all the way down the line. > Therefore, Herc's claim, even if it were true, would neither > prove nor disprove anything about Cantor. Please, say it ain't so! How could Herc ever be wrong about such a thing? MoeBlee === Subject: Re: the problem with Cantor posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU 5.1),gzip(gfe),gzip(gfe) > But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! > Really? Fascinating. You know, no one has been able to see that > before. Just to think that all this time we all thought Herc had it > absolutely correct all the way down the line. When I try to defend Herc or any of the other cranks, MoeBlee often criticizes me as being silly for trying to make a rigorous theory out of their jumbled claims. But in this post, I wasn't defending Herc -- I was actually attacking him. Yet MoeBlee still criticized me -- this time writing a sarcastic post about how I'm merely restating the obvious. So if MoeBlee will criticize me for defending Herc, and criticize me for attacking Herc, then is their anything I can say at all about Herc that won't lead to criticism from MoeBlee? Or am I damned if I do and damned if I don't? The reason for my post above is that here, I'm actually telling Herc that he's wrong -- but being more specific as to why he's wrong. It's refreshing to see, however flawed it may be, a different argument against Cantor rather than the same old infinite induction or only count the finite subsets of N arguments. Many of the standard mathematicians didn't try to figure out what Herc was doing or find out why he's wrong -- they simply knew that whatever Herc's doing is probably wrong (and rightly so). I bet some of the standard mathematicians would just wish that I would stop trying to defend Herc and the other cranks and attack them the same way that they themselves do. But why should I join the standard mathematicians in their attack against the cranks, when the likely response would be a sarcastic post similar to the one MoeBlee posted here? At least Nam Nguyen has already pointed out how MoeBlee has (apparently) contradicted himself (and there's no need to point out that cranks often contradict themselves, too). No matter what I post, someone's not going to like it, so there's no point in trying to please everyone when I post. === Subject: Re: the problem with Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! > Really? Fascinating. You know, no one has been able to see that > before. Just to think that all this time we all thought Herc had it > absolutely correct all the way down the line. When I try to defend Herc or any of the other cranks, > MoeBlee often criticizes me as being silly for trying > to make a rigorous theory out of their jumbled claims. But in this post, I wasn't defending Herc -- I was > actually _attacking_ him. Yet MoeBlee still criticized > me -- this time writing a sarcastic post about how I'm > merely restating the obvious. So if MoeBlee will criticize me for _defending_ Herc, > and criticize me for _attacking_ Herc, then is their > _anything_ I can say at _all_ about Herc that won't > lead to criticism from MoeBlee? Or am I damned if I do > and damned if I don't? What motivated my sarcasm was indeed the irony that we get these lectures from you about how the cranks could conceivably be vindicated but then you turn around to post exactly the kind of critique Herc has been getting for years already. I don't find fault - in itself - in ticking off specifics as to why Herc provides no challenge to the Cantor proofs. What I found worth sarcasm was the context in which you did that. > The reason for my post above is that here, I'm actually > telling Herc that he's wrong -- but being more specific > as to _why_ he's wrong. It's refreshing to see, however > flawed it may be, a different argument against Cantor > rather than the same old infinite induction or only > count the finite subsets of N arguments. Many of the > standard mathematicians didn't try to figure out what > Herc was doing or find out _why_ he's wrong -- they > simply knew that whatever Herc's doing is probably > wrong (and rightly so). See, again, that's where you go off making yourself out to be, in whatever sense, more intellectually inquisitive than standard mathematicians. Look, for SEVERAL years, people have been posting point by point, excuciatingly detailed explanations for Herc of his mistakes and fallacies. You may find yet another Herc variation to be refreshing (even if to refute), but, indeed, it is just another bozoid variation. Excuse us if, after reading years worth of that kind of stuff, we are not all as entertained by it as you are. > I bet some of the standard mathematicians would just > wish that I would stop trying to defend Herc and the > other cranks and attack them the same way that they > themselves do. You don't need you to attack (though you're welcome to do it). There are plenty of people who do that anyway. What I wish you'd stop doing is making sweeping and incorrect generalizations about standard mathematicians. > But why should I join the standard mathematicians in > their attack against the cranks, when the likely > response would be a sarcastic post similar to the > one MoeBlee posted here? See earlier in this post. > At least Nam Nguyen has > already pointed out how MoeBlee has (apparently) > contradicted himself No, Nam has not shown any contradiction in my remarks. I addressed that in the post just before yours here. MoeBlee === Subject: Re: the problem with Cantor MoeBlee, I have a hard time understanding how that big brain of yours ticks over. Please help me out by answering 1 question. I'll make it as brief as possible, a yes/no question. Q: Do you realise that the computable reals contain every possible digit sequence up to infinite length? that is all Herc === Subject: Re: the problem with Cantor <5nPlk.25589$IK1.13607@news-server.bigpond.net.au> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > MoeBlee, I have a hard time understanding how that big brain of yours ticks over. Please help me out by answering 1 question. I'll make it as brief as possible, a yes/no question. Q: Do you realise that the computable reals contain every possible digit sequence > æ æ æ up to infinite length? I can only answer your question if you tell me your axioms and rules of inference by which conclusions are drawn, and if you define your terms, back to your primitives. Such terms in your question include computable, real', possible, digit sequence, up to, and infinite length. MoeBlee === Subject: Re: the problem with Cantor > Q: Do you realise that the computable reals contain every possible digit > sequence up to infinite length? > I can only answer your question if you tell me your axioms and rules > of inference by which conclusions are drawn, and if you define your > terms, back to your primitives. Such terms in your question include > computable, real', possible, digit sequence, up to, and > infinite length. > between non-cranks) I'd say the problematic term here is up to. Does this mean: of finite, but unbound, length? Or does it include sequences of infinite length? In the latter case, the statement is wrong. Actually, not any digit sequence (if infinitely long sequences are admissible) are computable. So I'd propose not to realise that anti-fact. :-) B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor <8m3h94tpddo03n718etaa578lf1e15eo5p@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Q: Do you realise that the computable reals contain every possible digit > æ æsequence up to infinite length? > I can only answer your question if you tell me your axioms and rules > of inference by which conclusions are drawn, and if you define your > terms, back to your primitives. Such terms in your question include > computable, real', possible, digit sequence, up to, and > infinite length. between non-cranks) I'd say the problematic term here is up to. Does > this mean: of finite, but unbound, length? Or does it include > sequences of infinite length? In the latter case, the statement is wrong. Actually, not any digit > sequence (if infinitely long sequences are admissible) are computable. I agree. MoeBlee === Subject: Re: the problem with Cantor I agree. > Additional comment: > Q: Do you realise that the computable reals contain every possible digit > æ æsequence up to infinite length? > [...] I'd say the problematic term here is up to. Does > this mean: of finite, but unbound, length? Or does it include > sequences of infinite length? > One of the main problems when discussing mathematical topics with cranks seems to be there tendency to use unspecific terminology in their statements. (I reckon that there's a dependency relation between their usage of such terms and their problems to comprehend certain mathe- matical facts.) As a result very often such claims (expressed by cranks) are not even wrong. :-/ B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours ticks > over. Please help me out by answering 1 question. I'll make it as brief as possible, a yes/no question. Q: Do you realise that the computable reals contain every possible digit > sequence > up to infinite length? Up to but not including. Every finite binary digit sequence is, at least in theory, computable. In order for every infinite sequence to be computable as well, wouldn't have to be able to count the computation schemes required, one per sequence? And that is provably impossible. === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours ticks > over. > Please help me out by answering 1 question. > I'll make it as brief as possible, a yes/no question. > Q: Do you realise that the computable reals contain every possible digit > sequence > up to infinite length? Up to but not including. Every finite binary digit sequence is, at least in theory, computable. In order for every infinite sequence to be computable as well, wouldn't > have to be able to count the computation schemes required, one per > sequence? And that is provably impossible. Let me get this straight. You AGREE that ALL sequences up to oo length are computable, but you still think there's a NEW sequence of digits? How can you believe AS (all sequences) and ~AS at the same time? in the same breath? HOW DO YOU GET A NEW SEQUENCE when all sequences are computed? No finite part of your new sequence is unique. Therefore its not unique. Herc === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours > ticks > over. > Please help me out by answering 1 question. > I'll make it as brief as possible, a yes/no question. > Q: Do you realise that the computable reals contain every possible digit > sequence > up to infinite length? > Up to but not including. > Every finite binary digit sequence is, at least in theory, computable. > In order for every infinite sequence to be computable as well, wouldn't > have to be able to count the computation schemes required, one per > sequence? > And that is provably impossible. > Let me get this straight. You AGREE that ALL sequences up to oo length > are computable, but you still think there's a NEW sequence of digits? What deludes you to think I agree that all sequences are computable? As there are only countably many computation schemes but uncountably many binary sequences, such an agreement would be illogical. === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours > ticks > over. > Please help me out by answering 1 question. > I'll make it as brief as possible, a yes/no question. > Q: Do you realise that the computable reals contain every possible digit > sequence > up to infinite length? > Up to but not including. > Every finite binary digit sequence is, at least in theory, computable. > In order for every infinite sequence to be computable as well, wouldn't > have to be able to count the computation schemes required, one per > sequence? > And that is provably impossible. > > Let me get this straight. You AGREE that ALL sequences up to oo length > are computable, but you still think there's a NEW sequence of digits? What deludes you to think I agree that all sequences are computable? > As there are only countably many computation schemes but uncountably > many binary sequences, such an agreement would be illogical. You agreed above, up to infinite length. Do you agree or not? All sequences up to infinite length are computable. Herc === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours > ticks > over. > Please help me out by answering 1 question. > I'll make it as brief as possible, a yes/no question. > Q: Do you realise that the computable reals contain every possible > digit > sequence > up to infinite length? > Up to but not including. > Every finite binary digit sequence is, at least in theory, computable. > In order for every infinite sequence to be computable as well, wouldn't > have to be able to count the computation schemes required, one per > sequence? > And that is provably impossible. > Let me get this straight. You AGREE that ALL sequences up to oo length > are computable, but you still think there's a NEW sequence of digits? > What deludes you to think I agree that all sequences are computable? > As there are only countably many computation schemes but uncountably > many binary sequences, such an agreement would be illogical. You agreed above, up to infinite length. Up to infinite length is ambiguous, so I do not agree to it. All finite sequences are computable is not ambiguous, and to that I agree. Do you agree or not? I agree that all finite sequences are computable but not to up to infinite length. All sequences up to infinite length are computable. That is your mantra, not mine. Herc === Subject: Re: the problem with Cantor > All sequences up to infinite length are computable. All finite sequences are computable, obviously. So what? > Herc -- Alan Smaill === Subject: Re: the problem with Cantor > 1) Well, let's play along: I entered your link, pressed 1232323 and > got a left list of some 60 numbers, then an apaprently randomly > generated numbers, and a second list which apparently contains the > randly gen. number in its main diagoal entries...so? I don't > understand what is this supposed to show. I believe I know what Herc is trying to accomplish. Herc is making the following claim (quoted verbatim): Claim: Let L be a list of numbers in [0,1] such that for each d in > {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal > representation of L_i is d} is infinite. Let D be a random variable > uniform on [0,1]. Then almost surely there is a permutation L' of L > such that D is the diagonal of L'. But there are two big problems here: > 1) The claim most likely _fails_ in ZFC -- even for finite lists! > 2) Even if the claim does hold in ZFC, it has absolutely _no_ > bearing on the validity of Cantor in ZFC. Let's deal with 1) first. Herc's link is an applet consisting > of three steps: 1. Populate list -- clicking this causes a list of sixty real > numbers between zero and one, apparently in ternary since > only the digits 0,1,2, appear, each with sixty digits. > 2. Randomise diagonal -- clicking this causes a 61st real > number to appear. > 3. Generate list -- clicking this causes a reordering of the > first list to appear such that the real number chosen in the > second step appears on the diagonal. On the face of it, this should always work. We know that > there are 60! (60 factorial) permutations of the original list > and only 3^60 possible diagonals -- and factorials increase > much faster than exponentiation. (60! ~ 8*10^81, but > 3^60 ~ 4*10^28.) So there ought to be more than enough > permutations for all the possible diagonals. But I decided to try a little trick. I clicked on the first button > to populate the list -- but rather than clicking on the second > button, I manually typed in my own diagonal. What I did was > look at the first row, which happened to be: 0.021100001022012201100101102022102002102112020210111021110120 Then I used one of the usual tricks for generating an > antidiagonal (in Cantor's proof) from a diagonal to > make an antirow. I replaced the digits 1 and 2 with > 0 and the digit 0 with 1, to obtain the number: 0.100011110100100010011010010100010110010000101001000100001001 which I typed into the diagonal box. Then I clicked on the third button. Sure enough, only 59 > numbers appeared rather than 60. And of course, the > number that was present on the first list but missing > from the second list was the first number. It would be > _impossible_ for the first number to appear, since it > differs from the diagonal in _every_ digit! And it's easy to see that the same problem would > happen for _any_ list, whether finite or infinite, whether > saturated or unsaturated. Of course, Herc's claim is that almost surely such a > permutation exists. But what is almost surely? Does > Herc mean that the set of all real numbers for which a > permutation fails to exist has Lebesgue measure zero? If the new diagonal is a random variable in [0,1], the probability that it can be reordered with that diagonal is 1. SETS of reals with sufficient variety in the expansions don't really have diagonals, only lists do. A set can be ordered to have virtually any diagonal. There's infinite permutations to choose from. If so, then Herc may be right. So far, we've only shown > the antirows to be counterexamples -- where an > antirow is defined to be a number differing from every > digit from a number in a row. Suppose the first row contains the real number 1/2 -- > which is 0.111... in ternary. Then an antirow would be > any number consisting of only 0 and 2 in ternary -- and > the set of all such numbers is the famous Cantor > middle-third set, which has Lebesgue measure zero. Similarly, the set of all possible antirows for a given > row also has Lebesgue measure zero. And since this > is a _list_ of reals, there's exactly one real in this list > for every natural, so there are only countably many > such reals and their corresponding sets of antirows. So the set of all antirows for the entire list is the > countable union of null sets, and therefore itself null. Of course, this isn't to say that there doesn't exist a > counterexample to Herc's claim that doesn't happen to > be an antirow to any row at all. But the set of the only > _known_ counterexamples to Herc's claim does have > Lebesgue measure zero. But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! Indeed, let's compare my proof above to Cantor's: I show it's impossible for an _antirow_ to be a _diagonal_. > Cantor shows it's impossible for an _antidiagonal_ to be a _row_. If we think about this for a moment, we see that in my > proof, I only claim to show that there exists numbers > which can't be on the diagonal. That a counterexample > happens to be an antirow is irrelevant. I only care about > the diagonal, not rows or antirows. Similarly, Cantor claims that there exist real numbers > which can't be in any row of the list. That a counterexample > happens to be an antidiagonal is irrevelevant. Cantor cares > only about _rows_, not the diagonal or the antidiagonal. It's possible for a real number to be the diagonal of one > permutation but an antidiagonal of another permutation. Herc > was apparently hoping that this wouldn't be the case, but > we can even consider the example: 168 > 249 > 357 So an antidiagonal is 258, but if we reorder the list: 249 > 357 > 168 suddenly 258's the diagonal. Now 369's an antidiagonal, but: 357 > 168 > 249 now 369's the diagonal. Therefore, Herc's claim, even if it were true, would neither > prove nor disprove _anything_ about Cantor. > 2) What is superinfinity? And why is it mine? By superinfinity, most likely Herc (pejoratively) means > uncountable infinity. And it's yours (Tonio's), as much as > it belongs to every standard mathematician who believes > in its existence (i.e., who adheres to a set theory, such > as ZFC, which proves the existence of uncountable sets), > as opposed to Herc, who doesn't believe in the existence > of uncountable sets. ZFC doesn't prove the existence of uncountable sets, it finds a new number not on the list and human operaters interpret this as there must be some larger set type than infinity. It's like a young child arguing 0.999.. is different to 1. Every possible digit sequence up to infinite length is on the computable reals list so what new sequence of digits did it find? Herc === Subject: Re: the problem with Cantor ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? Second order ZF does demand the existence of uncountable sets. Finiteness is a second order property but finiteness is so obvious that nobody should have any problem using it as a starting assumption. http://en.wikipedia.org/wiki/Peano_axioms#Nonstandard_models Cantor's theorem can be proven in a variety of different ways. The diagonal proof is one way but Cantor's theorem can also be proven using membership. Below is a proof that was posted on this newsgroup a while back, using membership, and it holds for any set: Cantor's Theorem: For any set S, |S| < |P(S)|. If (a) |S|<=|P(S)| and (b) |S|=/=|P(S)| are both true then Cantor's theorem is proven. Letting q be the one to one function q:S-->P(S) such that s-->{s}, it is easy to see that (a) is true. To see that (b) is true, assume the contrary. That is, assume there is a bijection r:S-->P(S). Let T be the set: T = {s in S : s not in r(s)} Since r is assumed to be onto, there should be a t in S such that (c) T=r(t). If t is in T then, by definition, t is not in r(t) and this contradicts (c). So it must be that t is not in T but then (c) means that t is in T and this is another contradiction. Thus, r is not onto and therefore not a bijection. QED. k === Subject: Re: the problem with Cantor > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? Second order ZF does demand the existence of uncountable sets. Finiteness > is a second order property but finiteness is so obvious that nobody should > have any problem using it as a starting assumption. http://en.wikipedia.org/wiki/Peano_axioms#Nonstandard_models Cantor's theorem can be proven in a variety of different ways. The diagonal > proof is one way but Cantor's theorem can also be proven using membership. > Below is a proof that was posted on this newsgroup a while back, using > membership, and it holds for any set: Cantor's Theorem: For any set S, |S| < |P(S)|. If (a) |S|<=|P(S)| and (b) |S|=/=|P(S)| are both true then Cantor's theorem > is proven. Letting q be the one to one function q:S-->P(S) such that > s-->{s}, it is easy to see that (a) is true. To see that (b) is true, > assume the contrary. That is, assume there is a bijection r:S-->P(S). Let > T be the set: T = {s in S : s not in r(s)} Since r is assumed to be onto, there should be a t in S such that (c) > T=r(t). If t is in T then, by definition, t is not in r(t) and this > contradicts (c). So it must be that t is not in T but then (c) means that t > is in T and this is another contradiction. Thus, r is not onto and > therefore not a bijection. QED. > This one is my favorite! Which box contains the number of all the boxes that don't contain their own number? There is none - therefore - SUPERINFINITY Herc === Subject: Re: the problem with Cantor > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. In ZFC, one has the following: If A bijects with B then Card(A) = Card(B). If A injects into B then Card(A) <= Card(B) If A injects into B but B does not inject into A then Card(A) < Card(b) It is trivial in ZFC that any set injects into its power set. It is less trivial but still true in ZFC that no power set of a set injects back into that set. Let N be the set of naturals in ZFC and R be the set of lower Dedekind cut sets. One can show that in ZFC, Card(R) = Card(P(N)) and Card(N) < Card(P(N)). > It's like a > young child arguing 0.999.. is different to 1. It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? First prove that a list of all computable sequences is itself computable. If that can be proven at all, which is not obvious to me, then the new sequence must be an uncomputable sequence. And once we have a computable sequence of all computable sequences, we an easily create at least as many uncomputable sequences as there are computable ones. === Subject: Re: the problem with Cantor > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. In ZFC, one has the following: If A bijects with B then Card(A) = Card(B). > If A injects into B then Card(A) <= Card(B) > If A injects into B but B does not inject into A then Card(A) < Card(b) It is trivial in ZFC that any set injects into its power set. > It is less trivial but still true in ZFC that no power set of a set > injects back into that set. Let N be the set of naturals in ZFC and R be the set of lower Dedekind > cut sets. One can show that in ZFC, Card(R) = Card(P(N)) and Card(N) < > Card(P(N)). > It's like a > young child arguing 0.999.. is different to 1. It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? First prove that a list of all computable sequences is itself computable. If that can be proven at all, which is not obvious to me, then the new > sequence must be an uncomputable sequence. And once we have a computable sequence of all computable sequences, we > an easily create at least as many uncomputable sequences as there are > computable ones. Surely a TM could enumerate subsets. The output of a TM is a series of 1s and 0s. e.g. 000011111000111100011000000 Let the 0's be dividers between unary outputs. this becomes {5,4,2} Then the enumeration of subsets is merely UTM(n,0) 1 ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. > In ZFC, one has the following: > If A bijects with B then Card(A) = Card(B). > If A injects into B then Card(A) <= Card(B) > If A injects into B but B does not inject into A then Card(A) < Card(b) > It is trivial in ZFC that any set injects into its power set. > It is less trivial but still true in ZFC that no power set of a set > injects back into that set. > Let N be the set of naturals in ZFC and R be the set of lower Dedekind > cut sets. > One can show that in ZFC, Card(R) = Card(P(N)) and Card(N) < > Card(P(N)). > > It's like a > young child arguing 0.999.. is different to 1. > It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? > First prove that a list of all computable sequences is itself computable. > If that can be proven at all, which is not obvious to me, then the new > sequence must be an uncomputable sequence. > And once we have a computable sequence of all computable sequences, we > an easily create at least as many uncomputable sequences as there are > computable ones. > Surely a TM could enumerate subsets. Given a set of naturals, N, either in ZFC, NBG, or some generally similar set theory. let us see a description of any TM which will ennumerate ALL of its subsets. The output of a TM is a series of 1s and 0s. e.g. 000011111000111100011000000 Let the 0's be dividers between unary outputs. this becomes {5,4,2} Then the enumeration of subsets is merely UTM(n,0) 1 ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. > In ZFC, one has the following: > If A bijects with B then Card(A) = Card(B). > If A injects into B then Card(A) <= Card(B) > If A injects into B but B does not inject into A then Card(A) < Card(b) > It is trivial in ZFC that any set injects into its power set. > It is less trivial but still true in ZFC that no power set of a set > injects back into that set. > Let N be the set of naturals in ZFC and R be the set of lower Dedekind > cut sets. > One can show that in ZFC, Card(R) = Card(P(N)) and Card(N) < > Card(P(N)). > It's like a > young child arguing 0.999.. is different to 1. > It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? > First prove that a list of all computable sequences is itself computable. > If that can be proven at all, which is not obvious to me, then the new > sequence must be an uncomputable sequence. > And once we have a computable sequence of all computable sequences, we > an easily create at least as many uncomputable sequences as there are > computable ones. > > Surely a TM could enumerate subsets. Given a set of naturals, N, either in ZFC, NBG, or some generally > similar set theory. let us see a description of any TM which will > ennumerate ALL of its subsets. > The output of a TM is a series of 1s and 0s. > e.g. 000011111000111100011000000 > Let the 0's be dividers between unary outputs. > this becomes {5,4,2} > Then the enumeration of subsets is merely > UTM(n,0) 1 If so which is ennumerated first, the set of all even naturals or the > set of all odd naturals? > This will output every possible subset of N, i.e. the powerset, indexed by N. No it won't. OK, it will produce all the finite subsets of N. We need to input each natural into the UTM, and the UTM responds either YES (1), or NO (0), as to whether that natural is included in the subset. UTM(a,b) = 1 IFF b is a member of the ath subset. That will enumerate ALL subsets, finite and infinite of N. Herc === Subject: Re: the problem with Cantor > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. > In ZFC, one has the following: > If A bijects with B then Card(A) = Card(B). > If A injects into B then Card(A)<= Card(B) > If A injects into B but B does not inject into A then Card(A)< Card(b) > It is trivial in ZFC that any set injects into its power set. > It is less trivial but still true in ZFC that no power set of a set > injects back into that set. > Let N be the set of naturals in ZFC and R be the set of lower Dedekind > cut sets. > One can show that in ZFC, Card(R) = Card(P(N)) and Card(N)< > Card(P(N)). > It's like a > young child arguing 0.999.. is different to 1. > It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? > First prove that a list of all computable sequences is itself computable. > If that can be proven at all, which is not obvious to me, then the new > sequence must be an uncomputable sequence. > And once we have a computable sequence of all computable sequences, we > an easily create at least as many uncomputable sequences as there are > computable ones. > Surely a TM could enumerate subsets. > Given a set of naturals, N, either in ZFC, NBG, or some generally > similar set theory. let us see a description of any TM which will > ennumerate ALL of its subsets. > The output of a TM is a series of 1s and 0s. > e.g. 000011111000111100011000000 > Let the 0's be dividers between unary outputs. > this becomes {5,4,2} > Then the enumeration of subsets is merely > UTM(n,0) 1 Does this schema enumerate any of the infinite subsets of N? > If so which is ennumerated first, the set of all even naturals or the > set of all odd naturals? > This will output every possible subset of N, i.e. the powerset, indexed by N. > No it won't. > OK, it will produce all the finite subsets of N. > We need to input each natural into the UTM, and the UTM responds > either YES (1), or NO (0), as to whether that natural is included in the subset. > UTM(a,b) = 1 IFF b is a member of the ath subset. > That will enumerate ALL subsets, finite and infinite of N. If you want to get the set of all prime numbers, then what does 'a' > look like? > Has noone written a TM that determines if its input is prime? It would probably take 100 or so states to program, so 'a' would be a natural about 100 to 1000 digits long. The emulated TM would go through all numbers from d = 2 to b/2 and test if b div d has a remainder. If it finds a number that divides b with no remainder then b is not prime and it outputs 0. If it finds no divisors it outputs 1. UTM(a, 1) = 0 UTM(a, 2) = 1 UTM(a, 3) = 1 UTM(a, 4) = 0 UTM(a, 5) = 1 UTM(a, 6) = 0 ... Shouldn't be that hard to calculate a. Herc === Subject: Re: the problem with Cantor > If you want to get the set of all prime numbers, then what does 'a' > look like? > > Has noone written a TM that determines if its input is prime? It would > probably take 100 or so states to program, so 'a' would be a natural about > 100 to 1000 digits long. The emulated TM would go through all numbers from d = 2 to b/2 and test if > b div d has a remainder. If it finds a number that divides b with no > remainder > then b is not prime and it outputs 0. If it finds no divisors it outputs 1. UTM(a, 1) = 0 > UTM(a, 2) = 1 > UTM(a, 3) = 1 > UTM(a, 4) = 0 > UTM(a, 5) = 1 > UTM(a, 6) = 0 > ... Shouldn't be that hard to calculate a. Herc If you think it is so easy, then calculate it. === Subject: Re: the problem with Cantor UTM(a,b) = 1 IFF b is a member of the ath subset. That is only possible if one can list the subsets of N, so that every subset of N is an ath subset, which has been proved, in ZFC and the like, to be impossible. That will enumerate ALL subsets, finite and infinite of N. Not if there are more subsets that a-indices. === Subject: Re: the problem with Cantor > UTM(a,b) = 1 IFF b is a member of the ath subset. That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. Ahh, the old which box contains the numbers of all the boxes that don't contain their own number proof! Just by asking this question, which is impossible, or posing the theorem, you prove superinfinity. > That will enumerate ALL subsets, finite and infinite of N. Not if there are more subsets that a-indices. I've given you the computer program, what subset can all computer programs miscalculate? Herc === Subject: Re: the problem with Cantor > > UTM(a,b) = 1 IFF b is a member of the ath subset. > That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. Ahh, the old which box contains the numbers of all the boxes that don't > contain their own number proof! Just by asking this question, which is > impossible, or posing the theorem, you prove superinfinity. In ZFC It is rather more directly proved than herkimer admits. > That will enumerate ALL subsets, finite and infinite of N. > Not if there are more subsets that a-indices. I've given you the computer program, what subset can all computer > programs miscalculate? Why should any program miscalculate any of the sets that it can calculate? As for the many it cannot calculate at all, why should it miscalculate them either? === Subject: Re: the problem with Cantor <_DSlk.25641$IK1.251@news-server.bigpond.net.au> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > UTM(a,b) = 1 IFF b is a member of the ath subset. > That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. Ahh, the old which box contains the numbers of all the boxes that don't > contain their own number proof! æJust by asking this question, which is > impossible, or posing the theorem, you prove superinfinity. > ************************************************************** Second time, at least, that you've mentioned that thing about boxes and superinfinity, whatever that is. What part of Cantor's Theorem using the argument found in Russell's paradox isn't clear to you? The theorem is pretty succinct: it says that for any set X, we have |X| < |P(X)| , |.| meaning cardinal of set and P(X) the power set of X. cardinality higher than N's, meaning: there's a set, namely P(N), in which N can be embedded but from which there is no possible embedding into N. What's unclear here? Where are you stuck? Tonio === Subject: Re: the problem with Cantor > UTM(a,b) = 1 IFF b is a member of the ath subset. > That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. Ahh, the old which box contains the numbers of all the boxes that don't > contain their own number proof! Just by asking this question, which is > impossible, or posing the theorem, you prove superinfinity. > ************************************************************** -Second time, at least, that you've mentioned that thing about boxes -and superinfinity, whatever that is. - -What part of Cantor's Theorem using the argument found in Russell's -paradox isn't clear to you? The theorem is pretty succinct: it says -that for any set X, we have |X| < |P(X)| , |.| meaning cardinal of set -and P(X) the power set of X. - -cardinality higher than N's, meaning: there's a set, namely P(N), in -which N can be embedded but from which there is no possible embedding -into N. - -What's unclear here? Where are you stuck? I have given a description of a mapping from N to P(N). > UTM(a,b) = 1 IFF b is a member of the ath subset. This will compute all possible subsets (the powerset), finite and infinite of N. primes - its there odds - its there evens - its there {1,2,3} - its there How on earth can the set_of_all_computer_programs possibly miss some simple subset calculation on natural numbers? I have full trust in the capacity of a Universal Turing Machine to come up with a numerical list of all possible subsets of N. I don't believe in your proof 1 bit. All this proof does... > Cantor's Theorem: For any set S, |S| < |P(S)|. If (a) |S|<=|P(S)| and (b) |S|=/=|P(S)| are both true then Cantor's theorem > is proven. Letting q be the one to one function q:S-->P(S) such that > s-->{s}, it is easy to see that (a) is true. To see that (b) is true, > assume the contrary. That is, assume there is a bijection r:S-->P(S). Let > T be the set: T = {s in S : s not in r(s)} Since r is assumed to be onto, there should be a t in S such that (c) > T=r(t). If t is in T then, by definition, t is not in r(t) and this > contradicts (c). So it must be that t is not in T but then (c) means that t > is in T and this is another contradiction. Thus, r is not onto and > therefore not a bijection. QED is self reference and negate itself. Its just a mathematical tongue twister in disguise. Its exactly analagous to having an infinite sized room full of numbered boxes and asking which box contains the numbers of all the boxes that don't contain their own number?. It's just a non solvable puzzle its not a new subset. It reduces to which box (subset) contains its own number and also doesn't contain its own number?. You people are plugging rubbish into ZFC and you are getting rubbish out of it. Herc === Subject: Re: the problem with Cantor > I have given a description of a mapping from N to P(N). UTM(a,b) = 1 IFF b is a member of the ath subset. Such a UTM assumes a priori the existence of what herkimer wishes to conclude, that there is a surjective mapping from N to P(N). Such arguments are called circular, and are not accepted by the logically fastidious. This will compute all possible subsets (the powerset), finite and infinite of > N. > primes - its there > odds - its there > evens - its there > {1,2,3} - its there Not unless one has, a priori, a surjection from N to P(N). How on earth can the set_of_all_computer_programs possibly miss some > simple subset calculation on natural numbers? By being of smaller cardinality. I have full trust in the capacity of a Universal Turing Machine to come up > with a numerical list of all possible subsets of N. Your faith does not override the limitations of ZFC and other set theories in which no such Turing machines can be constructed. I don't believe in your proof 1 bit. And we do not believe in your circular arguments. You people are plugging rubbish into ZFC and you are getting rubbish out of > it. Whereas herkimer's rubbish has even less justification than ZFC provides. === Subject: Re: the problem with Cantor posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > UTM(a,b) = 1 IFF b is a member of the ath subset. > That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. > Ahh, the old which box contains the numbers of all the boxes that don't > contain their own number proof! Just by asking this question, which is > impossible, or posing the theorem, you prove superinfinity. ************************************************************** -Second time, at least, that you've mentioned that thing about boxes > -and superinfinity, whatever that is. > - > -What part of Cantor's Theorem using the argument found in Russell's > -paradox isn't clear to you? The theorem is pretty succinct: it says > -that for any set X, we have |X| < |P(X)| , |.| meaning cardinal of set > -and P(X) the power set of X. > - > -cardinality higher than N's, meaning: there's a set, namely P(N), in > -which N can be embedded but from which there is no possible embedding > -into N. > - > -What's unclear here? Where are you stuck? I have given a description of a mapping from N to P(N). > UTM(a,b) = 1 IFF b is a member of the ath subset. This will compute all possible subsets (the powerset), finite and infinite of N. > primes - its there > odds - its there > evens - its there > {1,2,3} - its there > **************************************************************** Where's that mapping from N to P(N)? All I can see is UTM(a,b) = 1 iff b is a member of a-th subset, and this is NOT a map with N as definition set: it appears to be that b is a natural number and a is some set, perhaps an element of P(N). Moreover: if a indeed is a way of counting elements of P(N), i.e. subsets of N, then we're already wrong: you can't count these guys! Meaning: you can't put the elements of P(N) in a 1-1 correspondence with elements of N, and then to talk of nth subset is meaningless ***************************************************************** > How on earth can the set of all computer programs possibly miss some > simple subset calculation on natural numbers? > ************************************************************* I don't really know, but (1) what is the set of all computer programs? (2) What is a simple subset calculation on natural numbers? You seem to believe that the set of all computer programs, whatever this means, is so powerful and mighty that it can't possible miss something that, apparently, is simple. Why do you believe that? ************************************************************* > I have full trust in the capacity of a Universal Turing Machine to come up > with a numerical list of all possible subsets of N. > *************************************************************** Why? Is this some kind of axiom or is this assumed in UTM's definition? And again: you seem to beleive you can ennumerate all possible subsets of N, but you really can't: this is EXACTLY what Cantor, among oither things, proved. *************************************************************** > I don't believe in your proof 1 bit. æAll this proof does... > Cantor's Theorem: For any set S, |S| < |P(S)|. > If (a) |S|<=|P(S)| and (b) |S|=/=|P(S)| are both true then Cantor's theorem > is proven. æLetting q be the one to one function q:S-->P(S) such that > s-->{s}, it is easy to see that (a) is true. æTo see that (b) is true, > assume the contrary. æThat is, assume there is a bijection r:S-->P(S). æLet > T be the set: > T = {s in S : s not in r(s)} > Since r is assumed to be onto, there should be a t in S such that (c) > T=r(t). æIf t is in T then, by definition, t is not in r(t) and this > contradicts (c). æSo it must be that t is not in T but then (c) means that t > is in T and this is another contradiction. æThus, r is not onto and > therefore not a bijection. QED is self reference and negate itself. æIts just a mathematical tongue twister in disguise. ************************************************************ Do you know the reductio ad absurdum method to prove mathematical statements? Is it clear to you? If it is not, then you have to research a little on it, and if it is then: do you reject this method? If you do, why? And if you don't, then what part of CT's proof you don't like/accept? ************************************************************ > Its exactly analagous to having an infinite sized room full of numbered boxes and > asking which box contains the numbers of all the boxes that don't contain their > own number?. ************************************************************ No, it is not analogous to this...not even close, leave alone exactly analogous. It could be so-so analogous if you'd say: there are some numbered balls in a room, and we have in another room all the possible boxes formed by introducing some (all, none) of the balls of the first room in a box (thus, we have boxes with one ball, with two balls, with no ball at all, with all the balls, etc), and these boxes are numbered too in some way. We're going to show that there are more boxes than balls, and we do this by assuming otherwise ==> there must be a box which contains all the balls which are not contained in the box with the same number as the balls. You see, the above example comes up pretty cumbersome, complex, and it even assumes that we have a huge number of copies of each ball (as many copies as possible boxes containing that specific ball), and after all this work we haven't attained anything better... ************************************************************ æIt's just a non solvable puzzle its not a new subset. æIt reduces to > which box (subset) contains its own number and also doesn't contain its own number?. You people are plugging rubbish into ZFC and you are getting rubbish out of it. > **************************************************************** So far ZFC has yielded pretty nice results. Have you ever thought, and I'm asking this without the slightest intention to offend, of the possibility that you don't REALLY grasp all this basic logic-set-theory stuff? Could it be that your mathematical education is lacking some basic training in these matters? Are you a mathematician? Of course, not being one doesn't automatically disqualifies you to give you opinion, but perhaps there's some stuff you haven't yet studied...? Tonio > Herc === Subject: Re: the problem with Cantor [...] > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? You can define a real number as the output of a program, as in You say that only computable reals exist. The accepted constructions of the real numbers don't mention computability. To a Platonist, the set of reals is something out there and the reals can't be counted with 1, 2, 3 .... , i.e. they form an uncountable set. You say the Platonists are wrong. To convince Platonists that they are wrong, one way would be to get a contradiction from ZFC, such as 1 =/= 1 . Besides arguments like that, it should be very hard to convert a Platonic set theorist. David Bernier === Subject: Re: the problem with Cantor > [...] > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? You can define a real number as the output of a program, as in You say that only computable reals exist. The accepted constructions > of the real numbers don't mention computability. To a Platonist, > the set of reals is something out there and the reals can't > be counted with 1, 2, 3 .... , i.e. they form an uncountable set. > Platonist and ZFC adherent aren't synonymous. > You say the Platonists are wrong. To convince Platonists that > they are wrong, one way would be to get a contradiction > from ZFC, such as 1 =/= 1 . Besides arguments like that, > it should be very hard to convert a Platonic set theorist. David Bernier Platonist set theorist and ZFC adherent aren't synonymous. ZF's universe, in any larger universe, one of which exists else ZF would be complete, and in being so by argument within itself inconsistent, is irregular and contains itself, thus ZF is inconsistent. That's not so say regular set theory isn't the most applicable foundation for finite combinatorics, because it's an excellent and more or less naturally consistent foundation for finite combinatorics. In the infinite, there's much to be considered about that it is the nature of infinity to have what might seem unintuitive (although not necessarily not intuitionistic) properties. the more closely they are measured, infinitesimal, and the universe is larger the more comprehensively it is measure, infinite, in nature, physics. In terms of particularly the unit interval of reals, in terms of real numbers vis-a-vis natural integers, that there is a way to construct them in their natural order from zero through one, while that was the obvious way to do it in terms of countable additivity in analysis, the density of the individua of the continuum leads to ready conflicts with the notion that they can be so arrayed. While that may be so, in terms of modeling the natural/unit equivalency function as a limit of real functions, much as the unit impulse function is modeled as a limit of real functions, leads to then these implicit mathematical objects, real numbers, only and all of which are on the unit interval, thus that the resulting construction of those numbers leads to that, for example, the antidiagonal and nested intervals results don't apply. I think more people should be interested in surpassing what is basically a limitation of adherence to the cumulative hierarchy of transfinite ordinals and their cardinals. Instead, by analyzing the polydimensional points of the real number line, in the penultimate and ultimate real number space(s), new applications will be discovered, in concordance with nature. Ross F. === Subject: Re: the problem with Cantor > [...] > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? You can define a real number as the output of a program, as in You say that only computable reals exist. The accepted constructions > of the real numbers don't mention computability. To a Platonist, > the set of reals is something out there and the reals can't > be counted with 1, 2, 3 .... , i.e. they form an uncountable set. You say the Platonists are wrong. To convince Platonists that > they are wrong, one way would be to get a contradiction > from ZFC, such as 1 =/= 1 . Besides arguments like that, > it should be very hard to convert a Platonic set theorist. David Bernier You just admitted you are all impervious to reason. I still have no comments on the fact that 1/ the computable reals compute all digit expansions to infinite length 2/ the diagonal of a real list with sufficient variety in the expansions is independent of the list Until YOU can address those issues, everyone in the future reading these historic posts No offence David you've done a pretty good job at being open, as have some others. Herc === Subject: Re: the problem with Cantor > [...] > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? > You can define a real number as the output of a program, as in > You say that only computable reals exist. The accepted constructions > of the real numbers don't mention computability. To a Platonist, > the set of reals is something out there and the reals can't > be counted with 1, 2, 3 .... , i.e. they form an uncountable set. > You say the Platonists are wrong. To convince Platonists that > they are wrong, one way would be to get a contradiction > from ZFC, such as 1 =/= 1 . Besides arguments like that, > it should be very hard to convert a Platonic set theorist. > David Bernier You just admitted you are all impervious to reason. On the contrary, he has just said that it is difficult to convince a Platonist that he is wrong without having any proof that that Platonist is wrong. I still have no comments on the fact that > 1/ the computable reals compute all digit expansions to infinite length That presumes falsely that all such expansions are computable. > 2/ the diagonal of a real list with sufficient variety in the expansions is > independent > of the list There are at least as many anti-diagonals not in list as expansion as in it, but each such anti-diagonal is dependent on the list from which it is produced. Until YOU can address those issues, everyone in the future reading > entrenched text book parrots you all are. There will always be those who stubbornly refuse to be persuaded by your own nonproofs of your own beliefs. === Subject: Re: the problem with Cantor > 1) Well, let's play along: I entered your link, pressed 1232323 and > got a left list of some 60 numbers, then an apaprently randomly > generated numbers, and a second list which apparently contains the > randly gen. number in its main diagoal entries...so? I don't > understand what is this supposed to show. I believe I know what Herc is trying to accomplish. Herc is making the following claim (quoted verbatim): Claim: Let L be a list of numbers in [0,1] such that for each d in > {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal > representation of L_i is d} is infinite. Let D be a random variable > uniform on [0,1]. Then almost surely there is a permutation L' of L > such that D is the diagonal of L'. But there are two big problems here: > 1) The claim most likely _fails_ in ZFC -- even for finite lists! > 2) Even if the claim does hold in ZFC, it has absolutely _no_ > bearing on the validity of Cantor in ZFC. Let's deal with 1) first. Herc's link is an applet consisting > of three steps: 1. Populate list -- clicking this causes a list of sixty real > numbers between zero and one, apparently in ternary since > only the digits 0,1,2, appear, each with sixty digits. But the Cantor 'anti-diagonal' construction does not work in ternary any more than it does in binary, as in ternary, one has to use either 0 or 2 in ternary to be different from 1, and allowing either a 0 or a 2 to appearin the anti-diagonal at least theoretically allows the anti-diagonal to be the dual representation of a listed number. With bases of four and larger, the anti-diagonal construction can always avoid using the digits, 0 and the base-minus-one, which are required in dual representations. > 2. Randomise diagonal -- clicking this causes a 61st real > number to appear. > 3. Generate list -- clicking this causes a reordering of the > first list to appear such that the real number chosen in the > second step appears on the diagonal. On the face of it, this should always work. We know that > there are 60! (60 factorial) permutations of the original list > and only 3^60 possible diagonals -- and factorials increase > much faster than exponentiation. (60! ~ 8*10^81, but > 3^60 ~ 4*10^28.) So there ought to be more than enough > permutations for all the possible diagonals. But I decided to try a little trick. I clicked on the first button > to populate the list -- but rather than clicking on the second > button, I manually typed in my own diagonal. What I did was > look at the first row, which happened to be: 0.021100001022012201100101102022102002102112020210111021110120 Then I used one of the usual tricks for generating an > antidiagonal (in Cantor's proof) from a diagonal to > make an antirow. I replaced the digits 1 and 2 with > 0 and the digit 0 with 1, to obtain the number: 0.100011110100100010011010010100010110010000101001000100001001 which I typed into the diagonal box. Then I clicked on the third button. Sure enough, only 59 > numbers appeared rather than 60. And of course, the > number that was present on the first list but missing > from the second list was the first number. It would be > _impossible_ for the first number to appear, since it > differs from the diagonal in _every_ digit! And it's easy to see that the same problem would > happen for _any_ list, whether finite or infinite, whether > saturated or unsaturated. Of course, Herc's claim is that almost surely such a > permutation exists. But what is almost surely? Does > Herc mean that the set of all real numbers for which a > permutation fails to exist has Lebesgue measure zero? If so, then Herc may be right. So far, we've only shown > the antirows to be counterexamples -- where an > antirow is defined to be a number differing from every > digit from a number in a row. Suppose the first row contains the real number 1/2 -- > which is 0.111... in ternary. Then an antirow would be > any number consisting of only 0 and 2 in ternary -- and > the set of all such numbers is the famous Cantor > middle-third set, which has Lebesgue measure zero. Similarly, the set of all possible antirows for a given > row also has Lebesgue measure zero. And since this > is a _list_ of reals, there's exactly one real in this list > for every natural, so there are only countably many > such reals and their corresponding sets of antirows. So the set of all antirows for the entire list is the > countable union of null sets, and therefore itself null. Of course, this isn't to say that there doesn't exist a > counterexample to Herc's claim that doesn't happen to > be an antirow to any row at all. But the set of the only > _known_ counterexamples to Herc's claim does have > Lebesgue measure zero. But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! Indeed, let's compare my proof above to Cantor's: I show it's impossible for an _antirow_ to be a _diagonal_. > Cantor shows it's impossible for an _antidiagonal_ to be a _row_. If we think about this for a moment, we see that in my > proof, I only claim to show that there exists numbers > which can't be on the diagonal. That a counterexample > happens to be an antirow is irrelevant. I only care about > the diagonal, not rows or antirows. Similarly, Cantor claims that there exist real numbers > which can't be in any row of the list. That a counterexample > happens to be an antidiagonal is irrevelevant. Cantor cares > only about _rows_, not the diagonal or the antidiagonal. It's possible for a real number to be the diagonal of one > permutation but an antidiagonal of another permutation. Herc > was apparently hoping that this wouldn't be the case, but > we can even consider the example: 168 > 249 > 357 So an antidiagonal is 258, but if we reorder the list: 249 > 357 > 168 suddenly 258's the diagonal. Now 369's an antidiagonal, but: 357 > 168 > 249 now 369's the diagonal. Therefore, Herc's claim, even if it were true, would neither > prove nor disprove _anything_ about Cantor. > 2) What is superinfinity? And why is it mine? By superinfinity, most likely Herc (pejoratively) means > uncountable infinity. And it's yours (Tonio's), as much as > it belongs to every standard mathematician who believes > in its existence (i.e., who adheres to a set theory, such > as ZFC, which proves the existence of uncountable sets), > as opposed to Herc, who doesn't believe in the existence > of uncountable sets. === Subject: Re: the problem with Cantor > I don't care Certainly not about truth. === Subject: Re: the problem with Cantor > I don't care Certainly not about truth. The truth is there is nothing larger than infinity. You have failed to give me a new sequence of digits not on the computable reals. Cantor's proof is an exercise in going round in circles. Herc === Subject: Re: the problem with Cantor > > I don't care > Certainly not about truth. The truth is there is nothing larger than infinity. > You have failed to give me a new sequence of digits not on the computable > reals. You have failed to give us any listing of all computable reals. The existence of such a listing would be proof that the computable reals are countable, and the non-existence of any such listing would be evidence off that set's uncountability. > Cantor's proof is an exercise in going round in circles. It is an eample of clearer thinking than Herkimer has yet demonstrated. === Subject: Re: the problem with Cantor > > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. > You have failed to give me a new sequence of digits not on the computable > reals. You have failed to give us any listing of all computable reals. > The existence of such a listing would be proof that the computable reals > are countable, and the non-existence of any such listing would be > evidence off that set's uncountability. > Cantor's proof is an exercise in going round in circles. It is an eample of clearer thinking than Herkimer has yet demonstrated. Use a Universal Turing Machine to compute all the reals. let the jth digit of the ith real = UTM(i,j) mod 10 where UTM has 2 parameters, the TM it is emulating and the input tape to the TM. UTM(TM, input) = output can continue on computing every computable real. Diagonalisation will not produce any new sequence of digits as I have demonstrated. You are far too stupid to admit that and maintain an argument. Herc === Subject: Re: the problem with Cantor > Use a Universal Turing Machine to compute all the reals. Which of the uncountably many incomputable reals will it compute first? === Subject: Re: the problem with Cantor > Use a Universal Turing Machine to compute all the reals. Which of the uncountably many incomputable reals will it compute first? I meant to say, use a UTM to compute all the computable reals, that's what you asked for. Although there are no incomputable reals. Herc === Subject: Re: the problem with Cantor > > Use a Universal Turing Machine to compute all the reals. > Which of the uncountably many incomputable reals will it compute first? I meant to say, use a UTM to compute all the computable reals, that's what > you asked for. Some of those computable reals will take all the time your UTM has. So it may never get past the first one. Although there are no incomputable reals. What exists outside UTMs is not necessarily computable. If Herkimer's universe is limited to what a UTM can produce, his life is pretty lifeless. === Subject: Re: the problem with Cantor > Use a Universal Turing Machine to compute all the reals. > Which of the uncountably many incomputable reals will it compute first? > I meant to say, use a UTM to compute all the computable reals, that's what > you asked for. Some of those computable reals will take all the time your UTM has. So it may never get past the first one. > Although there are no incomputable reals. What exists outside UTMs is not necessarily computable. If Herkimer's universe is limited to what a UTM can produce, his life is > pretty lifeless. brrrrr bing *pretty* it does not compute Herc === Subject: Re: the problem with Cantor > I don't care > Certainly not about truth. The truth is there is nothing larger than infinity. > You have failed to give me a new sequence of digits not on the computable reals. > Cantor's proof is an exercise in going round in circles. Herc Suppose we have a list LF of the numbers that have a terminating decimal expansion: 2.3, 3.14159, 2000000, 0.00000000000056, etc. 1/3 = 0.33333.... = __ 0.3 [ 0.3periodic]. The first 5 decimals of 1/3 are the same as those of 0.33333 , and so on for 5, 7, ... 100, ... 1000 decimals and more. But if we take a fixed number 'c' on LF and compare it to __ 0.3 [ 0.3periodic] , then either __ c < 0.3 or __ c>0.3 . Don't you think the same thing happens with Omega_{Herc} and LF? After all, you're only comparing a finite (but arbitrarily large) number of decimals. In other words, what works for Omega_{Herc} and LF should work for __ 0.3 and LF ... David Bernier === Subject: Re: the problem with Cantor > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. > You have failed to give me a new sequence of digits not on the computable reals. > Cantor's proof is an exercise in going round in circles. > Herc > > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c < 0.3 or > __ > c>0.3 . Don't you think the same thing happens with Omega_{Herc} and LF? After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... David Bernier You've shattered the illusion that Omega is on some list but the problem with Cantors proof still stands. Every finite prefix of possible sequences is computable, so the original diagonalisation technique fails to find a new sequence. 123 456 789 In this simple finite list, the diagonal is 159. The anti-diagonal is 261. It looks like diagonalisation actually finds new sequences of digits. but when you apply it to the set of computable reals it doesn't, the list is already saturated with every possible sequence. That's the claim, that there is a problem, not that Omega is on the list. The computable reals contains every possible sequence up to infinite length, and you tell me with a cloak and digger trick that you can find new sequences. You can't. All the sequences are computed, up to oo length, end of story. Have you taken a look at http://www.freewebs.com/namesort/linux.html It proves that a list of reals that contains every digit oo times in each column can be sorted to have any random diagonal. The diagonal is independent of such lists, e.g. the computable reals. That's another problem with Cantor's proof, the diagonal does not even depend on the list, it's just a random variable. Herc === Subject: Re: the problem with Cantor [...] > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c< 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... > David Bernier You've shattered the illusion that Omega is on some list but the problem > with Cantors proof still stands. Every finite prefix of possible sequences is computable, so the original > diagonalisation technique fails to find a new sequence. 123 > 456 > 789 In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. That's the claim, that there is a problem, not that Omega is on the list. The computable reals contains every possible sequence up to infinite length, > and you tell me with a cloak and digger trick that you can find new sequences. > You can't. All the sequences are computed, up to oo length, end of story. Let's say a Turing machine is decimal if, starting with a tape with k consecutive 1s , where k is a positive integer, it halts and leaves m(k) 1s on the tape, with 0<=m(k)<10 . This makes precise the idea of a procedure in a programming language with input = a positive integer and output = one of the 10 digits. If you want the diagonal and anti-diagonal to be computable and on the list, since diag and anti-diag depend on the ordering of the list, I think it's for you to explain how to order the list of decimal TMs so that diag and anti-diag are *computable* ; also the list of decimal TMs should be complete. How do you know that the entire anti-diagonal is computable? David Bernier > Have you taken a look at > http://www.freewebs.com/namesort/linux.html It proves that a list of reals that contains every digit oo times in each column > can be sorted to have any random diagonal. The diagonal is independent of > such lists, e.g. the computable reals. That's another problem with Cantor's proof, > the diagonal does not even depend on the list, it's just a random variable. Herc === Subject: Re: the problem with Cantor > [...] > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c< 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the problem > with Cantors proof still stands. > Every finite prefix of possible sequences is computable, so the original > diagonalisation technique fails to find a new sequence. > 123 > 456 > 789 > In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. > That's the claim, that there is a problem, not that Omega is on the list. > The computable reals contains every possible sequence up to infinite length, > and you tell me with a cloak and digger trick that you can find new sequences. > You can't. All the sequences are computed, up to oo length, end of story. Let's say a Turing machine is decimal if, starting with > a tape with k consecutive 1s , where k is a positive integer, it > halts and leaves m(k) 1s on the tape, with 0<=m(k)<10 . This makes precise the idea of a procedure in a programming language > with input = a positive integer and output = one of the 10 digits. If you want the diagonal and anti-diagonal to be computable > and on the list, since diag and anti-diag depend on the ordering > of the list, I think it's for you to explain how to order the list > of decimal TMs so that diag and anti-diag are > *computable* ; also the list of decimal TMs should > be complete. How do you know that the entire anti-diagonal is > computable? David Bernier The anti-diagonal is not computable, therefore it doesn't exist. let UTM(real, digit) mod 10 calculate the list of reals. (where real is deceptively an integer) the antidiagonal is UTM(digit, digit) + 1 mod 10 if this is computable, then some TM computes it and some emulated TM also computes it, UTM(ad, digit) mod 10 = UTM(digit, digit) + 1 mod 10 when digit = ad, UTM(ad, ad) mod 10 = UTM(ad, ad) + 1 mod 10 Contradiction! Therefore antidiag is an invalid formula. This is a more sensible conclusion than higher infinities exist as I've basically just rearranged Cantor's proof. One can argue its a valid specification of a sequence but it doesn't actually compute a new sequence of digits. Herc > Have you taken a look at > http://www.freewebs.com/namesort/linux.html > It proves that a list of reals that contains every digit oo times in each column > can be sorted to have any random diagonal. The diagonal is independent of > such lists, e.g. the computable reals. That's another problem with Cantor's proof, > the diagonal does not even depend on the list, it's just a random variable. > Herc > === Subject: Re: the problem with Cantor > [...] > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c< 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the problem > with Cantors proof still stands. Every finite prefix of possible sequences is computable, so the original > diagonalisation technique fails to find a new sequence. 123 > 456 > 789 In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. That's the claim, that there is a problem, not that Omega is on the list. The computable reals contains every possible sequence up to infinite length, > and you tell me with a cloak and digger trick that you can find new sequences. > You can't. All the sequences are computed, up to oo length, end of story. > Let's say a Turing machine is decimal if, starting with > a tape with k consecutive 1s , where k is a positive integer, it > halts and leaves m(k) 1s on the tape, with 0<=m(k)<10 . > This makes precise the idea of a procedure in a programming language > with input = a positive integer and output = one of the 10 digits. > If you want the diagonal and anti-diagonal to be computable > and on the list, since diag and anti-diag depend on the ordering > of the list, I think it's for you to explain how to order the list > of decimal TMs so that diag and anti-diag are > *computable* ; also the list of decimal TMs should > be complete. > How do you know that the entire anti-diagonal is > computable? > David Bernier The anti-diagonal is not computable, therefore it doesn't exist. let UTM(real, digit) mod 10 calculate the list of reals. (where real is deceptively an integer) With the mod 10, we can avoid talking about decimal TMs. For argument 'real', I'd put TM_n, the n'th TM that halts on any input. For 'digit', I'd put k and we get (after rewriting): UTM(TM_n, k)%10 // UTM a fixed universal TM (Indeed, TM_n and k will be presented to the UTM as input or 1s on the tape of the UTM just before it starts). for the k'th decimal of the n'th TM that halts under all inputs, with TMs: TM_1, TM_2 , .... and for k = 1, 2, 3 .... if we are only concerned with reals in [0, 1], and allowing say 0.5000... to appear for several always halting TMs: TM_a, TM_b and so on . If the anti-diag were computable, there would be a contradiction. I think what's going on is that the unsolvability of the halting problem prevents us from writing a program that finds the n'th always-halting TM. The anti-diagonal is not computable, therefore it doesn't exist. There was a long thread starting with: Uncomputable numbers are all in your head. The conclusion I see is that you're not a Platonist. You could abandon the Power set axiom for infinite sets and the axiom of choice. Constructivists don't draw conclusions using the law of the excluded middle. the antidiagonal is UTM(digit, digit) + 1 mod 10 if this is computable, then some TM computes it and some emulated TM also computes it, UTM(ad, digit) mod 10 = UTM(digit, digit) + 1 mod 10 when digit = ad, UTM(ad, ad) mod 10 = UTM(ad, ad) + 1 mod 10 Contradiction! Therefore antidiag is an invalid formula. This is a more sensible conclusion than higher infinities exist as I've basically just > rearranged Cantor's proof. One can argue its a valid specification of a sequence > but it doesn't actually compute a new sequence of digits. Then we can say you're a computist. David Bernier > Herc === Subject: Re: the problem with Cantor > [...] > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c< 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the problem > with Cantors proof still stands. > Every finite prefix of possible sequences is computable, so the original > diagonalisation technique fails to find a new sequence. > 123 > 456 > 789 > In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. > That's the claim, that there is a problem, not that Omega is on the list. > The computable reals contains every possible sequence up to infinite length, > and you tell me with a cloak and digger trick that you can find new sequences. > You can't. All the sequences are computed, up to oo length, end of story. > Let's say a Turing machine is decimal if, starting with > a tape with k consecutive 1s , where k is a positive integer, it > halts and leaves m(k) 1s on the tape, with 0<=m(k)<10 . > This makes precise the idea of a procedure in a programming language > with input = a positive integer and output = one of the 10 digits. > If you want the diagonal and anti-diagonal to be computable > and on the list, since diag and anti-diag depend on the ordering > of the list, I think it's for you to explain how to order the list > of decimal TMs so that diag and anti-diag are > *computable* ; also the list of decimal TMs should > be complete. > How do you know that the entire anti-diagonal is > computable? > David Bernier > The anti-diagonal is not computable, therefore it doesn't exist. > let UTM(real, digit) mod 10 calculate the list of reals. (where real is deceptively an integer) With the mod 10, we can avoid talking about decimal TMs. For > argument 'real', I'd put TM_n, the n'th TM that halts on any input. > For 'digit', I'd put k and we get (after rewriting): UTM(TM_n, k)%10 // UTM a fixed universal TM (Indeed, TM_n and k will be presented to the UTM as > input or 1s on the tape of the UTM just before it starts). for the k'th decimal of the n'th TM that halts under > all inputs, with TMs: TM_1, TM_2 , .... and for k = 1, 2, 3 .... > if we are only concerned with reals in [0, 1], and allowing > say 0.5000... to appear for several always halting TMs: > TM_a, TM_b and so on . If the anti-diag were computable, there would be a contradiction. > I think what's going on is that the unsolvability of the halting > problem prevents us from writing a program that finds > the n'th always-halting TM. The anti-diagonal is not computable, therefore it doesn't exist. There was a long thread starting with: HA I started that one! Read down a few posts... [Herc] A real number has to be enscribed somehow, computers are just a way of automating that Uncomputable numbers are all in your head. The conclusion I see is that you're not a Platonist. > You could abandon the Power set axiom for > infinite sets and the axiom of choice. > Constructivists don't draw conclusions using > the law of the excluded middle. > the antidiagonal is UTM(digit, digit) + 1 mod 10 > if this is computable, then some TM computes it and some emulated TM also computes it, > UTM(ad, digit) mod 10 = UTM(digit, digit) + 1 mod 10 > when digit = ad, > UTM(ad, ad) mod 10 = UTM(ad, ad) + 1 mod 10 > Contradiction! > Therefore antidiag is an invalid formula. > This is a more sensible conclusion than higher infinities exist as I've basically just > rearranged Cantor's proof. One can argue its a valid specification of a sequence > but it doesn't actually compute a new sequence of digits. Then we can say you're a computist. David Bernier I just think numbers have to be enscribed somehow, the only noncomputable things I've seen are self referencing negating tricks, they define themselves to be undoable. MATHS define an object type define a self reference in the object type and negate it find the contradiction claim something whymsical and far fetched I don't fall for any of it. Computable numbers are very very expressive, there's no limit whatsoever. Infinitely long random strings aren't numbers, its just noise. The anti-diagonal is just a facet of lists. Each digit position in the real list contains infinite of each digit, so you can reorder the list and construct any diagonal you want, its just a random variable. Herc === Subject: Re: the problem with Cantor [Herc] > A real number has to be enscribed somehow, computers are just a way > of automating that Numerals may have to be enscribed, numbers don't. === Subject: Re: the problem with Cantor The anti-diagonal is not computable, therefore it doesn't exist. In what axiom system? Unless you can state exactly what assumptions you are making (and you are making quite a few) there is no way to determine what does or does not exist in the system formed by those assumptions. let UTM(real, digit) mod 10 calculate the list of reals. (where real is > deceptively an integer) the antidiagonal is UTM(digit, digit) + 1 mod 10 Not in any standard decimal system. So the rest of Herk's garbage is just that. === Subject: Re: the problem with Cantor > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. You have > failed to give me a new sequence of digits not on the computable > reals. Cantor's proof is an exercise in going round in circles. > Herc > > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: 2.3, 3.14159, 2000000, 0.00000000000056, > etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , > and so on for 5, 7, ... 100, ... 1000 decimals and more. But if we > take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c < 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) > number of decimals. In other words, what works for Omega_{Herc} > and LF should work for > __ > 0.3 and LF ... > David Bernier You've shattered the illusion that Omega is on some list but the > problem with Cantors proof still stands. Every finite prefix of possible sequences is computable, so the > original diagonalisation technique fails to find a new sequence. Then Herkimer must be claiming that the diagonal appears at some finite position in the list. Which position is it in, dumbkopf? 123 456 789 In this simple finite list, the diagonal is 159. The anti-diagonal is 261. For the above simple finite list, there are 729 different base 10 anti-diagonals. It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. Is any list of all computable reals itself computable? And given any arbitrary infinite list of computable reals, is the diagonal for that list computable? Until Herk the Jerk can definitively answer those, one way or the other, there are holes in his arguemnts. > The computable reals contains every possible sequence up to infinite > length, and you tell me with a cloak and digger trick that you can > find new sequences. I don't claim that any such new sequence is necessarily computable, merely that for any list it exists. > You can't. All the sequences are computed, up to oo length, end of > story. There is nothing in Cantor's proof that requires the 'anti-diagonal to be computable, merely existing in that non-physical way that numbers have of existing. Have you taken a look at http://www.freewebs.com/namesort/linux.html It proves that a list of reals that contains every digit oo times in > each column can be sorted to have any random diagonal. It does not prove anything at all about such lists, since it only works with finite lists of finite lengths. > The diagonal is independent of such lists > That's another problem with Cantor's > proof, the diagonal does not even depend on the list, it's just a > random variable. Nonsense. Cantor's original 'diagonal' proof applies only to lists of infinite binary sequences with values in {m,w}. So anything not valid for that original proof is irrelevant. === Subject: Re: the problem with Cantor > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. You have > failed to give me a new sequence of digits not on the computable > reals. Cantor's proof is an exercise in going round in circles. > Herc > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: 2.3, 3.14159, 2000000, 0.00000000000056, > etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , > and so on for 5, 7, ... 100, ... 1000 decimals and more. But if we > take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c < 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) > number of decimals. In other words, what works for Omega_{Herc} > and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the > problem with Cantors proof still stands. > Every finite prefix of possible sequences is computable, so the > original diagonalisation technique fails to find a new sequence. Then Herkimer must be claiming that the diagonal appears at some finite > position in the list. Which position is it in, dumbkopf? You can't correctly define an anti-diagonal. You also can't cite any sequence of numbers that is new to the list. Oh sure it works for 123 456 789 but it doesn't work for the infinite set of computable reals. The computable reals contains all finite sequences of digits up to oo length. Every pattern possible is computable. > 123 456 789 > In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. For the above simple finite list, there are 729 different base 10 > anti-diagonals. If usenet posts had audio you'd hear clapping right now. Did you remember the antidiagonals for the 6 permutations of the list? > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. Is any list of all computable reals itself computable? And given any > arbitrary infinite list of computable reals, is the diagonal for that > list computable? yes computable things are computable yes there is no contradiction in computing the diagonal somewhere in the list Until Herk the Jerk can definitively answer those, one way or the > other, there are holes in his arguemnts. > The computable reals contains every possible sequence up to infinite > length, and you tell me with a cloak and digger trick that you can > find new sequences. I don't claim that any such new sequence is necessarily computable, > merely that for any list it exists. > You can't. All the sequences are computed, up to oo length, end of > story. There is nothing in Cantor's proof that requires the 'anti-diagonal to > be computable, merely existing in that non-physical way that numbers > have of existing. Numbers exist because they can be computed. If they couldn't be computed you wouldn't know what they were. > Have you taken a look at http://www.freewebs.com/namesort/linux.html > It proves that a list of reals that contains every digit oo times in > each column can be sorted to have any random diagonal. It does not prove anything at all about such lists, since it only works > with finite lists of finite lengths. Oh you had a look. Claim: Let L be a list of numbers in [0,1] such that for each d in {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal representation of L_i is d} is infinite. Let D be a random variable uniform on [0,1]. Then almost surely there is a permutation L' of L such that D is the diagonal of L'. > The diagonal is independent of such lists > That's another problem with Cantor's > proof, the diagonal does not even depend on the list, it's just a > random variable. Nonsense. Did you randomise the diagonal at the website, and produce an equivalent set in the second list with that new diagonal? What does that mean? Cantor's original 'diagonal' proof applies only to lists of infinite > binary sequences with values in {m,w}. So anything not valid for that original proof is irrelevant. Anything valid for an equivalent proof is relevant. This brings up an important point. Virgil, like most of you, believe numbers and mathematical things exist without being computable? Tell me if you subtracted the theories of mathematics that are computable from the total set of theories in mathematics that you believe in, what are you left with? Herc === Subject: Re: the problem with Cantor > > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. You have > failed to give me a new sequence of digits not on the computable > reals. Cantor's proof is an exercise in going round in circles. > Herc > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: 2.3, 3.14159, 2000000, 0.00000000000056, > etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , > and so on for 5, 7, ... 100, ... 1000 decimals and more. But if we > take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c < 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) > number of decimals. In other words, what works for Omega_{Herc} > and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the > problem with Cantors proof still stands. > Every finite prefix of possible sequences is computable, so the > original diagonalisation technique fails to find a new sequence. > Then Herkimer must be claiming that the diagonal appears at some finite > position in the list. Which position is it in, dumbkopf? You can't correctly define an anti-diagonal. Cantor did, at least for any list of binary sequences (which is all he used, the decimal model came later and was done by others. > You also can't cite any > sequence of numbers that is new to the list. I can cite a rule which will produce a sequence not in a given list. but it doesn't work for the infinite set of computable reals. Then the rule must define an incomputable sequnce > The computable > reals > contains all finite sequences of digits up to oo length. Every pattern > possible is > computable. Since there are uncountably more real numbers than computable real numbers, you must explain how that works. > 123 456 789 > In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > For the above simple finite list, there are 729 different base 10 > anti-diagonals. If usenet posts had audio you'd hear clapping right now. > Did you remember the antidiagonals for the 6 permutations of the list? It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. > Is any list of all computable reals itself computable? And given any > arbitrary infinite list of computable reals, is the diagonal for that > list computable? yes computable things are computable There are things that one can speak of which are not, like the set of all Dedekind cuts of the rationals or the set of all equivalence classes of Cauchy sequences mod the null sequences. Either of which makes the set of reals to computable and ssome members of that set also not computable. > yes there is no contradiction in computing the diagonal somewhere in the list Until Herk the Jerk can definitively answer those, one way or the > other, there are holes in his arguemnts. > The computable reals contains every possible sequence up to infinite > length, and you tell me with a cloak and digger trick that you can > find new sequences. > I don't claim that any such new sequence is necessarily computable, > merely that for any list it exists. > You can't. All the sequences are computed, up to oo length, end of > story. There can, and must be be uncomputable sequences, at least unless there are uncountably many computable sequences. > There is nothing in Cantor's proof that requires the 'anti-diagonal to > be computable, merely existing in that non-physical way that numbers > have of existing. Numbers exist because they can be computed. > If they couldn't be computed > you wouldn't know what they were. That's what uncomputable numbers are like, you don't know exactly what they are. > Have you taken a look at http://www.freewebs.com/namesort/linux.html > It proves that a list of reals that contains every digit oo times in > each column can be sorted to have any random diagonal. > It does not prove anything at all about such lists, since it only works > with finite lists of finite lengths. Oh you had a look. Claim: Let L be a list of numbers in [0,1] such that for each d in > {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal > representation of L_i is d} is infinite. Let D be a random variable > uniform on [0,1]. Then almost surely there is a permutation L' of L > such that D is the diagonal of L'. Almost surely in any infinite set allows for infinitely many exceptions. Almost surely in an uncountably infinite set allows uncountably many exceptions. > > The diagonal is independent of such lists > > That's another problem with Cantor's > proof, the diagonal does not even depend on the list, it's just a > random variable. > Nonsense. Did you randomise the diagonal at the website, and produce an equivalent set > in the second list with that new diagonal? What does that mean? That each list has its own private diagonal(s) which need not work for any other list. Cantor's original 'diagonal' proof applies only to lists of infinite > binary sequences with values in {m,w}. > So anything not valid for that original proof is irrelevant. Anything valid for an equivalent proof is relevant. This brings up an important point. Virgil, like most of you, believe numbers > and mathematical things exist without being computable? I believe that when working with a set of axioms, whatever the axioms require to exist does exist, at least within that axiom system. Tell me if you > subtracted the theories of mathematics that are computable from the total > set of theories in mathematics that you believe in, what are you left with? Most of mathematics, except for, say, finite groups. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > As always, I believe that there can be rigorous theories in > which the claims made by so-called cranks such as Herc > can hold. Not just so-called. > There are two ways to make card(N) = card(R) -- > either by making N bigger or by making R smaller. Since all Peano systems are isomorphic with one another, and since all complete ordered fields are isomorphic with one another, what you're asking for is at least a pretty tall order. Anyway, cranks aren't interested in rigorous theories. MoeBlee Perhaps, in the spirit of this thread, we could make R smaller > by insisting that only computable numbers exist. Then there > would be no Omega or any other numbers missing from the > list, and so Herc's claim that there exist only countably many > reals can still hold. > of === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au Since all Peano systems are isomorphic with one another ... Not sure what you mean here. By the upward LS theorem, there are models of PA of every infinite cardinality. -- hz === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <4893EDC1.4D1756EF@gmail.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Since all Peano systems are isomorphic with one another ... Not sure what you mean here. æBy the upward LS theorem, there are > models of PA of every infinite cardinality. I didn't say all models of PA are isomorphic. I said all Peano systems (also known as 'Dedekind systems') are isomorphic. Df: P is a Peano system <-> ESfz(P = & zeS & f:S->S~{z} & f is 1-1 & Ax((x subset of S & zex & x closed under f) -> x=S)) Th: All Peano systems are isomporhic. Th: Between any two Peano systems there exists exactly one isomorphism. MoeBlee === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au Since all Peano systems are isomorphic with one another ... > Not sure what you mean here. By the upward LS theorem, there are > models of PA of every infinite cardinality. I didn't say all models of PA are isomorphic. I said all Peano systems > (also known as 'Dedekind systems') are isomorphic. Df: > P is a Peano system <- ESfz(P = & > zeS & > f:S->S~{z} & > f is 1-1 & > Ax((x subset of S & zex & x closed under f) -> x=S)) Th: All Peano systems are isomporhic. Th: Between any two Peano systems there exists exactly one > isomorphism. -- hz === Subject: Re: the problem with Cantor > There are two ways to make card(N) = card(R) -- > either by making N bigger or by making R smaller. > [lwal..., designated crank] > Does this asshole have a brain? If we either make N bigger or R smaller, the resulting set(s) would not be N and/or R anymore. *sigh* (Clearly there's a REASON why we write R* and N* in non-standard analysis instead of N and R.) Hence there's NO WAY to make card(N) = card(R) IF N denotes the set of natural numbers, R the set of real numbers, Card cardinality (defined in the spirit of Cantor and Frege) and = identity (in the way WE -i.e. non-cranks- know it). Right, there are two way to make 1 = 2 -- either by making 1 bigger or by making 2 smaller. (Actually, there's a third way: by redefining = as =/=.) Why oh why are mathematical cranks just so one-dimensional? [Consider] the potentially infinite set N. If we add another element, the set remains the same. (W. Mueckenheim) B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <7eh694plfn9etuavv9gqv1mpepq0ukkd01@4ax.com> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > If we either make N bigger or R > smaller, the resulting set(s) would not be N and/or R anymore. > *sigh* (Clearly there's a REASON why we write R* and N* in > non-standard analysis instead of N and R.) Hence there's NO WAY to > make card(N) = card(R) IF N denotes the set of natural numbers, R > the set of real numbers, Card cardinality (defined in the spirit of > Cantor and Frege) and = identity (in the way WE -i.e. non-cranks- know > it). > æ æ æ æ Right, there are two way to make 1 = 2 -- > æ æ æ æ either by making 1 bigger or by making 2 smaller. > æ æ æ æ (Actually, there's a third way: by redefining > æ æ æ æ æ= as =/=.) In this post, Balthasar clearly compares the formulae card(N) = card(R) and 1 = 2. We know that the the perspective of Balthasar -- and most standard mathematicians -- any poster who claims that card(N) = card(R) might as well be claiming that 1 = 2 -- either claim merits the crank label. But I disagree and feel that there are worlds of difference between card(N) = card(R) and 1 = 2, even though (card(N) = card(R)) <-> (1 = 2) is a theorem of ZFC. Formulae involving infinite sets are open to various interpretations, such as classical, constructivism, intuitionism -- whereas those involving finite sets are not subject to these types of interpretations. Thus I consider ~(1 = 2) to be absolutely true in a way that the formula ~(card(N) = card(R)) isn't -- since the former holds for both classical and constructivist mathematicians, unlike the latter where there is some debate as to what N and R actually are . Herc has mentioned computable reals in this thread, as if he were a sort of constructivist who only believes in the existence of computable reals. To a constructivist, Omega simply isn't a real number -- it doesn't even exist . Omega is thus not an element of R. To someone who only philosophically believes in the existence of computable reals, there exist only countably many reals -- and therefore there's a bijection between N and R -- found by enumerating every Turing algorithm that computes a real. Perhaps, just as Balthasar has intimated with his example of *N and *R, a constructivist shouldn't use the symbol R to denote the set of real numbers that he believes exists, but to use some other notation such as Rc -- so that Herc's claim becomes card(N) = card(Rc). Then the symbol R would be reserved for the full, classical set R -- anyone who doesn't believe in the existence of the classical set R would have no right to use the symbol R in a formula. Maybe it's too bad that this isn't the case (cf. the subthread of the WM thread in which Balthasar, among others, discussed pedantry vs. abuse of notation re: definition of card). > Why oh why are mathematical cranks just so one-dimensional? When I first started posting at sci.math about a year ago, I came in fully prepared that someone would call me a crank. By the standard mathematicians' definition of crank, I am indeed a borderline crank, simply because I do not state that Herc and other posters like him are 100% wrong. I may not like the label, but I cannot deny that I satisfy their definition of the term. And I'd rather be labeled a borderline crank than blindly accept ZFC and classical mathematics as the only theories worth discussing and call anyone who posts anything that contradicts ZFC or classical analysis wrong. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > If we either make N bigger or R > smaller, the resulting set(s) would not be N and/or R anymore. > *sigh* (Clearly there's a REASON why we write R* and N* in > non-standard analysis instead of N and R.) Hence there's NO WAY to > make card(N) = card(R) IF N denotes the set of natural numbers, R > the set of real numbers, Card cardinality (defined in the spirit of > Cantor and Frege) and = identity (in the way WE -i.e. non-cranks- know > it). > æ æ æ æ Right, there are two way to make 1 = 2 -- > æ æ æ æ either by making 1 bigger or by making 2 smaller. > æ æ æ æ (Actually, there's a third way: by redefining > æ æ æ æ æ= as =/=.) In this post, Balthasar clearly compares the formulae > card(N) = card(R) and 1 = 2. We know that the > the perspective of Balthasar -- and most standard > mathematicians -- any poster who claims that > card(N) = card(R) might as well be claiming that > 1 = 2 -- either claim merits the crank label. But I disagree and feel that there are worlds of > difference between card(N) = card(R) and 1 = 2, > even though (card(N) = card(R)) <-> (1 = 2) is a > theorem of ZFC. Formulae involving infinite sets are > open to various interpretations, such as classical, > constructivism, intuitionism -- whereas those > involving finite sets are not subject to these > types of interpretations. Thus I consider ~(1 = 2) > to be absolutely true in a way that the formula > ~(card(N) = card(R)) isn't -- since the former > holds for both classical and constructivist > mathematicians, unlike the latter where there is > some debate as to what N and R actually are . The distinction between finitistic and infinitistic statements is well understood by many mathematicians who still work in a classical framework. > Herc has mentioned computable reals in this thread, > as if he were a sort of constructivist who only > believes in the existence of computable reals. You checked with Herc on that, right? > To a constructivist, Omega simply isn't a real > number -- it doesn't even exist . That's fine, but does not refute Baltasar's point. IF N and R are taken to exist in the sense of the set of natural numbers and the set of real numbers, then they don't have the same cardinality. > Omega is thus > not an element of R. Who said it is? > To someone who only > philosophically believes in the existence of > computable reals, there exist only countably > many reals -- and therefore there's a bijection > between N and R -- found by enumerating every > Turing algorithm that computes a real. Then that is just taking 'R' in a DIFFERENT sense. > Perhaps, just as Balthasar has intimated with > his example of *N and *R, a constructivist > shouldn't use the symbol R to denote the set > of real numbers that he believes exists, but to > use some other notation such as Rc -- so that > Herc's claim becomes card(N) = card(Rc). Then > the symbol R would be reserved for the full, > classical set R -- anyone who doesn't believe > in the existence of the classical set R would > have no right to use the symbol R in a formula. It's not a question of rights. It's just a matter of being clear. > And I'd rather be labeled a borderline crank > than blindly accept ZFC and classical mathematics > as the only theories worth discussing And who blindly accepts ZFC and classical mathematics as the only theories worth discussing? You keep referring to such people. And I keep challenging you on it. PLEASE, already, say who these people are. > and call > anyone who posts anything that contradicts ZFC > or classical analysis wrong. Who does that? What are wrong are certain (and many) incorrect claims and misconceptions about ZFC. MoeBlee === Subject: Re: the problem with Cantor > To a constructivist [...]. > That's fine, but does not refute Baltasar's point. IF N and R are > taken to exist in the sense of the set of natural numbers and the set > of real numbers, then they don't have the same cardinality. > Right, there are two ways to make 1 = 2 -- either by making 1 bigger or by making 2 smaller. (Actually, there's a third way: by redefining = to mean =/=.) Since this mirrors exactly one of the statements of this guy. SURE: IF wishes were horses, THEN beggars would ride. (But since...) > Omega is thus _not_ an element of R. > Who said it is? > Well at least I would dare to claim that (Chaitin's) Omega (Omega_U) is a real number. ;-) Omega_U is perhaps the most obvious specific example of an uncomputable number. It is also known to be a transcendental number. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ http://mathworld.wolfram.com/ChaitinsConstant.html B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=UJeUTgkAAADYai-ULU41ORCvNnkXmdRu Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) |But I disagree and feel that there are worlds of |difference between card(N) = card(R) and 1 = 2, |even though (card(N) = card(R)) <-> (1 = 2) is a |theorem of ZFC. Formulae involving infinite sets are |open to various interpretations, such as classical,constructivism, intuitionism -- whereas those |involving finite sets are not subject to these |types of interpretations. They're subject to reinterpretation if you work at it. ;-) |Thus I consider ~(1 = 2) |to be absolutely true in a way that the formula |~(card(N) = card(R)) isn't -- since the former |holds for both classical and constructivist |mathematicians, unlike the latter where there is |some debate as to what N and R actually _are_. I can't think offhand of any seriously intended theory or interpretation that has a continuum R, and doesn't render ~(card(N)=card(R)) valid. |Herc has mentioned computable reals in this thread, |as if he were a sort of constructivist who only |believes in the existence of computable reals. | |To a constructivist, Omega simply _isn't_ a real |number -- it doesn't even _exist_. Well, be a little careful. Errett Bishop in his famous textbook on constructive analysis has an aside about how many of these classical constructions can be treated constructively as instances of what he calls fickle reals. A sequence of rationals r_i has fickle convergence if for each epsilon>0 there exists an N such that there does not exist a sequence i1epsilon for j=1,...,N-1. You could say, for each epsilon>0 there is an upper bound on how many times the sequence moves by epsilon. Nonconstructively, fickle convergence implies convergence, but it doesn't follow constructively. Fickle convergence contradicts the claim that for each N, there are i,j>N for which |r_i-r_j|>epsilon, but doesn't necessarily provide you with an example of an N for which this doesn't happen. Bishop doesn't claim there (and as far as I know never did claim) that the fickle continuum was worth studying very much. It presumably would go something like the studies in nonconstructive mathematics of computable sequences of rationals that converge but not necessarily at a computable rate of convergence. |Omega is thus |_not_ an element of R. Depending on your school of constructive mathematics, this not might get put into quotation marks, or else you might say that its being an element of R implies a standard nonconstructive principle, that every Turing machine computation either terminates or does not terminate. |To someone who only |philosophically believes in the existence of |computable reals, there exist only countably |many reals -- and therefore there's a bijection |between N and R -- found by enumerating every |Turing algorithm that computes a real. No, this is incorrect. Cantor's first proof that there's no surjection from N to R (and hence also no bijection) is perfectly constructive. His second proof suffers only from a technical weakness, in that it assumes each real has a decimal expansion, although if we are given a real, we cannot necessarily compute the decimal expansion. There have been serious mathematicians who apparently were willing to assume that every real is computable, and not just as a quaint hypothesis. Beeson describes the Moscow school of constructive mathematics under Markov as having accepted the computability of every sequence of natural numbers as an axiom, which implies that every real is computable too. The same hypothesis implies however that there's no way of enumerating just the Turing machines that compute reals. The Turing machines can be numbered, and there's a subset of them that compute reals, but there's no computable enumeration of the ones that do. |Perhaps, just as Balthasar has intimated with |his example of *N and *R, a constructivist |shouldn't use the symbol R to denote the set |of real numbers that he believes exists, but to |use some other notation such as Rc -- so that |Herc's claim becomes card(N) = card(Rc). Then |the symbol R would be reserved for the full, |classical set R -- anyone who doesn't believe |in the existence of the classical set R would |have no right to use the symbol R in a formula. I disagree with this, especially since not all constructivists assume that all reals are computable. I propose that anybody who uses the usual definition of real line is entitled to refer to the set satisfying the definition as the real line R. Perhaps you're thinking specifically of people who does assume that the reals are all computable. If a person who doesn't assume that the reals are all computable is working with the set of computable reals, then it's often appropriate for them to use a distinctive notation like R_c. In that case, they really aren't referring to R when they talk about R_c. But to someone who actually believes that R = R_c, talking about R and talking about R_c are just the same thing. It's only appropriate for them to make note of the fact that they're assuming that R = R_c (since this is an unusual assumption) but to ask them to use special notation throughout their work is akin to asking them to write as though they were wrong about what R is like, and actually talking about something else. It's a little like in theological discussion when people try to argue that the God of some people is not the same as the God of others, when they all believe in a supreme being but have different beliefs about what God is like. I would say a change in notation is even less appropriate a suggestion in the case of those constructivists who do not assume that all reals are computable. Logicians have been known to cook up exotic theories of the reals, but generally speaking the theories of the reals all share a common core of constructive principles. The so-called Bishop school more or less works with this shared core of principles, treating all the stuff that doesn't follow from it in a sort of agnostic manner, like you might talk about statements you know are independent of your favorite set theory. People working this way are surely entitled to describe their results as being about the real line, since not only are they valid results about the real line, they don't require any assumptions that aren't normally made, so they're acceptable to as large a group as can reasonably be expected. Those results include in particular ~(card(N)=card(R)). talking about when they talk about the real line are those things that are reals, if the law of excluded middle and the axiom of choice are true. It would be tempting to ask mathematicians to make note of the fact that they're working with this expanded version of the real line, that includes Chaitin's Omega and so on. That battle was lost a long time ago, however. Keith Ramsay === Subject: Re: the problem with Cantor > If we either make N bigger or R > smaller, the resulting set(s) would not be N and/or R anymore. > *sigh* (Clearly there's a REASON why we write R* and N* in > non-standard analysis instead of N and R.) Hence there's NO WAY to > make card(N) = card(R) IF N denotes the set of natural numbers, R > the set of real numbers, Card cardinality (defined in the spirit of > Cantor and Frege) and = identity (in the way WE -i.e. non-cranks- know > it). > æ æ æ æ Right, there are two way to make 1 = 2 -- > æ æ æ æ either by making 1 bigger or by making 2 smaller. > æ æ æ æ (Actually, there's a third way: by redefining > æ æ æ æ æ= as =/=.) In this post, Balthasar clearly compares the formulae > card(N) = card(R) and 1 = 2. We know that the > the perspective of Balthasar -- and most standard > mathematicians -- any poster who claims that > card(N) = card(R) might as well be claiming that > 1 = 2 -- either claim merits the crank label. But I disagree and feel that there are worlds of > difference between card(N) = card(R) and 1 = 2, > even though (card(N) = card(R)) <-> (1 = 2) is a > theorem of ZFC. Formulae involving infinite sets are > open to various interpretations, such as classical, > constructivism, intuitionism -- whereas those > involving finite sets are not subject to these > types of interpretations. Thus I consider ~(1 = 2) > to be absolutely true in a way that the formula > ~(card(N) = card(R)) isn't -- since the former > holds for both classical and constructivist > mathematicians, unlike the latter where there is > some debate as to what N and R actually _are_. Herc has mentioned computable reals in this thread, > as if he were a sort of constructivist who only > believes in the existence of computable reals. To a constructivist, Omega simply _isn't_ a real > number -- it doesn't even _exist_. Omega is thus > _not_ an element of R. To someone who only > philosophically believes in the existence of > computable reals, there exist only countably > many reals -- and therefore there's a bijection > between N and R -- found by enumerating every > Turing algorithm that computes a real. I am of the opinion that there is more than one Turing algorithm to compute any computable real, for some computable reals there are very likely countably many such algorithms. So that constructability of such a bijection is, at best, problematical. Perhaps, just as Balthasar has intimated with > his example of *N and *R, a constructivist > shouldn't use the symbol R to denote the set > of real numbers that he believes exists, but to > use some other notation such as Rc -- so that > Herc's claim becomes card(N) = card(Rc). Then > the symbol R would be reserved for the full, > classical set R -- anyone who doesn't believe > in the existence of the classical set R would > have no right to use the symbol R in a formula. Maybe it's too bad that this isn't the case (cf. > the subthread of the WM thread in which > Balthasar, among others, discussed pedantry vs. > abuse of notation re: definition of card). Why oh why are mathematical cranks just so one-dimensional? When I first started posting at sci.math about a > year ago, I came in fully prepared that someone > would call me a crank. By the standard > mathematicians' definition of crank, I am > indeed a borderline crank, simply because I do > not state that Herc and other posters like him > are 100% wrong. I may not like the label, but I > cannot deny that I satisfy their definition of > the term. And I'd rather be labeled a borderline crank > than blindly accept ZFC and classical mathematics > as the only theories worth discussing and call > anyone who posts anything that contradicts ZFC > or classical analysis wrong. What I object to is WM, and others, telling me that I am not allowed to accept ZFC as a theory worth discussing. I have no objections to someone preferring not to accept ZFC or whatever, but I do object to their attempting to control what I am allowed to think about. Particularly in the case of those like WM who are both mathematically and logically incompetent. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=-eQqtQoAAACZVM-kNEsOn3k7GSvoJoS4 CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent Or intentionally and maliciously dishonest. I'm still not sure about WM, but my gut tells me that he knows his position is untenable, and he gets his jollies out of riling up everyone else. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; FunWebProducts; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. ********************************************************** In the case of WM, I honestly think he embraces all four talents: mathematically and logically incompetent, and also intentionally and maliciously dishonest. I've had a hard time swallowing the fact that he's actually teaching people stuff. A pity. Tonio === Subject: Re: the problem with Cantor What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. Well said! Particularly in the case of those like WM who are both mathematically > and logically incompetent Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. WM has too many published papers supporting his position to be doing it for fun. If those papers were as publicly discredited as they ought to be, it could have serious effects on his job. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; FunWebProducts; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. > Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent > Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** Where has WM publish any of his papers on this stuff? Is there any reachable link where one can acceed to some of them? Thanx Tonio === Subject: Re: the problem with Cantor > > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. > Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent > Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. > WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** Where has WM publish any of his papers on this stuff? Is there any > reachable link where one can acceed to some of them? > Thanx Tonio Try http://arxiv.org/find/math/1/au:+Mueckenheim_W/0/1/0/all/0/1 You might be amused by #8 of the list shown there, in which WM claims to establish that the set of naturals is non-denumerable by reason of there being no bijection between N and itself. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=-eQqtQoAAACZVM-kNEsOn3k7GSvoJoS4 CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** > Where has WM publish any of his papers æon this stuff? Is there any > reachable link where one can acceed to some of them? > Thanx > Tonio Tryhttp://arxiv.org/find/math/1/au:+Mueckenheim W/0/1/0/all/0/1 You might be amused by #8 of the list shown there, in which WM claims to > establish that the set of naturals is non-denumerable by reason of there > being no bijection between N and itself Yeah, but none of those are what I would call mainstream mathematics. You have a crank giving a talk at the meeting of the German Math. Soc. --- happens all the time. (Note: no proceedings.) You have a paper on Physical Constraints of Numbers. I didn't look at it, but he is not alone in feeling unease about this, and the rest is unpublished. He has tenure, I presume, and if he chooses not to subject his ravings to peer review, and his university is happy with his level of output, he is not going to get into trouble. His students give him great reviews. It's clearly not ideal, but there is nothing so blatant that he could lose his job over. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=1U85NgoAAABKAgcWPdpT0VtXQLBqKIly SIMBAR={5671DD94-1EBC-4971-8627-EEDFF9B1090D}; Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; IEMB3; IEMB3),gzip(gfe),gzip(gfe) > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. > Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent > Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. > WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** > Where has WM publish any of his papers æon this stuff? Is there any > reachable link where one can acceed to some of them? > Thanx > Tonio Tryhttp://arxiv.org/find/math/1/au:+Mueckenheim W/0/1/0/all/0/1 You might be amused by #8 of the list shown there, in which WM claims to > establish that the set of naturals is non-denumerable by reason of there > being no bijection between N and itself.- Hide quoted text - - Show quoted text - What is your opinion about it Virgil. Zuhair === Subject: Re: the problem with Cantor > > What I object to is WM, and others, telling me that I am not > allowed to > accept ZFC as a theory worth discussing. I have no objections to > someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. > Well said! > Particularly in the case of those like WM who are both > mathematically > and logically incompetent > Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. > WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** > Where has WM publish any of his papers æon this stuff? Is there any > reachable link where one can acceed to some of them? > Thanx > Tonio > Tryhttp://arxiv.org/find/math/1/au:+Mueckenheim_W/0/1/0/all/0/1 > You might be amused by #8 of the list shown there, in which WM claims to > establish that the set of naturals is non-denumerable by reason of there > being no bijection between N and itself.- Hide quoted text - > - Show quoted text - What is your opinion about it Virgil. Zuhair WM seems to think that a natural number can have a decimal representation with more than finitely many non-zero digits in it. There are such number systems allowing infinitely many non-zero digits to the left of the radix point, such the p-adics, but even in them, the numerals for naturals don't have infinitely many non-zero digits to the left of the radix point. Also, anyone who claims that there cannot be a bijection of a set with itself, as WM does in this paper, is obviously not in touch with any sort of set theoretical reality or simple logic. === Subject: Re: the problem with Cantor There are two ways to make card(N) = card(R) -- > either by making N bigger or by making R smaller. [lwal..., designated crank] Does this asshole have a brain? > And if so - why doesn't he use it? B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <7eh694plfn9etuavv9gqv1mpepq0ukkd01@4ax.com> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Does this asshole have a brain? > And if so - why doesn't he use it? In Balthasar's opinion, if I were to use my brain, I would see that Herc is 100% wrong and that there's no possible interpretation in which Herc could be right. Of course, I'm fully aware that in that in ZFC, Herc's reasoning is invalid. I already know that the negations of Herc's claims are theorems of ZFC. But Herc already has enough people to tell him that he's wrong. Why should I simply repeat what others are telling him? Indeed, I'd claim that for me to keep repeating Herc, you're wrong! over and over again isn't fully using my brain either. I believe that repeating You're wrong seldom wins anyone to one's point of view. I prefer to say I disagree instead -- though I'm not perfect and have probably called several people wrong -- both standard and crank -- during my first year of posting here. When Herc makes a claim that N and R have the same cardinality, I point out that his claim doesn't hold in ZFC, but perhaps there's some other rigorous theory in which his claim could possibly hold. In other words, I'm trying to find the shortest distance between the claim and something that does hold. Indeed, Balthasar has already mentioned *N and *R. The sets *N and *R do have the same cardinality, namely c. So this is a short path from the claim to something that does hold (though I don't believe that Herc had *N and *R in mind, but this may be similar to WM's ideas, since WM has mentioned Robinson in his posts). Since Herc mentions computable reals, we see how there are only countably many reals that are computable. So I use my brain and realize that to someone who only believes in the existence of computable reals, there are exactly as many (computable) reals as there are naturals. This may not be exactly what Herc has in mind, but it's the shortest distance between his claim that fails in ZFC and something that does hold. It's the difference between insulting someone and constructive criticism. Many standard mathematicians prefer the former -- since it is easier -- while I strive for the latter. I believe that it's better to find a theory in which Herc's claims hold rather than simply repeat how he's wrong. And I have found such a theory -- the theory of computable reals. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <7eh694plfn9etuavv9gqv1mpepq0ukkd01@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Does this asshole have a brain? > And if so - why doesn't he use it? In Balthasar's opinion, if I were to use my > brain, I would see that Herc is 100% wrong > and that there's no possible interpretation > in which Herc could be right. Of course, I'm fully aware that in that in > ZFC, Herc's reasoning is invalid. I already > know that the negations of Herc's claims are > theorems of ZFC. But Herc already has enough people to tell > him that he's wrong. Why should I simply > repeat what others are telling him? Indeed, > I'd claim that for me to keep repeating Herc, > you're wrong! over and over again isn't > fully using my brain either. I believe that repeating You're wrong > seldom wins anyone to one's point of view. I > prefer to say I disagree instead -- though > I'm not perfect and have probably called > several people wrong -- both standard and > crank -- during my first year of posting here. When Herc makes a claim that N and R have the > same cardinality, I point out that his claim > doesn't hold in ZFC, but perhaps there's some > other rigorous theory in which his claim could > possibly hold. Herc doesn't care about rigorous theories; he has barely a notion of a rigorous thoery. Moreover, it's possible that all kinds of statements are theorems of various rigorous theories. So what? If there is substance here with regard to a rigorous theory, then we'd like to know at least SOMETHING about such a theory and about what it can prove in mathematics. > In other words, I'm trying to > find the shortest distance between the claim > and something that does hold. No, you're not. Rather, you're trying to make yourself out to be some kind of spokesman for fairness and open-mindedness. If what you're interested in is a rigorous theory, then, please, by all means, give us at least a sketch or even a glimmer of a notion of a particular one. > Indeed, Balthasar has already mentioned *N > and *R. The sets *N and *R do have the same > cardinality, namely c. So this is a short > path from the claim to something that does > hold (though I don't believe that Herc > had *N and *R in mind, but this may be > similar to WM's ideas, since WM has mentioned > Robinson in his posts). Just about anyone somewhat informed about these matters understands that if 'N' and 'R' refer to something other than the set of natural numbers and the set of real numbers, then we can say different things about N and R. > Since Herc mentions computable reals, we see > how there are only countably many reals that > are computable. So I use my brain and realize > that to someone who only believes in the > existence of computable reals, there are > exactly as many (computable) reals as there > are naturals. This is well known. > This may not be exactly what Herc has in > mind, but it's the shortest distance between > his claim that fails in ZFC and something that > does hold. So what? > It's the difference between insulting someone > and constructive criticism. Do you have any IDEA how many thousands and thousands of words have been typed in wasted effort trying to communicate with Herc? > Many standard > mathematicians prefer the former Well, usually, after all other efforts have failed. > -- since it > is easier No, since it is all that is LEFT after years and years of trying to reason with someone impervious to reason. > -- while I strive for the latter. I > believe that it's better to find a theory in > which Herc's claims hold rather than simply > repeat how he's wrong. And I have found such > a theory -- the theory of computable reals. WHAT THEORY of computable reals? What SPECIFIC axioms and rules of inference do you have in mind? Anyway, the notion of computable reals is formalizable in classical mathematics and it is a theorem that the set of computable reals is 1-1 with omega. MoeBlee === Subject: Re: the problem with Cantor > When Herc makes a claim that N and R have the > same cardinality, I point out that his claim > doesn't hold in ZFC, but perhaps there's some > other rigorous theory in which his claim could > possibly hold. > [...] it's possible that all kinds of statements > are theorems of various rigorous theories. So what? > Good point! Consider the following theory (formulated in the context of FOPL with identity). (Axiom of difference) 0 =/= 1. Now I can prove the following theorem in this theory: 0 =/= 1. Wow! You see, we NORMALLY would be tempted to claim that it is not the case that 0 =/= 1, but of course there's some other rigorous theory in which his claim could possibly hold - as has just been proved! :-) B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor > Moreover, it's possible that all kinds of statements > are theorems of various rigorous theories. So what? If there is > substance here with regard to a rigorous theory, then we'd like to > know at least SOMETHING about such a theory and about what it can > prove in mathematics. Who in the real world are the ones who would have the right to *define* what substance *must* be? And how would such a definition help or hurt mathematical reasoning in general? > No, you're not. Rather, you're trying to make yourself out to be some > kind of spokesman for fairness and open-mindedness. You sounded as if speaking for fairness and open-mindedness were some kind of a bad thing people should avoid doing? > If what you're > interested in is a rigorous theory, then, please, by all means, give > us at least a sketch or even a glimmer of a notion of a particular > one. Didn't you just say a moment ago So what? to a rigorous theory, why now encouraging one to talk about such a thing? MoeBlee > -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <7eh694plfn9etuavv9gqv1mpepq0ukkd01@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Moreover, it's possible that all kinds of statements > are theorems of various rigorous theories. So what? If there is > substance here with regard to a rigorous theory, then we'd like to > know at least SOMETHING about such a theory and about what it can > prove in mathematics. Who in the real world are the ones who would have the right to *define* > what substance *must* be? It's an informal notion. Anyone is welcome to propose whatever he or she feels is or is not substantive. > And how would such a definition help or hurt > mathematical reasoning in general? It's up to each individual to decide for him or herself whether any given bit of writing has enough substance in it to be of mathematical interest. > No, you're not. Rather, you're trying to make yourself out to be some > kind of spokesman for fairness and open-mindedness. You sounded as if speaking for fairness and open-mindedness were some kind > of a bad thing people should avoid doing? No, I don't say it's a bad thing or that one should avoid doing it. Rather, in the present case, I find that the poster seems more interested in giving the impression that he is doing it (or even feels that he is doing it) than in showing us any formal theories he keeps talking of the possibility of. And also, that in the present case, the poster is misguided as to who is and isn't being fair in certain of these discussions. I consider it quite fair to mention that anyone is free to propose theories and that they can be evaluated publically and individually on whatever basis of evaluation the evaluator mentions, meanwhile being careful to keep straight just what is the case as to questions of what is or is not provable in such theories as ZFC. And I find not so fair or open-minded the continual misinformed and misconceived dogma of cranks that such theories as ZFC are wrong. (And I don't even hold that one may not have reasonable grounds for not preferring ZFC or even thinking that ZFC is fundamentally wrong; except that cranks don't give reasonable grounds, as instead they give dogma.) > If what you're > interested in is a rigorous theory, then, please, by all means, give > us at least a sketch or even a glimmer of a notion of a particular > one. Didn't you just say a moment ago So what? to a rigorous theory, why now > encouraging one to talk about such a thing? I've said so what to the MERE mention that something may be a theorem of some formal theory. That doesn't mean that I am unconcerned about formal theories. MoeBlee === Subject: Re: the problem with Cantor > It's the difference between insulting someone > and constructive criticism. Many standard > mathematicians prefer the former -- since it > is easier -- But please don't mistake 'sci.math regulars' for 'standard mathematicians'. > ... while I strive for the latter. I > believe that it's better to find a theory in > which Herc's claims hold rather than simply > repeat how he's wrong. And I have found such > a theory -- the theory of computable reals. Reactions that suggest that standard mathematics can't make sense of certain ideas may indeed be the main reason why crackpots keep thinking they've something valuable to contribute. Unfortunately, experience learns that referring a crackpot to some relevant parts of respectable mathematics doesn't work. Oddly enough, it also doesn't stop anti-crackpots from continuing their bashing. In short: don't spoil your time; it's hopeless either way. -- Herman Jurjus === Subject: Re: the problem with Cantor > Indeed, Balthasar has already mentioned *N > and *R. The sets *N and *R do have the same > cardinality, namely c. So this is a short > path from the claim to something that does > hold (though I don't believe that Herc > had *N and *R in mind, but this may be > similar to WM's ideas, since WM has mentioned > Robinson in his posts). I think I asked this before: *N and *R are not uniquely defined in Robinson's approach. So what do you mean when you say that *N and *R have the cardinality c? There are certainly non-standard models of PA that are countable. Of course, there are countable models of the reals with their arithmetic properties also, from Lowenheim-Skolem. -- Alan Smaill === Subject: Re: the problem with Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > at best classical constructivism. classical constructivism, funny. And Herc hardly represents any coherent version of constructivism. > despite not accepted @ sci.math not new either. Most mathematicians are not constructivists. But that direct that explanations from a constructivist viewpoint are unwelcome merely for being constructivist. On the contrary, much work in mathematical logic concerns results about the ramifications of constructivism even for classical mathematics. MoeBlee === Subject: Re: the problem with Cantor <2ah49414i6l7cihuf20qabo9ukjlfasppu@4ax.com> posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > In a fully-computable realm, ... Do we have a formal definition of /fully-computable realm/, Moe? Moreover does this idiot ASSUME that the context of our considerations > is a fully-computable realm? - Whatever that may be. Now that the discussion is over and all results are clear, i can take the time for you: Apart from your absolute cluelessness, you are by far the most revolting piece of around. Keep enjoying your conversantion. -LV > ... any formalizable property does express a set [...]. Fascinating. One might think that (at least in a standard realm) > /being not an element of itself/ is a formalizable property: æ æ æ æ x !e x. Still it's hard to see how in a fully-computable realm the Russell-set > can exist, i.e. a set y such that æ æ æ æ Ax(x e y <-> x !e x). Note that this property (in a class theory) does express (determine) > a proper class. (Though in NF, where there are only sets, the predicate > x !e x is not admissible, since it's not stratified.) > It is rather the accepted approach that is problematic in this respect. Crank speak. B. -- For every line of Cantor's list it is true that this line does not > æcontain the diagonal number. æNevertheless the diagonal number may > æbe in the infinite list. (WM, sci.logic) === Subject: first eigenvalue i am still looking for the first eigenvalue of the bilaplacian on the unit ball in R^N with zero Dirichlet boundary conditions. Does anyone know it in terms of known or computable constants? (clearly it is known and is the first zero of some special function or something) craig === Subject: Re: first eigenvalue <21335622.1217629592027.JavaMail.jakarta@nitrogen.mathforum.org>, > i am still looking for the first eigenvalue of the bilaplacian on the unit > ball in R^N with zero Dirichlet boundary conditions. Does anyone know it in terms of known or computable constants? If not does anyone have any suggestion where i look or what i should type in (clearly it is known and is the first zero of some special function or > something) Have you considered posting to sci.math.research? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: old hat? or just not true? No, I think Prof. Edgar's proof was correct. Empirically it's tantalizing. === Subject: ? derivative of max norm posting-account=H-IscAoAAABkDNrURGSxo9jPN3MJ3a8A 1.0.3705; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) The def of max norm of a vector is: max( abs(x_i), i = 1 to N ). But how does one determine its derivative? === Subject: Re: ? derivative of max norm > The def of max norm of a vector is: max( abs(x_i), i = 1 to N ). But how does one determine its derivative? > By using a precise definition of derivative and then thinking about it. Really, how much thought have you put into this? Care to share some ideas you've tried? === Subject: Re: ? derivative of max norm > The def of max norm of a vector is: max( abs(x_i), i = 1 to N ). But how does one determine its derivative? > The partial deriv wrt x_i is 1 if ||x|| = x_i > all other |x_j| -1 if ||x|| = -x_i > all other |x_j| 0 if |x_i| < ||x|| undefined otherwise -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: New Produced Luxury Residence - Classified Preview 2008/11 posting-account=uWkSGAoAAABcv1Qo8K_q913gt5Szmp28 Hotbar 4.5.1.0),gzip(gfe),gzip(gfe) 28 Juli, 05:01 Title: Munifus Landmark Estate » House Manufacturing Worldwide Project Contracting City: Stockholm, London, Moscow, Los Angeles, Washington, New York, Miami, Chicago, Austin, Tokyo, Jordan, Yemen, Oslo, Helsinki, Berlin, Vienna, Bejing, Istanbul, Alger, Gibraltar, Monte Carlo, Paris, Brussels, Amsterdam, Hamburg, Frankfurt, Zurich, Barcelona, Madrid, State: NON-US Description: Munifus Landmark Estate » represents a building-contractor located in many of the most beautiful and highly attractive locations around the world. The following represents a detailed description for one selection of Munifus Luxury Property ». We have chosen to present a European-style residence to fit an extraordinary home residence in architectural richness with panoramic views, a lovely design and custom built home. Munifus House » has since it suit its first master safe home seems a perfect family living and grand entertaining.With plans and permits in place Munifus House » has found to the comfortable architectural style and setting on a peaceful inviting skilfully ground with wealth of charming details. Completely in the perfect featuring Munifus Luxury Property » is been aimed highest degree for quality to a superb living. Spacious and luxurious family living has been a typical decorative Munifus House » impression. Information: Prize Indices Analysis, Insurance, Inspection, Finance. Agent: Mr Roger K. Olsson Int call: +46 (0) 705474830 Address: Mr Roger K. Olsson PL 25 40 Kuttainen SE: 980 16 Karesuando Real estate website: http://www.rapidsellers.com/molsson/ Prospect: http://publishing.yudu.com/Freedom/Ak8i8/DevelopmentProject/ === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 1, 7:16æam, David Formosa (aka ? the Platypus) [...] > | [...] > | > | > existence exists : æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ tautology > | > nonexistence is nonexistent : æ æ æ æ æ æ æ æ æ ætautology > | > indeterminate existence exists indeterminately : tautology > | > | These are not tautologies, these are type errors. [...] > Admittedly, my usage of existence was a bit liberal here. What do you mean by existence? æTypically we use exists to mean has > the property of being within the concrete universe. æBy this > definition abstractions like red, two and existence don't exist as > they are not within the concrete universe. æRather we all we can do is > see examples of objects that have these properties. > However, I > think that the overall picture that is being painted is valid, Could you give me an argument to support your argument that this is a > valid thesis? [...] My very liberal usage of the words (1)existence, (2)nonexistence, and (3) existentially indeterminate, can be thought of or even defined more rigorously as follows: Definition of existence (as I am using it) The sum totality of all things which are said to exist. Definition of nonexistence (as I am using it) The sum totality of all things which are said to be nonexistent. This includes both those things which cannot exist, and those things which could exist but simply do not. I dont think that it is logical to attempt to draw a distinction between those two classes (which are'nt even classes btw). Definition of the existially indeterminate (as I am using it) The sum totality of all things which may or may not exist. I'll post some analysis below to justify this, a worked problem. But to reword things, [1] The sum totality of all things which are said to exist, exists. [2] The sum totality of all things which are said to be nonexistent, clearly do not exist. [3] The sum totality of all things which are existially indeterminate, obviously may or may not exist. That's a bit better I think. One strange implication of [3] is that no matter how much work you do in this area, nothing is provable and the whole thing can simply evaporate. > [1] Existence exists. Somehow, this forms the basis of all known > mathematics and allows logic to function like a classical Newtonian > Clock. No logic I know of depends on this. æMathematics can function quite > well as a formal game that has no existance. > [2] Nonexistence does not exist. There is a paradox here, and the > reason becomes quite obvious. Normally a paradox is an indercator that your axioms have problems. I think that the paradox here is caused because you cannot perform acts of logic (like math) upon the nonexistent. But, because nonexistence is an existential form, if we expect it to have this property of being self referential, then a paradox must result. This also sheds some light on zero and the problems which can be devised, e.g. zero apples is zero oranges etc. > You cannot make statements about the nonexistent. In standard mathmatics one can make statements about the properties of > the empty set, however they all end up being trivally true. Empty set is very much like zero. It creates no problems, but I am pretty sure that there are some things that you are not supposed to do with it to avoid division by zero type issues. What you cant do is calculate the area of a 4-sided triangle. > You cant say that it exists or not, because it does not > exist. One could put that argument as X doesn't exist so therefor X's > existance can't be established, however you have just established > truth value of X's existance in the first part of the sentence. > Nor would it make sense to calculate the area of a 4sided > triangle because it does not exist. 4 sided triangles are logically inconstant, this means that you will > not find any objects that both possess the triangle property and the 4 > sided property. > The presence of paradox implies the reflexivity of the property. Can you please explain this? Some analysis and a worked problem. First, the central motivation for this approach. [1] Length can be graduated by Plancklengths. [2] The precise position of graduations is indeterminate. The graduations can slide back and forth freely. [3] Therefore length is probabilistic, and any length contained in R can exist in space, despite Plancklength. To consider the idea of random length, you would have one of two possible cases : Case1 A random segment is added to a given length, L = |--------------------| + |~ ~ ~| Case2 The randomness is multiplied into the length and the whole thing stretches like a rubberband L' = |~~~~~~~~~~~~~~~~~| For Case2, you must have existential indeterminacy. You must be able to say that points exist with probability a,b,c.... So, thats the motivation. -------------------------------------------------------------------- Examples: Calculate the distance between 2 points (0,0) and (1,1), where the existential potential function is given as z = f(x,y) = 1/2. What this means is that each point in the region in question has a 50:50 probability of existing, and we want the distance from 0,0) to (1,1). Our answer will be an expected length. Ordinarily, the distance is given by SQRT(2)/2. But in this case the expected length is SQRT(2)/4. The math is simple enough, and I could haul out all kinds of integrals but I dont want to muddle it up. The philosophy is the hard part, not the algebra. If you want the area of the unit square where the existential potential function is given as z = f(x,y) = 1/2, the expected area of the unit square is simply 1/2 * 1/2 = 1/4. Again, you could do this all with integrals (which resemble PDFs), but we'll just stick to simple cases for the time being. We can talk about these simple examples, but what I am moving toward is explaining the precession of perihelion of planets. I think that it can be calculated pretty easily with just vector calculus and an appropriate choice of existential potential function for 3 or 4 space. === Subject: Re: New math. Dont read this. > I think that the paradox here is caused because you > cannot perform > acts of logic (like math) upon the nonexistent. Nope, not true. Example: Penrose triangle. > But, > because > nonexistence is an existential form, if we expect it > to have this > property of being self referential, then a paradox > _must_ result. Nope. Self reference is not paradox. It merely limits the number of true statements one can extract from a system. Self reference can lead to paradox (such as Russell's antinomy); that is, however, a result of the inherent limitation. > This also sheds some light on zero and the problems > which can be > devised, e.g. zero apples is zero oranges etc. No problem here. One apple is also equivalent to one orange. As is 23 apples to 23 oranges. And so on. Ordinal identity is independent of object properties. You cannot make statements about the nonexistent. > We do it all the time. Proof by double negation depends on our ability to make statements of that which cannot exist. > In standard mathmatics one can make statements > about the properties of > the empty set, however they all end up being > trivally true. > The term trivial in mathematics (though often overused) refers to conclusions, not assumptions. Zero has a long history of development (See Charles Seife, Zero: The Biography of a Dangerous Idea) and is far from trivial. The empty set is subtler still. Empty set is very much like zero. It creates no > problems, but I am > pretty sure that there are some things that you are > not supposed to do > with it to avoid division by zero type issues. > Such as? > What you cant do is calculate the area of a 4-sided > triangle. > But you can count the legs on a three legged horse. Your point is vacuous. You cant say that it exists or not, because it > does not > exist. > You just DID say that it exists or not. > One could put that argument as X doesn't exist so > therefor X's > existance can't be established, however you have > just established > truth value of X's existance in the first part of > the sentence. > No, you haven't. Tom > Nor would it make sense to calculate the area of > a 4sided > triangle because it does not exist. > 4 sided triangles are logically inconstant, this > means that you will > not find any objects that both possess the triangle > property and the 4 > sided property. > The presence of paradox implies the reflexivity > of the property. > Can you please explain this? > Some analysis and a worked problem. First, the central motivation for this approach. > [1] Length can be graduated by Plancklengths. > [2] The precise position of graduations is > indeterminate. The > graduations can slide back and forth freely. > [3] Therefore length is probabilistic, and _any_ > length contained in R > can exist in space, despite Plancklength. To consider > the idea of > random length, you would have one of two possible > cases : Case1 > A random segment is _added_ to a given length, > L = |--------------------| + |~ ~ ~| Case2 > The randomness is _multiplied_into_ the length and > the whole thing > stretches like a rubberband > L' = |~~~~~~~~~~~~~~~~~| For Case2, you must have existential indeterminacy. > You must be able > to say that points exist with probability a,b,c.... So, thats the motivation. ------------------------------------------------------ > -------------- Examples: Calculate the distance between 2 points (0,0) and > (1,1), where the > existential potential function is given as z = f(x,y) > = 1/2. What this > means is that each point in the region in question > has a 50:50 > probability of existing, and we want the distance > from 0,0) to (1,1). > Our answer will be an expected length. Ordinarily, the distance is given by SQRT(2)/2. But > in this case the > expected length is SQRT(2)/4. The math is simple enough, and I could haul out all > kinds of integrals > but I dont want to muddle it up. The philosophy is > the hard part, not > the algebra. > If you want the area of the unit square where the > existential > potential function is given as z = f(x,y) = 1/2, the > expected area > of the unit square is simply 1/2 * 1/2 = 1/4. Again, > you could do this > all with integrals (which resemble PDFs), but we'll > just stick to > simple cases for the time being. > We can talk about these simple examples, but what I > am moving toward > is explaining the precession of perihelion of > planets. I think that it > can be calculated pretty easily with just vector > calculus and an > appropriate choice of existential potential function > for 3 or 4 > space. > === Subject: Re: New math. Dont read this. [...] > But, > because > nonexistence is an existential form, if we expect it > to have this > property of being self referential, then a paradox > _must_ result. Nope. Self reference is not paradox. It merely > limits the number of true statements one can extract from > a system. Indeed most of the systems that mathmatics deals with (at least thouse powerfull enought to embed natural numbers) can be made into a self referencial system. === Subject: Re: New math. Dont read this. > [...] > But, > because > nonexistence is an existential form, if we expect > it > to have this > property of being self referential, then a > paradox > _must_ result. > Nope. Self reference is not paradox. It merely > limits the number of true statements one can > extract from > a system. Indeed most of the systems that mathmatics deals with > (at least thouse > powerfull enought to embed natural numbers) can be > made into a self > referencial system. > Exactly. Tautology is a useful tool in this context. Tom === Subject: Re: New math. Dont read this. <27358300.1217673383465.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 2, 7:11æpm, David Formosa (aka ? the Platypus) > [...] > But, > because > nonexistence is an existential form, if we expect it > to have this > property of being self referential, then a paradox > must result. > Nope. æSelf reference is not paradox. æIt merely > limits the number of true statements one can extract from > a system. The paradox is not caused by the self reference. It it caused because the self reference is being forced to act upon nonexistence, which is not a valid thing to do from a logical standpoint. The fact that nonexistence is indeed paradoxical, this seems to validate the thesis that it is acting upon itself vis-a-vis self reference. === Subject: Re: New math. Dont read this. > On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > _must_ result. > Nope. Self reference is not paradox. It merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical standpoint. The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. > Do some study into what paradox means. Tom === Subject: Re: New math. Dont read this. <19917341.1217764210902.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > must result. > Nope. æSelf reference is not paradox. æIt merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical standpoint. > The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. Do some study into what paradox means. Tom- Hide quoted text - - Show quoted text - Is an apple an orange, or is an apple not an orange. Perhaps you could give me an answer, either yes or no. Should be a very simple thing to do. Is an apple an orange ? Please check the appropriate box. [ ] yes [ ] no === Subject: Re: New math. Dont read this. > On Aug 3, 6:49 am, T.H. Ray On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > _must_ result. > Nope. Self reference is not paradox. It > merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. > It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical > standpoint. > The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. > Do some study into what paradox means. > Tom- Hide quoted text - > - Show quoted text - > Is an apple an orange, or is an apple not an orange. > Perhaps you could > give me an answer, either yes or no. Should be a very > simple thing to > do. Is an apple an orange ? Please check the appropriate box. [ ] yes > [ ] no > Sorry, your naive conceptions of identity are way off from mathematical modeling or logical tractability. Let's say A and O are equal (i.e., the same)when A=O. Then your zero apples is the same as zero oranges claim is simply trivial, isn't it? No paradox, and nothing even of interest. Tom === Subject: Re: New math. Dont read this. <21685791.1217777047116.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Aug 3, 6:49 am, T.H. Ray On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > must result. > Nope. æSelf reference is not paradox. æIt > merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. > It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical > standpoint. > The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. > Do some study into what paradox means. > Tom- Hide quoted text - > - Show quoted text - > Is an apple an orange, or is an apple not an orange. > Perhaps you could > give me an answer, either yes or no. Should be a very > simple thing to > do. > Is an apple an orange ? > Please check the appropriate box. > [ ] yes > [ ] no Sorry, your naive conceptions of identity are way > off from mathematical modeling or logical tractability. > Let's say A and O are equal (i.e., the same)when A=O. > Then your zero apples is the same as zero oranges > claim is simply trivial, isn't it? æNo paradox, and > nothing even of interest. Tom- Hide quoted text - - Show quoted text - Actually Tom, we need to distinguish between the abstract and the physical. Lets talk math for a second. You could easily refute me by saying that apples are never oranges, because you need to divide by zero in order to yield the statement that apples = oranges. You could easily refute me that way. Even though I would argue that it is in fact the act of multiplying that yields the contradiction. Regardless, that is the abstract case and it is irresolvable in my opinion - and I know you disagree. But lets consider the physical case instead. Imagine your kitchen table. There are no apples, and there are no oranges. You have nothing on the table. So, do you have zero apples on the table ? Yes or No ? That is where the paradox lives. Now, if math were a valid model of reality, you would'nt just invent rules in order to duct tape things together to keep arithmetic from blowing up. Math is supposed to model reality. I dont think that it is valid to invent rules simply because we dont like paradoxes and we want to illegalize them from a particular model. Truth is not a referrendum. === Subject: Re: New math. Dont read this. [...] > Now, if math were a valid model of reality, you would'nt just invent > rules in order to duct tape things together to keep arithmetic from > blowing up. Math is supposed to model reality. Math is supposed to amuse mathimatisions. If it happens to model reality thats a happy coincidence. > I dont think that it is > valid to invent rules simply because we dont like paradoxes and we > want to illegalize them from a particular model. For the nonconstructivest amoung us concluding that a theorm is true from the fact that maths blows up if its false is quite legitlimate reasoning. === Subject: Re: New math. Dont read this. <21685791.1217777047116.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 3, 4:49æpm, David Formosa (aka ? the Platypus) [...] > Now, if math were a valid model of reality, you would'nt just invent > rules in order to duct tape things together to keep arithmetic from > blowing up. Math is supposed to model reality. Math is supposed to amuse mathimatisions. æIf it happens to model > reality thats a happy coincidence. > I dont think that it is > valid to invent rules simply because we dont like paradoxes and we > want to illegalize them from a particular model. For the nonconstructivest amoung us concluding that a theorm is true > from the fact that maths blows up if its false is quite legitlimate > reasoning. Try this on. You can prove that a photon is a wave. You can also prove that a If I have a segment of length and I can demonstrate that it is continuous, but I can also demonatrste that it is discrete.....do you really think that reducto ad absurdum is going to help you out here ? I say that if you can prove it to be a wave, and you can prove it to whether it is one or the other. Is an apple an orange ? Or, is it indeterminate whether you have an apple or an orange ? === Subject: Re: New math. Dont read this. > On Aug 3, 4:49 pm, David Formosa (aka ? the > Platypus) > [...] > Now, if math were a valid model of reality, you > would'nt just invent > rules in order to duct tape things together to > keep arithmetic from > blowing up. Math is supposed to model reality. > Math is supposed to amuse mathimatisions. If it > happens to model > reality thats a happy coincidence. > I dont think that it is > valid to invent rules simply because we dont like > paradoxes and we > want to illegalize them from a particular model. > For the nonconstructivest amoung us concluding that > a theorm is true > from the fact that maths blows up if its false is > quite legitlimate > reasoning. > Try this on. You can prove that a photon is a wave. You can also > prove that a > one can measure simultaneously both the position and Nothing mysterious here. > If I have a segment of length and I can demonstrate > that it is > continuous, but I can also demonatrste that it is > discrete.....do you > really think that reducto ad absurdum is going to > help you out here ? > You can only demonstrate that a length is continuous, to arbitrary accuracy. > I say that if you can prove it to be a wave, and you > can prove it to > _indeterminate_ > whether it is one or the other. > Google quantum superposition. > Is an apple an orange ? Or, is it _indeterminate_ > whether you have an > apple or an orange ? No more than it is indeterminate whether you have a Tom === Subject: Re: New math. Dont read this. Actually Tom, we need to distinguish between the > abstract and the > physical. > We need to distinguish between mathematical logic and nonsense. > Lets talk math for a second. That'd be a switch. >You could easily refute > me by saying that > apples are never oranges, because you need to divide > by zero in order > to yield the statement that apples = oranges. Apples could equal _anything_ if one could divide by zero. Did you not understand that? > You > could easily refute > me that way. Even though I would argue that it is in > fact the act of > multiplying that yields the contradiction. Your argument would, however, be wrong. It is in the fact that there is no multiplicative inverse (i.e., division) that division by zero is forbidden. I explained this. > Regardless, that is the > abstract case and it is irresolvable in my opinion - > and I know you > disagree. > I disagree, because it is meaningless. > But lets consider the physical case instead. > Imagine your kitchen table. There are no apples, and > there are no > oranges. You have nothing on the table. So, do you > have zero apples on > the table ? Yes or No ? > I also have zero platypuses and zero dingbats. > That is where the paradox lives. > Argh. Stop doing violence to the word paradox. > Now, if math were a valid model of reality, you > would'nt just invent > rules in order to duct tape things together to keep > arithmetic from > blowing up. Math is supposed to model reality. I dont > think that it is > valid to invent rules simply because we dont like > paradoxes and we > want to illegalize them from a particular model. > Truth is not a > referrendum. > Sigh. The rules are not arbitrary or whimsical. They are imposed for the sake of the system's self consistency. Tom > === Subject: Re: New math. Dont read this. <7186082.1217788891533.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Actually Tom, we need to distinguish between the > abstract and the > physical. We need to distinguish between mathematical logic > and nonsense. That is exactly it Tom, exactly. We do need to distinguish between the sensical and the nonsensical. My thesis is that there might be regarded a grey area in between where it is indeterminate whether one is performing sense or nonsense. And I say might be ragarded because it is indeterminate. I know that what I am saying sounds ridiculous but I dont think that you guys have refuted me, Im just not convinced that Im wrong, Perhaps I am, and I remain open to that but I just dont see any solid reason why I would be refuted at this point. === Subject: Re: New math. Dont read this. > On Aug 3, 1:41 pm, T.H. Ray Actually Tom, we need to distinguish between the > abstract and the > physical. > We need to distinguish between mathematical logic > and nonsense. > That is exactly it Tom, exactly. We do need to distinguish between the sensical and > the nonsensical. My > thesis is that there might be regarded a grey area > in between where > it is indeterminate whether one is performing sense > or nonsense. And I say might be ragarded because it is > indeterminate. I know that what I am saying sounds ridiculous but I > dont think that > you guys have refuted me, Im just not convinced that > Im wrong, Perhaps > I am, and I remain open to that but I just dont see > any solid reason > why I would be refuted at this point. You fail to understand that to be refuted (i.e., falsified), your proposition must provide the means of falsification. To merely make a claim equivalent to God created the world. Refute that. has nothing to do with logic or mathematics. Tom === Subject: Re: New math. Dont read this. <16634840.1217857338102.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > On Aug 3, 1:41 pm, T.H. Ray Actually Tom, we need to distinguish between the > abstract and the > physical. > We need to distinguish between mathematical logic > and nonsense. > That is exactly it Tom, exactly. > We do need to distinguish between the sensical and > the nonsensical. My > thesis is that there might be regarded a grey area > in between where > it is indeterminate whether one is performing sense > or nonsense. > And I say might be ragarded because it is > indeterminate. > I know that what I am saying sounds ridiculous but I > dont think that > you guys have refuted me, Im just not convinced that > Im wrong, Perhaps > I am, and I remain open to that but I just dont see > any solid reason > why I would be refuted at this point. You fail to understand that to be refuted (i.e., > falsified), your proposition must provide the means of > falsification. æTo merely make a claim equivalent to > God created the world. æRefute that. has nothing to > do with logic or mathematics. Tom- Hide quoted text - - Show quoted text - That's the existential dichotomy doin all the talkin'. === Subject: Re: New math. Dont read this. <21685791.1217777047116.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Aug 3, 6:49 am, T.H. Ray On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > must result. > Nope. æSelf reference is not paradox. æIt > merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. > It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical > standpoint. > The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. > Do some study into what paradox means. > Tom- Hide quoted text - > - Show quoted text - > Is an apple an orange, or is an apple not an orange. > Perhaps you could > give me an answer, either yes or no. Should be a very > simple thing to > do. > Is an apple an orange ? > Please check the appropriate box. > [ ] yes > [ ] no Sorry, your naive conceptions of identity are way > off from mathematical modeling or logical tractability. > Let's say A and O are equal (i.e., the same)when A=O. > Then your zero apples is the same as zero oranges > claim is simply trivial, isn't it? æNo paradox, and > nothing even of interest. Tom- Hide quoted text - - Show quoted text - What I meant is that zero apples is indistinguishable from zero oranges. Now either apples are oranges, or they are not oranges. But you still have the case that zero apples cannot be distinguished from zero oranges unless you create special rules. We cannot say that apples are oranges. But we can say that zero apples is zero oranges, and I did not divide by zero. I multiplied. === Subject: Re: New math. Dont read this. What I meant is that zero apples is indistinguishable > from zero > oranges. > So what? > Now either apples are oranges, or they are not > oranges. > Again, so what? The ordinal number of apples or oranges says nothing about the properties of apples and oranges. > But you still have the case that zero apples _cannot_ > be distinguished > from zero oranges unless you create special rules. > Huh? You mean, like describing the properties of apples and oranges? Man, you are confused. > We cannot say that apples are oranges. > We can say anything we want. > But we can say that zero apples is zero oranges, > and I did not > divide by zero. I multiplied. > And the result was, no surprise, zero. ???? Tom === Subject: Re: New math. Dont read this. <27358300.1217673383465.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > I think that the paradox here is caused because you > cannot perform > acts of logic (like math) upon the nonexistent. Nope, not true. æExample: æPenrose triangle. Visual illusion. > But, > because > nonexistence is an existential form, if we expect it > to have this > property of being self referential, then a paradox > must result. Nope. æSelf reference is not paradox. æIt merely > limits the number of true statements one can extract from > a system. æSelf reference can lead to paradox (such > as Russell's antinomy); that is, however, a result of > the inherent limitation. Self reference is indeed paradox if you have a situation where (1) It must be self referential, AND (2) It cannot be self referential. Nonexistence satisfies both (1) and (2). > This also sheds some light on zero and the problems > which can be > devised, e.g. zero apples is zero oranges etc. No problem here. One apple is also equivalent > to one orange. æAs is 23 apples to 23 oranges. æAnd > so on. æOrdinal identity is independent of object > properties. Yikes. I'll never send you to the store for anything..... > You cannot make statements about the nonexistent. We do it all the time. æProof by double negation depends > on our ability to make statements of that which > cannot exist. You are in fact making statements and manipulations on the existent, and then deriving by proof that things dont exist. Proof is a decision making process. You are not really causing logic to act upon nonexistence. You cannot do that anymore than you can divide by zero. > In standard mathmatics one can make statements > about the properties of > the empty set, however they all end up being > trivally true. The term trivial in mathematics (though often > overused) refers to conclusions, not assumptions. > Zero has a long history of development (See Charles > Seife, Zero: The Biography of a Dangerous Idea) and > is far from trivial. æThe empty set is subtler still. > Empty set is very much like zero. It creates no > problems, but I am > pretty sure that there are some things that you are > not supposed to do > with it to avoid division by zero type issues. Such as? AOC > What you cant do is calculate the area of a 4-sided > triangle. But you can count the legs on a three legged horse. > Your point is vacuous. If I said that you cannot dovide by zero - is that also vacuous ? In fact, it is indeterminate whether this whole thesis is vacuous. My point is that indeterminacy does not constitute negation. And so, even though the whole thesis is wholly and thoroughly indeterminate, it is indeed not negated any more than it is validated. It is permanently stuck right where it belongs, in a state of indeterminacy. Neither negated, nor validated. Neither negatable, nor validateable. > You cant say that it exists or not, because it > does not > exist. You just DID say that it exists or not. True. I did say it. I said it because I had to, not because I wanted to. === Subject: Re: New math. Dont read this. >[snip the circular nonsense] > > If I said that you cannot dovide by zero - is that > also vacuous ? > Yes. If you knew the _reason_ that division by zero is a prohibited operation, you would know that the rule is chosen to prevent arithmetic from blowing up; i.e., to allow consistent answers to arithmetic problems. In fact, there is mathematics in which division by zero is allowed. It is not indeterminate. Tom > In fact, it is indeterminate whether this whole > thesis is vacuous. My > point is that indeterminacy does not constitute > negation. And so, even > though the whole thesis is wholly and thoroughly > indeterminate, it is > indeed not negated any more than it is validated. It > is permanently > stuck right where it belongs, in a state of > indeterminacy. Neither > negated, nor validated. Neither negatable, nor > validateable. You cant say that it exists or not, because > it > does not > exist. > You just DID say that it exists or not. > True. I did say it. I said it because I had to, not > because I wanted > to. > === Subject: Re: New math. Dont read this. <10325230.1217691464822.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) >[snip the circular nonsense] > If I said that you cannot dovide by zero - is that > also vacuous ? Yes. If you knew the reason that division by zero is a > prohibited operation, you would know that the rule > is chosen to prevent arithmetic from blowing up; i.e., > to allow consistent answers to arithmetic problems. > In fact, there is mathematics in which division by > zero is allowed. æIt is not indeterminate. Tom And you think that you know the reason ? The real reason ? Perhaps there is a deeper reason than just arithmetic ? Or that would be impossible ? The only place where division by zero is allowed is in QM and I dont find it very gratifying. === Subject: Re: New math. Dont read this. > On Aug 2, 10:37 am, T.H. Ray [snip the circular nonsense] > If I said that you cannot dovide by zero - is > that > also vacuous ? > Yes. > If you knew the _reason_ that division by zero is a > prohibited operation, you would know that the rule > is chosen to prevent arithmetic from blowing up; > i.e., > to allow consistent answers to arithmetic problems. > In fact, there is mathematics in which division by > zero is allowed. It is not indeterminate. > Tom > And you think that you know the reason ? The real > reason ? Perhaps > there is a deeper reason than just arithmetic ? Or > that would be > impossible ? > Yes, Huang, I know the real reason. It is not mystical nor difficult to understand. It is simply that multiplication of any term by zero is zero. Therefore, division being the inverse of multiplication, there is no accommodation for division by zero. 1/0 times 0 does not equal 1. If it did, one could prove anything mathematically and numbers would be useless. There would be no logical structure at all. (In Seife's book I reference earlier, he uses division by zero to prove that Winston Churchill is a carrot.) > The only place where division by zero is allowed is > in QM and I dont > find it very gratifying. You need to come up with an example before anyone can determine what you're trying to say here. Tom === Subject: Re: New math. Dont read this. <31376811.1217764074350.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > The only place where division by zero is allowed is > in QM and I dont > find it very gratifying. You need to come up with an example before anyone can > determine what you're trying to say here. Tom- Hide quoted text - > I already did but the conversation seems to have shifted away from worked examples, and more toward foundational issues. === Subject: Re: New math. Dont read this. > On Aug 1, 7:16 am, David Formosa (aka ? the Platypus) [...] > However, I > think that the overall picture that is being painted is valid, > Could you give me an argument to support your argument that this is a > valid thesis? > [...] > My very liberal usage of the words (1)existence, (2)nonexistence, and > (3) existentially indeterminate, can be thought of or even defined > more rigorously as follows: Definition of existence (as I am using it) > The sum totality of all things which are said to exist. By sum totality you mean something equilverlent to class in the mathmatical sence of the word, if not what is the diffrence? I'm also wondering how you define exist do abstractions exist? You state that sum totalities can exist, what does this mean for one to exist? > Definition of nonexistence (as I am using it) > The sum totality of all things which are said to be nonexistent. This > includes both those things which cannot exist, and those things > which could exist but simply do not. I dont think that it is logical > to attempt to draw a distinction between those two classes (which > are'nt even classes btw). If something is not included in the sum totality of things that do not exist does that mean it exists? Can a sum totality that does not exist include members? Why are they not considered classes? > Definition of the existially indeterminate (as I am using it) > The sum totality of all things which may or may not exist. Given that (a / not(a)) is a tautology everything is existially indeterminate. > I'll post some analysis below to justify this, a worked problem. But to reword things, > [1] The sum totality of all things which are said to exist, exists. Ok this implies that abstractions do exist. > [2] The sum totality of all things which are said to be nonexistent, > clearly do not exist. Its not at all clear to me. Not clearly knowing what you mean by sum totality nor what you mean by a sum totality existing or not existing. It is my view that using the verb exist to discuss abstract objects is an example of fallacy of reifiction and doesn't lead to usefull resoning. > [3] The sum totality of all things which are existially indeterminate, > obviously may or may not exist. Again it is not obvious to mean. [...] [...] > Normally a paradox is an indercator that your axioms have problems. > I think that the paradox here is caused because you cannot perform > acts of logic (like math) upon the nonexistent. Logic and mathics has very strong tools to deal with nonexistent objects, I would argue that mathatics only deals with abstractions and that abstractions don't have any concrete existence[1]. [...] > You cannot make statements about the nonexistent. > In standard mathmatics one can make statements about the properties of > the empty set, however they all end up being trivally true. Empty set is very much like zero. It creates no problems, but I am > pretty sure that there are some things that you are not supposed to do > with it to avoid division by zero type issues. What you cant do is calculate the area of a 4-sided triangle. You can't calculate the area of a 4-sided triangle because the sentence is meaningless. Likewise the sentence The colour of the quadratic equation is also meaningless. [...] > The presence of paradox implies the reflexivity of the property. > Can you please explain this? > Some analysis and a worked problem. First, the central motivation for this approach. > [1] Length can be graduated by Plancklengths. The length of a physical object may be graduated by plancklengths. However length itself is an abstract quality that has no graduations. > [2] The precise position of graduations is indeterminate. The > graduations can slide back and forth freely. > [3] Therefore length is probabilistic, and _any_ length contained in R > can exist in space, despite Plancklength. To consider the idea of > random length, you would have one of two possible cases : That doesn't answer my question, how does a paradox show that a property is reflexive? === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 2, 4:55æam, David Formosa (aka ? the Platypus) > On Aug 1, 7:16 am, David Formosa (aka ? the Platypus) [...] > However, I > think that the overall picture that is being painted is valid, > Could you give me an argument to support your argument that this is a > valid thesis? > [...] > My very liberal usage of the words (1)existence, (2)nonexistence, and > (3) existentially indeterminate, can be thought of or even defined > more rigorously as follows: > Definition of existence (as I am using it) > The sum totality of all things which are said to exist. By sum totality you mean something equilverlent to class in the > mathmatical sence of the word, if not what is the diffrence? I'm also wondering how you define exist do abstractions exist? æ You state that sum totalities can exist, what does this mean for one > to exist? Well, math is a logical model. An abstraction. And so the usage of the concept of existence will always have two very distinct usages. One is abstract, and one is physical. I think that my usage applies to both the physical and the abstract, and so I usually do not bother to draw the distinction. The model is so close to reality that observing this technicality becomes cumbersome, but the distinction will always be there. As far as physical existence of abstractions goes, I have no idea, that happens in the mind somehow and Im not really concerned with whether an abstract mathematical object has some aspect which makes it exist physically... completely different issue and Im not really concerned with it. For your last question, all I can do is explain what I'm seeing. I believe that physical existence is bounded by extreme scales. Everything which exists relative to us is within about +/- 40 orders of magnitude. The boundaries are where you find existential indeterminacy. This is the precise place where math and physics must part and go their separate ways, because mathematics has no Plancklength. To have such a thing as a Plancklength in mathematics, you have to construct it deliberately. So, if the universe were an Oreo cookie, the crispy chocolate outer cracker would be existentially indeterminate, and existence would be defined as the sugary cream filling. Yum. > Definition of nonexistence (as I am using it) > The sum totality of all things which are said to be nonexistent. This > includes both those things which cannot exist, and those things > which could exist but simply do not. I dont think that it is logical > to attempt to draw a distinction between those two classes (which > are'nt even classes btw). If something is not included in the sum totality of things that do not > exist does that mean it exists? Can a sum totality that does not exist include members? Why are they not considered classes? (i) You cannot consider them to be distinct classes because that would constitute performing a logical or mathematical procedure on the nonexistent. If you consider all of the nonexistent cubes, and you have that some are red and some are blue, you cannot divide them into separate classes according to color because that is a mathematical act. That would be the equivalent of dividing by zero. > Definition of the existially indeterminate (as I am using it) > The sum totality of all things which may or may not exist. Given that (a / not(a)) is a tautology everything is existially > indeterminate. > I'll post some analysis below to justify this, a worked problem. > But to reword things, > [1] The sum totality of all things which are said to exist, exists. Ok this implies that abstractions do exist. It might, but I really dont intend for that to be the case. Because the brain is made of molecules, it is very likely that the abstract and the physical do intersect somehow, but I am not really interested in trying to say that abstractions exist physically. > [2] The sum totality of all things which are said to be nonexistent, > clearly do not exist. Its not at all clear to me. æNot clearly knowing what you mean by sum > totality nor what you mean by a sum totality existing or not > existing. æIt is my view that using the verb exist to discuss abstract > objects is an example of fallacy of reifiction and doesn't lead to > usefull resoning. Well, this definition is a bit paradoxical for the same reason as (i) above. There is no sum totality of things nonexistent, for the reason explained in (i). However, it is possible for beings such as ourselves to contemplate the nonexistent, or to determine that something does not exist (real or abstract). This creates a bit of an illusion that there is some collection of objects which are nonexistent, but clearly there is no such collection or set, it is impossible as explained in (i). One cannot attempt to define nonexistence without creating a fallacy of reification. That is because you are trying to define a paradox. Nonexistence is self referential, just like the other existential forms. As an existent being, I have the ability to enumerate many things which do not exist. I could create a very long list which becomes infinitely long. This creates the illusion that there is such a thing as a sum totality of things nonexistent when in fact there is none. You are correct, I did commit the fallacy of reification, but there is no way to avoid it and here is why : You cannot define nonexistence. To define it it to perform a logical act upon it, and it is immune to logic. My fallacy was in attempting to define it in the first place. But this does not alter the thesis that nonexistence is self-referential, and that this property of being self referential is consistent among all of these supposed existential forms. > [3] The sum totality of all things which are existially indeterminate, > obviously may or may not exist. Again it is not obvious to mean. [...] [...] > Normally a paradox is an indercator that your axioms have problems. > I think that the paradox here is caused because you cannot perform > acts of logic (like math) upon the nonexistent. Logic and mathics has very strong tools to deal with nonexistent > objects, I would argue that mathatics only deals with abstractions > and that abstractions don't have any concrete existence[1]. I would agree completely, with the caveat that there may or may not be some kind of intersection of these worlds in the mind (which is not understood). But it seems that what you said would also be true even if there was no such thing as human beings. So, I dont think that the mind is neccesarily requisite to any of this either. > You cannot make statements about the nonexistent. > In standard mathmatics one can make statements about the properties of > the empty set, however they all end up being trivally true. > Empty set is very much like zero. It creates no problems, but I am > pretty sure that there are some things that you are not supposed to do > with it to avoid division by zero type issues. > What you cant do is calculate the area of a 4-sided triangle. You can't calculate the area of a 4-sided triangle because the > sentence is meaningless. æLikewise the sentence The colour > of the quadratic equation is also meaningless. That is true. I just use that as a prop. > The presence of paradox implies the reflexivity of the property. > Can you please explain this? > Some analysis and a worked problem. > First, the central motivation for this approach. > [1] Length can be graduated by Plancklengths. The length of a physical object may be graduated by plancklengths. > However length itself is an abstract quality that has no graduations. Length can be physical or abstract. I would argue that physical length is a tangible thing which is composed of dimension. Mathematical length is merely a model of the physical variety. > [2] The precise position of graduations is indeterminate. The > graduations can slide back and forth freely. > [3] Therefore length is probabilistic, and any length contained in R > can exist in space, despite Plancklength. To consider the idea of > random length, you would have one of two possible cases : That doesn't answer my question, how does a paradox show that a > property is reflexive? Lets assume that all existential forms are self referential. Then you have that nonexistence is nonexistent. We just performed (or attempted to perform) a logical act upon the nonexistent by saying that it does not exist, which is not valid logically. So, the assumption that all existential forms are self referential points to this situation with the particular case of nonexistence where you have paradox. The usage of existential indeterminacy allows us to do quantum mechanics without saying that things pop in and out of existence - which is nonsense. You can also create an analysis based on these ideas but you sacrifice the ability to prove anything because it is quasi-logic. === Subject: Re: New math. Dont read this. > On Aug 2, 4:55æam, David Formosa (aka ? the Platypus) > On Aug 1, 7:16 am, David Formosa (aka ? the Platypus) [...] > My very liberal usage of the words (1)existence, (2)nonexistence, and > (3) existentially indeterminate, can be thought of or even defined > more rigorously as follows: > Definition of existence (as I am using it) > The sum totality of all things which are said to exist. > By sum totality you mean something equilverlent to class in the > mathmatical sence of the word, if not what is the diffrence? > I'm also wondering how you define exist do abstractions exist? æ > You state that sum totalities can exist, what does this mean for one > to exist? Well, math is a logical model. An abstraction. And so the usage of the > concept of existence will always have two very distinct usages. One is > abstract, and one is physical. Now if something physically exists it means that I can (at least in principal) physically interact with it. I can see it, feel it, or detect via verious pysical means. What does it mean for an abstraction to exist? > I think that my usage applies to both the physical and the abstract, > and so I usually do not bother to draw the distinction. In order to communicate clearly and allow us to reason about this without resulting in error we have to be careful about this distinction. [...] > As far as physical existence of abstractions goes, I have no idea, > that happens in the mind somehow and Im not really concerned with > whether an abstract mathematical object has some aspect which makes it > exist physically... completely different issue and Im not really > concerned with it. The problem is that you are making claims that implicitly assumes this. For example you claim that The sum totality of objects that exist exists however this makes a claim about an abstraction (the sum totality of objects that exist) and its existance. Now if you had said something on the lines of Every object that exists, exists this would by tautological. > For your last question, all I can do is explain what I'm seeing. I > believe that physical existence is bounded by extreme scales. > Everything which exists relative to us is within about +/- 40 orders > of magnitude. The boundaries are where you find existential > indeterminacy. This is the precise place where math and physics must > part and go their separate ways, because mathematics has no > Plancklength. To have such a thing as a Plancklength in mathematics, > you have to construct it deliberately. So, if the universe were an Oreo cookie, the crispy chocolate outer > cracker would be existentially indeterminate, and existence would be > defined as the sugary cream filling. Yum. [...] > If something is not included in the sum totality of things that do not > exist does that mean it exists? > Can a sum totality that does not exist include members? > Why are they not considered classes? > (i) You cannot consider them to be distinct classes because that > would constitute performing a logical or mathematical procedure on the > nonexistent. There is nothing in the rules of mathmatics that prevents it from dealing with objects that do not exist. > If you consider all of the nonexistent cubes, and you > have that some are red and some are blue, you cannot divide them into > separate classes according to color because that is a mathematical > act. That would be the equivalent of dividing by zero. One can constract a mathmatical system that permits the division by zero. [...] > But to reword things, > [1] The sum totality of all things which are said to exist, exists. > Ok this implies that abstractions do exist. It might, but I really dont intend for that to be the case. Can you construct your statement in a way that doesn't carry this implication. Indeed can you write your sentence in a way that unambiguously convays your meaning preferably in a way that makes use of commonly shared definitions. > Because > the brain is made of molecules, it is very likely that the abstract > and the physical do intersect somehow, The map is not the landscape. [...] > [2] The sum totality of all things which are said to be nonexistent, > clearly do not exist. > Its not at all clear to me. æNot clearly knowing what you mean by sum > totality nor what you mean by a sum totality existing or not > existing. æIt is my view that using the verb exist to discuss abstract > objects is an example of fallacy of reifiction and doesn't lead to > usefull resoning. Well, this definition is a bit paradoxical for the same reason as (i) > above. There is no sum totality of things nonexistent, for the > reason explained in (i). You still haven't told me what a sum totality is. How can we have meaningful discussions about anything with out knowing what the words we are using mean? [...] > One cannot attempt to define nonexistence without creating a fallacy > of reification. How does defining nonexistence treat an abstraction like its a real thing? [...] > You are correct, I did commit the fallacy of reification, but there is > no way to avoid it and here is why : If your resoning depends on a fallacy then its invalid. If its invalid then there is nothing to support it being truthful. > You cannot define nonexistence. To define it it to perform a logical > act upon it, and it is immune to logic. There is no reson for it to be immune to logic, all standard logics don't have this restriction. [...] > The presence of paradox implies the reflexivity of the property. > Can you please explain this? [...] > That doesn't answer my question, how does a paradox show that a > property is reflexive? Lets assume that all existential forms are self referential. Then you > have that nonexistence is nonexistent. We just performed (or attempted to perform) a logical act upon the > nonexistent by saying that it does not exist, which is not valid > logically. That sequence leads to the conclusion that your inital assumption was not true. So the logical conculsion you get from this is It is not true that all existential forms are self referential or a little less clunky There exist existential forms that are not self referential. > So, the assumption that all existential forms are self referential > points to this situation with the particular case of nonexistence > where you have paradox. However again this doesn't show that the existance property is reflexive. === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 2, 8:42æpm, David Formosa (aka ? the Platypus) > On Aug 2, 4:55æam, David Formosa (aka ? the Platypus) > On Aug 1, 7:16 am, David Formosa (aka ? the Platypus) > [...] > My very liberal usage of the words (1)existence, (2)nonexistence, and > (3) existentially indeterminate, can be thought of or even defined > more rigorously as follows: > Definition of existence (as I am using it) > The sum totality of all things which are said to exist. > By sum totality you mean something equilverlent to class in the > mathmatical sence of the word, if not what is the diffrence? > I'm also wondering how you define exist do abstractions exist? æ > You state that sum totalities can exist, what does this mean for one > to exist? > Well, math is a logical model. An abstraction. And so the usage of the > concept of existence will always have two very distinct usages. One is > abstract, and one is physical. Now if something physically exists it means that I can (at least in > principal) physically interact with it. æI can see it, feel it, or > detect via verious pysical means. æWhat does it mean for an > abstraction to exist? Whather abstractions actually exist or not in the mind, I dont know and am not concerned. I consider math and physics to be distinct. What happens in the mind is beyond the scope of my claims. > I think that my usage applies to both the physical and the abstract, > and so I usually do not bother to draw the distinction. In order to communicate clearly and allow us to reason about this > without resulting in error we have to be careful about this > distinction. [...] Agreed. There is a distinction, but my usage is intended to apply to both physical and abstract cases equally well. If you have points which are existentially indeterminate, then you can assign probabilities to their existence quite easily. You can explain the bending of physical space pretty easy that way. > As far as physical existence of abstractions goes, I have no idea, > that happens in the mind somehow and Im not really concerned with > whether an abstract mathematical object has some aspect which makes it > exist physically... completely different issue and Im not really > concerned with it. The problem is that you are making claims that implicitly assumes > this. æFor example you claim that The sum totality of objects that > exist exists however this makes a claim about an abstraction (the sum > totality of objects that exist) and its existance. æNow if you had > said something on the lines of Every object that exists, exists > this would by tautological. You could consider these objects individually, or collectively. I dont think it makes a difference until you start talking about thinks like Russel's paradox. But that is correct, regarding existent things I have been referring to the totality of all existent things. Technically, it makes no sense to speak of the totality of nonexistent things because there is no such totality, I only do this for convenience. > For your last question, all I can do is explain what I'm seeing. I > believe that physical existence is bounded by extreme scales. > Everything which exists relative to us is within about +/- 40 orders > of magnitude. The boundaries are where you find existential > indeterminacy. This is the precise place where math and physics must > part and go their separate ways, because mathematics has no > Plancklength. To have such a thing as a Plancklength in mathematics, > you have to construct it deliberately. > So, if the universe were an Oreo cookie, the crispy chocolate outer > cracker would be existentially indeterminate, and existence would be > defined as the sugary cream filling. Yum. [...] > If something is not included in the sum totality of things that do not > exist does that mean it exists? > Can a sum totality that does not exist include members? > Why are they not considered classes? > (i) æYou cannot consider them to be distinct classes because that > would constitute performing a logical or mathematical procedure on the > nonexistent. There is nothing in the rules of mathmatics that prevents it from > dealing with objects that do not exist. There are many situations where something can be shown to not exist. This is quite different than performing logical operations on the nonexistent. In fact, the only thing that allows logic to function properly at all is that things are existing nicely and so everything behaves like a perfect Newtonian Clock. Of course math does deal with the nonexistent, but only at a distance. You cannot make rules and expect nonexistent objects to obey them in accordance to any logic, nonexistent objects are nonsensical. That which is existentially indeterminate is somewhere in between. You dont know if it is neccesarily logical, or nonsensical. This is indeterminate in the world of existential indeterminacy. And that it because when you make rules and logical statements, you might be talking about the existent or the nonexistent. You dont know which, and cant. And dont need to. But you do sacrifice the ability to ever prove anything. > If you consider all of the nonexistent cubes, and you > have that some are red and some are blue, you cannot divide them into > separate classes according to color because that is a mathematical > act. That would be the equivalent of dividing by zero. One can constract a mathmatical system that permits the division by > zero. [...] > But to reword things, > [1] The sum totality of all things which are said to exist, exists. > Ok this implies that abstractions do exist. > It might, but I really dont intend for that to be the case. Can you construct your statement in a way that doesn't carry this > implication. æIndeed can you write your sentence in a way that > unambiguously convays your meaning preferably in a way that makes use > of commonly shared definitions. What I really need to do is study some predicate calculus and try to formalize my babble symbolically. Well, I know Meinong was a quack, Russell kicked his ass gloriously and deservedly. Whether abstractions exist or not, I wont attempt to answer that. But I do think that the thesis makes sense physically, and so there must be some way to make sense of it mathematically. While I think that this approach brings math and physics much closer to each other, they will always be distinct unless one is talking about what happens in the conscious mind. I would never make assertions regarding the physical nature of abstractions. > Because > the brain is made of molecules, it is very likely that the abstract > and the physical do intersect somehow, The map is not the landscape. Agreed completely - well stated. > [...] > [2] The sum totality of all things which are said to be nonexistent, > clearly do not exist. > Its not at all clear to me. æNot clearly knowing what you mean by sum > totality nor what you mean by a sum totality existing or not > existing. æIt is my view that using the verb exist to discuss abstract > objects is an example of fallacy of reifiction and doesn't lead to > usefull resoning. > Well, this definition is a bit paradoxical for the same reason as (i) > above. There is no sum totality of things nonexistent, for the > reason explained in (i). You still haven't told me what a sum totality is. æHow can we have > meaningful discussions about anything with out knowing what the words > we are using mean? Ive been applying induction to existent things to derive a totality of all thing existent. It is handy to also apply this to the nonexistent for purposes of illustrating a more important point. But, clearly, one cannot perform induction on the nonexistent to obtain such a sum totality, and so yes I took a shortcut. > One cannot attempt to define nonexistence without creating a fallacy > of reification. How does defining nonexistence treat an abstraction like its a real > thing? [...] > You are correct, I did commit the fallacy of reification, but there is > no way to avoid it and here is why : If your resoning depends on a fallacy then its invalid. æIf its > invalid then there is nothing to support it being truthful. Im not sure that I committed the fallacy, we might be talking about 2 different reifications. I think that you're usage is in the sense of the fallacy that Meinong made, and I would certainly plead innocent to that. > You cannot define nonexistence. To define it it to perform a logical > act upon it, and it is immune to logic. There is no reson for it to be immune to logic, all standard logics > don't have this restriction. So then I should be able to calculate the area of a round square ? > [...] > The presence of paradox implies the reflexivity of the property. > Can you please explain this? [...] > That doesn't answer my question, how does a paradox show that a > property is reflexive? > Lets assume that all existential forms are self referential. Then you > have that nonexistence is nonexistent. > We just performed (or attempted to perform) a logical act upon the > nonexistent by saying that it does not exist, which is not valid > logically. That sequence leads to the conclusion that your inital assumption was > not true. æSo the logical conculsion you get from this is It is not > true that all existential forms are self referential or a little less > clunky There exist existential forms that are not self referential. Disagree. > So, the assumption that all existential forms are self referential > points to this situation with the particular case of nonexistence > where you have paradox. However again this doesn't show that the existance property is > reflexive.- Hide quoted text - It does indeed. === Subject: Re: New math. Dont read this. > On Aug 2, 8:42æpm, David Formosa (aka ? the Platypus) [...] > Now if something physically exists it means that I can (at least in > principal) physically interact with it. æI can see it, feel it, or > detect via verious pysical means. æWhat does it mean for an > abstraction to exist? Whather abstractions actually exist or not in the mind, I dont know > and am not concerned. I consider math and physics to be distinct. What > happens in the mind is beyond the scope of my claims. The problem is that you keep making claims that carry the implication that such abstractions exists. [...] > In order to communicate clearly and allow us to reason about this > without resulting in error we have to be careful about this > distinction. > [...] Agreed. There is a distinction, but my usage is intended to apply to > both physical and abstract cases equally well. Now I'm compleately confused. In the responce just above here you say Whather asbtractions actually exist or not in the mind, I don't know and am not concerned then you claim that your usage of the word exists is intended to apply both to physical and abstract cases. [...] > The problem is that you are making claims that implicitly assumes > this. æFor example you claim that The sum totality of objects that > exist exists however this makes a claim about an abstraction (the sum > totality of objects that exist) and its existance. æNow if you had > said something on the lines of Every object that exists, exists > this would by tautological. You could consider these objects individually, or collectively. I dont > think it makes a difference until you start talking about thinks like > Russel's paradox. Is the sum totality of objects that ... a collection of some sort or a way of saying All objects that have property ... have property .... This is not some technical distinction but a core and fundermental key to understanding what you are writing about. Unless you can make this clear your words can't convay meaning. [...] > There is nothing in the rules of mathmatics that prevents it from > dealing with objects that do not exist. There are many situations where something can be shown to not exist. > This is quite different than performing logical operations on the > nonexistent. Sure you can. I'll given an example. In the theory of computation there exists a device called a halting oracle. However if the strong turing-church therom holds (and I beleave that it does) it isn't possable for a halting oracle to exist. However nothing prevents us from resoning about the behavour of halting oricals and discussing the results of there existance. [...] > Of course math does deal with the nonexistent, but only at a distance. > You cannot make rules and expect nonexistent objects to obey them in > accordance to any logic, nonexistent objects are nonsensical. Yes we can and we do. Even if the objects contain logical inconsitancies we can make use of a paraconsitant logic. [...] > Can you construct your statement in a way that doesn't carry this > implication. æIndeed can you write your sentence in a way that > unambiguously convays your meaning preferably in a way that makes use > of commonly shared definitions. What I really need to do is study some predicate calculus and try to > formalize my babble symbolically. Please do. Formalizing your reasoning is a great way to find loop holes and overlooked fallacies. > Well, I know Meinong was a quack, Russell kicked his ass gloriously > and deservedly. Russell had quite a bit of respect for Meinong. [...] > You still haven't told me what a sum totality is. æHow can we have > meaningful discussions about anything with out knowing what the words > we are using mean? Ive been applying induction to existent things to derive a totality of > all thing existent. I'm sorry you still haven't defined totality or sum totality in a way I can understand. Nor are you using the word induction in a mannor that is standard in mathmatics. Please step back and start defining your terms. [...] > You cannot define nonexistence. To define it it to perform a logical > act upon it, and it is immune to logic. > There is no reson for it to be immune to logic, all standard logics > don't have this restriction. So then I should be able to calculate the area of a round square ? Calculating the area of a round square isn't a logical operation. Logical operations involve words like For every, For all,Not, And, Or and Predicate. [...] > Lets assume that all existential forms are self referential. Then you > have that nonexistence is nonexistent. > We just performed (or attempted to perform) a logical act upon the > nonexistent by saying that it does not exist, which is not valid > logically. > That sequence leads to the conclusion that your inital assumption was > not true. æSo the logical conculsion you get from this is It is not > true that all existential forms are self referential or a little less > clunky There exist existential forms that are not self referential. Disagree. I'm sorry but what you showed me there was a pritty standard reductio ad abserdum (spelling?). This takes the form of Assume X Show that X leads to an absurdity. This proves not(X) is true. > However again this doesn't show that the existance property is > reflexive. It does indeed. What meaning are you using for the word reflexive in this context? === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 3, 5:55æam, David Formosa (aka ? the Platypus) > On Aug 2, 8:42æpm, David Formosa (aka ? the Platypus) [...] > Now if something physically exists it means that I can (at least in > principal) physically interact with it. æI can see it, feel it, or > detect via verious pysical means. æWhat does it mean for an > abstraction to exist? > Whather abstractions actually exist or not in the mind, I dont know > and am not concerned. I consider math and physics to be distinct. What > happens in the mind is beyond the scope of my claims. The problem is that you keep making claims that carry the implication > that such abstractions exists. Existence in mathematics is abstract. Existence in physics is physical reality. Mathematical existence models the physical variety. And my thesis is quite simply that you can refine the model to a great extent by employing existential indeterminacy which is an abstract kind of existence where one assigns probabilities of existence to abstract objects such as points, areas, volumes, whatever. In no way does this imply or even hint at the idea that abstractions exist, or that abstractions are physically real. It is merely an expansion of the concept of existence, which is abstract. And the utility of this method can be demonstrated in a physics lab. It would be pretty funny if physics actually surpassed mathematics due to the limitations of our imagination. Math is usually accused of creating frivolous knowledge, it amazes me that the reverse is true in this case. > [...] > In order to communicate clearly and allow us to reason about this > without resulting in error we have to be careful about this > distinction. > [...] > Agreed. There is a distinction, but my usage is intended to apply to > both physical and abstract cases equally well. Now I'm compleately confused. æIn the responce just above here you say > Whather asbtractions actually exist or not in the mind, I don't know > and am not concerned then you claim that your usage of the word > exists is intended to apply both to physical and abstract cases. I do not confuse abstractions with physically real objects. What I am doing creating an abstract model which models physics more closely. It is a better model, and the model is still abstract. There should be no confusion on that. > [...] > The problem is that you are making claims that implicitly assumes > this. æFor example you claim that The sum totality of objects that > exist exists however this makes a claim about an abstraction (the sum > totality of objects that exist) and its existance. æNow if you had > said something on the lines of Every object that exists, exists > this would by tautological. > You could consider these objects individually, or collectively. I dont > think it makes a difference until you start talking about thinks like > Russel's paradox. Is the sum totality of objects that ... a collection of some sort or > a way of saying All objects that have property ... have property > .... æThis is not some technical distinction but a core and > fundermental key to understanding what you are writing about. æUnless > you can make this clear your words can't convay meaning. Yes. And there are two cases, abstract and physical. To establish the negation of your rebuttal regarding reification and restate properly : All [physical] objects [that exist] that have property [they exist]. All [abstract] objects [that exist] that have property [they exist]. All [physical] objects [that ~exist] that have property [they ~exist]. All [abstract] objects [that ~exist] that have property [they ~exist]. All [physical] objects [that exist probabilistically] that have property [they exist probabilistically]. All [abstract] objects [that exist probabilistically] that have property [they exist probabilistically]. > There is nothing in the rules of mathmatics that prevents it from > dealing with objects that do not exist. > There are many situations where something can be shown to not exist. > This is quite different than performing logical operations on the > nonexistent. Sure you can. æI'll given an example. æIn the theory of computation > there exists a device called a halting oracle. æHowever if the strong > turing-church therom holds (and I beleave that it does) it isn't > possable for a halting oracle to exist. æHowever nothing prevents us > from resoning about the behavour of halting oricals and discussing the > results of there existance. [...] > Of course math does deal with the nonexistent, but only at a distance. > You cannot make rules and expect nonexistent objects to obey them in > accordance to any logic, nonexistent objects are nonsensical. Yes we can and we do. æEven if the objects contain logical > inconsitancies we can make use of a paraconsitant logic. [...] > Can you construct your statement in a way that doesn't carry this > implication. æIndeed can you write your sentence in a way that > unambiguously convays your meaning preferably in a way that makes use > of commonly shared definitions. > What I really need to do is study some predicate calculus and try to > formalize my babble symbolically. Please do. æFormalizing your reasoning is a great way to find loop > holes and overlooked fallacies. > Well, I know Meinong was a quack, Russell kicked his ass gloriously > and deservedly. Russell had quite a bit of respect for Meinong. Im sure he did, but he disagreed completely with his ideas regarding the existence of abstractions. > [...] > You still haven't told me what a sum totality is. æHow can we have > meaningful discussions about anything with out knowing what the words > we are using mean? > Ive been applying induction to existent things to derive a totality of > all thing existent. I'm sorry you still haven't defined totality or sum totality in a > way I can understand. æNor are you using the word induction in a > mannor that is standard in mathmatics. æPlease step back and start > defining your terms. [...] > You cannot define nonexistence. To define it it to perform a logical > act upon it, and it is immune to logic. > There is no reson for it to be immune to logic, all standard logics > don't have this restriction. > So then I should be able to calculate the area of a round square ? Calculating the area of a round square isn't a logical operation. > Logical operations involve words like For every, For all,Not, > And, Or and Predicate. But you just said earlier that I could do math on the nonexistent ? If There is no reson for it to be immune to logic,..., then I should be able to perform the operation. > [...] > Lets assume that all existential forms are self referential. Then you > have that nonexistence is nonexistent. > We just performed (or attempted to perform) a logical act upon the > nonexistent by saying that it does not exist, which is not valid > logically. > That sequence leads to the conclusion that your inital assumption was > not true. æSo the logical conculsion you get from this is It is not > true that all existential forms are self referential or a little less > clunky There exist existential forms that are not self referential. > Disagree. I'm sorry but what you showed me there was a pritty standard reductio > ad abserdum (spelling?). æThis takes the form of Assume X > Show that X leads to an absurdity. > This proves not(X) is true. Reducto ad absurdum is fine in a mathematical or logical system where you have that things either exist or not. If your system is built on just these two choices, then yes, reducto ad absurdum would falsify the premise. But if you have existential indeterminacy, then reducto ad absurdum is actually what you want to achieve. I am not pulling your leg. If I can prove that a premise is both true and false, then in my scheme this does NOT invalidate the premise, but rather proves the presence if indeterminacy. This method is not possible or sensible without existential indeterminacy. === Subject: Re: New math. Dont read this. > On Aug 3, 5:55æam, David Formosa (aka ? the Platypus) [...] > Yes. And there are two cases, abstract and physical. To establish the > negation of your rebuttal regarding reification and restate properly : All [physical] objects [that exist] that have property [they exist]. > All [abstract] objects [that exist] that have property [they exist]. All [physical] objects [that ~exist] that have property [they > ~exist]. > All [abstract] objects [that ~exist] that have property [they > ~exist]. All [physical] objects [that exist probabilistically] that have > property [they exist probabilistically]. > All [abstract] objects [that exist probabilistically] that have > property [they exist probabilistically]. Now we have got to a point where your terms are written in a clear unambiguous way I can say that I aggry that the above are true. [...] > There is no reson for it to be immune to logic, all standard logics > don't have this restriction. > So then I should be able to calculate the area of a round square ? > Calculating the area of a round square isn't a logical operation. > Logical operations involve words like For every, For all,Not, > And, Or and Predicate. But you just said earlier that I could do math on the nonexistent ? Yes but not all forms of maths, logic one can do with nonexistent things, as for the area of a round square its undefined. > If There is no reson for it to be immune to logic,..., then I should > be able to perform the operation. But you didn't ask me to perform an act in the part of maths that is called logic. [...] > I'm sorry but what you showed me there was a pritty standard reductio > ad abserdum (spelling?). æThis takes the form of > Assume X > Show that X leads to an absurdity. > This proves not(X) is true. Reducto ad absurdum is fine in a mathematical or logical system where > you have that things either exist or not. If your system is built on > just these two choices, then yes, reducto ad absurdum would falsify > the premise. So what you are suggesting is some sort of tristate logic? Or some form of constructivism? > But if you have existential indeterminacy, then reducto ad absurdum is > actually what you want to achieve. I am not pulling your leg. If I can > prove that a premise is both true and false, then in my scheme this > does NOT invalidate the premise, but rather proves the presence if > indeterminacy. This method is not possible or sensible without > existential indeterminacy. Have you read up on probility logic and Bayesian reasoning? === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > So what you are suggesting is some sort of tristate logic? æOr some > form of constructivism? That is what would be implied, yes. But it's very strange. Not the customary kind of logic, and I dont know much about that aspect oy my approach. I'm definately no logician. But if you start from the point of view of those who invented random variables 80 years ago, just try to put yourself in their shoes. They were trying to avoid certain pitfalls, and instead of sidestepping those pitfalls I have embraced them and attempted to formalize them. That's all it is. > But if you have existential indeterminacy, then reducto ad absurdum is > actually what you want to achieve. I am not pulling your leg. If I can > prove that a premise is both true and false, then in my scheme this > does NOT invalidate the premise, but rather proves the presence if > indeterminacy. This method is not possible or sensible without > existential indeterminacy. Have you read up on probility logic and Bayesian reasoning? I know a little but not enough to brag about. First, I dont know if I am even doing math or not because that is just the nature of the approach. If you cant prove anything, if you are holding logic in one hand and nonsense in the other, you cant really call that math. But, the analysis makes sense. So I spend most of my time in this area trying to create a kind of calculus and explain why it would make any sense at all, and not surprisingly I have been able to convince myself that there are many things that seem sensible, and even usable to physics. And I am probably lucky that I dont have a career in math because I feel that my methods actually violate certain mores of the field and I do feel a bit guilty for pursuing voodoo mathematics. And while I am confident in saying that almost every area of known and accepted mathematics would have some counterpart in an existentially indeterminate system of pseudo-math.....I could not possibly hope to explain all of it in those terms. There's a hell of alot that I dont even know, and so I focus my efforts on existential calculus and foundational issues. If these things make sense, then I'd probably start looking at set theory of something. Maybe differential equations or something. === Subject: Re: Relativity and Lorentz transformations > Ian Parker says... >http://ianparker.g3z.com/Relativity/aviation.htmfor an account of > disc aircraft (Completely classical) tells you how they REALLY work. > File not found >http://ianparker.g3z.com/Relativity/hoax.htmTells you the REAL > motivations for The Einstein Hoax > File not found > You fell into their little free website trap, didn't you? > Soon Alex Trebek and Jessie Ventura will be arriving at your house to > have a little talk about how deceptive Venus can appear to be in a dark > sky...particularly during a solar eclipse, when gravity bends its light > waves. ;-) > Pat >No, the error was in case. The Website is case sensitive. Let us >however start discusing content. Okay, the content seems to be that you are either creating a > parody website or are making fun of an actual website. In either > case, you seem to be pointing out the lunacy of anti-relativity > cranks. To take one line: > Dr. Einstein's Special Relativity (which is easily seen to be a mathematical > copy of the earlier Lorentz Transformations Aether Theory) to propagate the idea > that our physical reality was too subtle to be understood by mortal men and > could only be dealt with using sophisticated mathematics. The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. Another way of looking at it is that the Lorentz transformations do not have to be derived at all. Assume the principle of relativity for inertial frames (what ever they are :). Signaling is with electromangnetic fields. Electormagnetism is governed by experimentally determined laws, such as Faraday's law. The experimentally determined laws can be formulated as a set of partial differential equations. The PDEs imply a wave equation in which the wave disturbance propogates with constant speed, independently of relative, unaccelerated motion. The speed is experimentally derivable from the behavior of magnetic and electric fields. The transformation group for the PDEs is the Lorentz group. Therefore measurements transform by the Lorentz group from frame to frame. A frame is one where the PDEs of electromagnetism govern the behavior of matter. The structure of matter is governed by the PDEs, therefore matter is governed by the Lorentz group. -- Michael Press === Subject: Re: Relativity and Lorentz transformations posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs > Ian Parker says... >http://ianparker.g3z.com/Relativity/aviation.htmfor an account of > disc aircraft (Completely classical) tells you how they REALLY work. > File not found >http://ianparker.g3z.com/Relativity/hoax.htmTells you the REAL > motivations for The Einstein Hoax > File not found > You fell into their little free website trap, didn't you? > Soon Alex Trebek and Jessie Ventura will be arriving at your house to > have a little talk about how deceptive Venus can appear to be in a dark > sky...particularly during a solar eclipse, when gravity bends its light > waves. ;-) > Pat >No, the error was in case. The Website is case sensitive. Let us >however start discusing content. > Okay, the content seems to be that you are either creating a > parody website or are making fun of an actual website. In either > case, you seem to be pointing out the lunacy of anti-relativity > cranks. > To take one line: > Dr. Einstein's Special Relativity (which is easily seen to be a mathematical > copy of the earlier Lorentz Transformations Aether Theory) to propagate the idea > that our physical reality was too subtle to be understood by mortal men and > could only be dealt with using sophisticated mathematics. > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. Another way of looking at it is that the Lorentz > transformations do not have to be derived at all. Assume > the principle of relativity for inertial frames (what > ever they are :). Signaling is with electromangnetic > fields. Electormagnetism is governed by experimentally > determined laws, such as Faraday's law. < http://en.wikipedia.org/wiki/Faraday_paradox See also: Multiple integral Lorentz force Retarded potentials Sue... > Michael Press === Subject: Re: Relativity and Lorentz transformations > Ian Parker says... >http://ianparker.g3z.com/Relativity/aviation.htmfor an account of > disc aircraft (Completely classical) tells you how they REALLY work. > File not found >http://ianparker.g3z.com/Relativity/hoax.htmTells you the REAL > motivations for The Einstein Hoax > File not found > You fell into their little free website trap, didn't you? > Soon Alex Trebek and Jessie Ventura will be arriving at your house to > have a little talk about how deceptive Venus can appear to be in a dark > sky...particularly during a solar eclipse, when gravity bends its light > waves. ;-) > Pat >No, the error was in case. The Website is case sensitive. Let us >however start discusing content. > Okay, the content seems to be that you are either creating a > parody website or are making fun of an actual website. In either > case, you seem to be pointing out the lunacy of anti-relativity > cranks. > To take one line: > Dr. Einstein's Special Relativity (which is easily seen to be a mathematical > copy of the earlier Lorentz Transformations Aether Theory) to propagate the idea > that our physical reality was too subtle to be understood by mortal men and > could only be dealt with using sophisticated mathematics. > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. > Another way of looking at it is that the Lorentz > transformations do not have to be derived at all. Assume > the principle of relativity for inertial frames (what > ever they are :). Signaling is with electromangnetic > fields. Electormagnetism is governed by experimentally > determined laws, such as Faraday's law. < examples, where Faraday's law does not work. A rectangle of photoconductive material slides along > a pair of wires. At a fixed location a strong light > and a strong magnetic field create a narrow > immovable strip of conducting material subject > to a Lorentz force. Figure 4 shows a translating rectangle of material > with a narrow conducting strip subject to a > conducting at a fixed location by, for example, > a strong light beam at this location. The magnetic > field also is confined to the same strip. > The Lorentz force drives a current from the top > rail to the bottom rail through this strip, and the > circuit is completed through leads attached to > the top and bottom conducting rails. In this example, > the circuit does not move, and the magnetic flux > through the circuit is not changing, so Faraday's > law suggests no current flows. However, the > Lorentz force law suggests a current does flow. > This example is based upon one devised by > Richard Feynman to illustrate the inapplicability > of Faraday's law of induction to certain situations > (that is, the version of Faraday's law of induction > which relates EMF to magnetic flux, which he > terms the flux rule). Referring to his example, > Feynman said:[3] > http://en.wikipedia.org/wiki/Faraday_paradox See also: Multiple integral > Lorentz force > Retarded potentials Yes, ok. I quoted Faraday's law as a discovery and formulation in electromagnetism. Early formulations had to be sharpened. The essay gives two examples that purport to falsify FL. One fails to properly apply Leibniz's rule for differentiating a definite integral, and one attempts a shell game with physical components of the circuit. curl E(r, t) = @ /@t B(r, t). -- Michael Press === Subject: Re: Relativity and Lorentz transformations posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs Gecko/20071201 Epiphany/2.20 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > Another way of looking at it is that the Lorentz > transformations do not have to be derived at all. Assume > the principle of relativity for inertial frames (what > ever they are :). Signaling is with electromangnetic > fields. Electormagnetism is governed by experimentally > determined laws, such as Faraday's law. > Inapplicability of Faraday's law Figure 4: An example, based on one of Feynman's examples, where Faraday's law does not work. http://en.wikipedia.org/wiki/Faraday_paradox > See also: > Multiple integral > Lorentz force > Retarded potentials Yes, ok. I quoted Faraday's law as a discovery > and formulation in electromagnetism. Early > formulations had to be sharpened. The essay > gives two examples that purport to falsify FL. > One fails to properly apply Leibniz's rule for > differentiating a definite integral, and one > attempts a shell game with physical components > of the circuit. Charges are not moving points but rather superpositioned volumes of space. When that is considered, it becomes apparent that the integral form is requured for a complete expression. http://en.wikipedia.org/wiki/Multiple_integral#Some_practical_applications This puts the imaginary current (reactive) where it belongs, in the near-field, rather that along the entire path as a Lorentz transformation does. Compare: Retarded potentials http://farside.ph.utexas.edu/teaching/em/lectures/node50.html with Maxwell equations (with moved bodies) http://www.wolfram-stanek.de/maxwell_equations.htm#maxwell_classic_extended The implication are profound for resolving the constancy of light speed and the principle of relativity. < http://www.bartleby.com/173/7.html << The key to understanding special relativity is Einstein's relativity principle, which states that: All inertial frames are totally equivalent for the performance of all physical experiments. In other words, it is impossible to perform a physical experiment which differentiates in any fundamental sense between different inertial frames. By definition, Newton's laws of motion take the same form in all inertial frames. Einstein generalized this result in his special theory of relativity by asserting that all laws of physics take the same form in all inertial frames. > http://farside.ph.utexas.edu/teaching/em/lectures/node108.html See also: Covariant formulation of classical electromagnetism http://en.wikipedia.org/wiki/Formulation_of_Maxwell's_equations_in_special_r elativity Also note that the Purcell derivation of magnetism is circular and widely dicredited. http://en.wikipedia.org/wiki/Relativistic_electromagnetism http://physics.weber.edu/schroeder/mrr/MRRtalk.html Things should be made as simple as possible, but not any simpler. --Albert Einstein Sue... > -- > Michael Press === Subject: Re: Relativity and Lorentz transformations says... > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. Another way of looking at it is that the Lorentz >transformations do not have to be derived at all. I think you are missing the point: no matter how much experimental basis you have -- and we have a lot of it -- at some point, you must do derivation to get the Lorentz transformation. >Assume >the principle of relativity for inertial frames (what >ever they are :). Signaling is with electromangnetic >fields. Electormagnetism is governed by experimentally >determined laws, such as Faraday's law. The experimentally >determined laws can be formulated as a set of partial >differential equations. True, but you are leaving out Maxwell's displacement current, which was NOT experimentally derived. See http://farside.ph.utexas.edu/teaching/em/lectures/node46.html >The PDEs imply a wave equation >in which the wave disturbance propogates with constant >speed, independently of relative, unaccelerated motion. Once you have the term for displacement current, yes. >The speed is experimentally derivable from the behavior >of magnetic and electric fields. The transformation group >for the PDEs is the Lorentz group. Yes, but this is why people were skeptical of Maxwell's equations for so uncomfortably long. They realized this contradicted Newtonian physics, which had been around for much longer than Maxwell and his equations. And then when they finally did accept them, they tried to explain it wiht a physical Lorentz contraction, rather than with Relativity. It was only when Einstein made it clear that ALL the then known laws of physics have to be preserved by the transformation group, and that this could be done without sacrificing Newtonian physics for v << c -- it was then people realized the Relativity route was much better than Lorentc contractions and Galilean relativity. >Therefore measurements >transform by the Lorentz group from frame to frame. But this, of course, is exactly what the Relativity deniers will continue to deny. === Subject: Re: Relativity and Lorentz transformations > says... > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. >Another way of looking at it is that the Lorentz >transformations do not have to be derived at all. I think you are missing the point: no matter how much experimental basis you > have -- and we have a lot of it -- at some point, you must do derivation to get > the Lorentz transformation. Assume >the principle of relativity for inertial frames (what >ever they are :). Signaling is with electromangnetic >fields. Electormagnetism is governed by experimentally >determined laws, such as Faraday's law. The experimentally >determined laws can be formulated as a set of partial >differential equations. True, but you are leaving out Maxwell's displacement current, which was NOT > experimentally derived. No, but it is experimentally verifiable, though not easily, and maybe not with nineteenth century apparatuses. curl B = mu_0 j is an attempt to formulate Faraday's findings. Turns out the equation only holds when div j = 0, which is not always. Yes, I left out much. I cited Faraday's law as an example of an experimental law that was folded into a set of PDEs describing electric and magnetic fields. > See > http://farside.ph.utexas.edu/teaching/em/lectures/node46.html Yes, clear. >The PDEs imply a wave equation >in which the wave disturbance propogates with constant >speed, independently of relative, unaccelerated motion. Once you have the term for displacement current, yes. Yes, of course. I was worried you were telling me something I overlooked. Turns out I had not overlooked anything. Is there something wrong with the view that the Lorentz transformation is implicit in the PDEs of EM? That LT does not have to be derived? Yes the LT are derived from assumptions on ideal bodies and signals. Real bodies and signals are governed by EM, and EM is formulated in a system whose transformation group is the Lorentz group. -- Michael Press === Subject: Re: Relativity and Lorentz transformations | Press | > says... | > | > | > The mathematics used in Einstein's derivation of the Lorentz | > transformations involves nothing more than high school algebra. | > being intentionally dishonest. | > | >Another way of looking at it is that the Lorentz | >transformations do not have to be derived at all. | > | > I think you are missing the point: no matter how much experimental basis you | > have -- and we have a lot of it -- at some point, you must do derivation to get | > the Lorentz transformation. | > | >Assume | >the principle of relativity for inertial frames (what | >ever they are :). Signaling is with electromangnetic | >fields. Electormagnetism is governed by experimentally | >determined laws, such as Faraday's law. The experimentally | >determined laws can be formulated as a set of partial | >differential equations. | > | > True, but you are leaving out Maxwell's displacement current, which was NOT | > experimentally derived. | | No, but it is experimentally verifiable, though not easily, | and maybe not with nineteenth century apparatuses. | | curl B = mu_0 j | is an attempt to formulate Faraday's findings. | Turns out the equation only holds when div j = 0, | which is not always. | | Yes, I left out much. I cited Faraday's law as an example of | an experimental law that was folded into a set of PDEs | describing electric and magnetic fields. | | > See | > http://farside.ph.utexas.edu/teaching/em/lectures/node46.html | | Yes, clear. | | >The PDEs imply a wave equation | >in which the wave disturbance propogates with constant | >speed, independently of relative, unaccelerated motion. | > | > Once you have the term for displacement current, yes. | | Yes, of course. I was worried you were telling me something | I overlooked. Turns out I had not overlooked anything. | Is there something wrong with the view that the Lorentz | transformation is implicit in the PDEs of EM? That LT | does not have to be derived? | | Yes the LT are derived from assumptions on ideal bodies | and signals. Real bodies and signals are governed by EM, | and EM is formulated in a system whose transformation | group is the Lorentz group. | | -- | Michael Press What a load of drooling, babbling, ing NONSENSE, you ranting idiot! Why did Einstein say the speed of light from A to B is c-v, the speed of light from B to A is c+v, the time each way is the same? Your answer goes here: ________________________________________________________ Other answers have been: According to Ian Parker: We are not talking about the speed of light here we are talking classical stability theory. -- Idiot Ian Parker. ______________________________________________________ According to cretin harald.vanlintelButNotThis@epfl.ch Easy: he did NOT say that. ______________________________________________________ According to xxein: It is an artefactual/superficially imposed yin-yang of sorts. ______________________________________________________ According to Lamenting Shubert: Why do you want to know? ______________________________________________________ According to Imbecile Jimmy Black: In neither system (meaning frame of reference in modern-day terminology) is the speed of light c-v or c+v. In both systems the speed of light is c. According to the imbecile Jimmy Black, Einstein did not write the equation ______________________________________________________ According to Dork Bruere ______________________________________________________ According to Spirit of Truth: that math is correct but WRONG ______________________________________________________ 'we establish by definition that the time required by light to travel from A to B equals the time it requires to travel from B to A' because I SAY SO and you have to agree because I'm the great genius, STOOOPID, don't you dare question it. -- Rabbi Albert Einstein === Subject: Re: Relativity and Lorentz transformations : Androcles : Why did Einstein say : the speed of light from A to B is c-v, : the speed of light from B to A is c+v, : the time each way is the same? What he said was moves relatively to the initial point of k, when measured in the stationary system, with velocity c-v (etc). Which, of course, is not the same thing as a velocity in coordinate system k. A distinction the paper pages thoroughly clear in the context you snipped away. He also didn't say the time in the stationary system was the same in those two cases; he substituted the values into two different places on the previous page; the values x'/(c-v) and x'/(c+v) are not equal. The times in the moving system are equal, and the UNequal times in the stationary system are used to derive just how stationary and moving coordinates are related. Stationary and moving being, of course, arbitrary labels, as is also made thoroughly clear. Aaaaaand this is the point where you foam at the mouth and call me names and say various things you think are insulting about Einstein and accuse me of trimming your sacred post and so on and on and on. Good luck with that. Wayne Throop throopw@sheol.org http://sheol.org/throopw === Subject: Re: Relativity and Lorentz transformations > says... > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. >Another way of looking at it is that the Lorentz >transformations do not have to be derived at all. I think you are missing the point: no matter how much experimental basis you > have -- and we have a lot of it -- at some point, you must do derivation to get > the Lorentz transformation. Assume >the principle of relativity for inertial frames (what >ever they are :). Signaling is with electromangnetic >fields. Electormagnetism is governed by experimentally >determined laws, such as Faraday's law. The experimentally >determined laws can be formulated as a set of partial >differential equations. True, but you are leaving out Maxwell's displacement current, which was NOT > experimentally derived. See > http://farside.ph.utexas.edu/teaching/em/lectures/node46.html The PDEs imply a wave equation >in which the wave disturbance propogates with constant >speed, independently of relative, unaccelerated motion. Once you have the term for displacement current, yes. The speed is experimentally derivable from the behavior >of magnetic and electric fields. The transformation group >for the PDEs is the Lorentz group. Yes, but this is why people were skeptical of Maxwell's equations for so > uncomfortably long. They realized this contradicted Newtonian physics, which had > been around for much longer than Maxwell and his equations. And then when they > finally did accept them, they tried to explain it wiht a physical Lorentz > contraction, rather than with Relativity. Do you take the Lorentz transformation to be a convenient bookkeeping tool? I used to take it that way, but eventually adopted the view of Larmor, Lorentz, Einstein, and Poincare'. Insofar as physics deals with the physical, then bodies in paper detailing these matters. How to teach special relativity. Progress in Scientific Culture, Vol 1, No 2, summer 1976. Reprinted in Speakable and unspeakable in quantum mechanics, 1987. > It was only when Einstein made it clear that ALL the then known laws of physics > have to be preserved by the transformation group, and that this could be done > without sacrificing Newtonian physics for v << c -- it was then people realized > the Relativity route was much better than Lorentc contractions and Galilean > relativity. Therefore measurements >transform by the Lorentz group from frame to frame. But this, of course, is exactly what the Relativity deniers will continue to > deny. They have their own frame for reference. Happily, the myriad of laws they labour under do not all apply to me. -- Michael Press === Subject: Re: Relativity and Lorentz transformations says... > True, but you are leaving out Maxwell's displacement current, which was NOT > experimentally derived. See > http://farside.ph.utexas.edu/teaching/em/lectures/node46.html You are welcome. I see the author of those lecture notes also used one of my favorite E&M text, Griffiths, which describes the invention as follows: The problem is in the right side of equation (7.31) [divergence of Ampere'e Law], which should be zero, but isn't. Applying the continuity equation (5.25) and Gauss's law, the offending term can be rewritten: divJ = -dRho/dt = -d(esp[0]divE) =-div(eps[0]dE/dt) [In typing this in, I am using d for the 'round d' that indicates partial differentiation that appears in the text; I also replaced nabla. and nabla-cross with div] It might occur to you that if we were to add the quantity eps[0](dE/dt) to J, in Ampere's law, it would be just right to kill off the extra divergence: curlB = Mu[0]J + mu[0]eps[0]dE/dt. (Maxwell himself had other reasons for wanting to add this quanitty to Ampere's law. To him the rescue of the continuity equation was a happy divident rather than a primary motive. But today we regognize this argument as a far more compellingone than Maxwell's, which was based on a now-discredited model of the ether). p274 Introduction to Electrodynamics, Griffiths, David; Prentice Hall 1981.] >Do you take the Lorentz transformation to be a convenient >bookkeeping tool? Oh, no. I take it as much more than that. It is the group under which the laws of physics are invariants. You see, I take the invention of relativity as a vindication of Hilbert's Erlangen program, the grand idea of founding all the kinds of geometry on the transformation groups, their representations, and especially, their invariants. This is the mathematical idea that lies behind discovering all the symmetries of nature that show up in physics. It does not, however, explain symmetry breaking. > But this, of course, is exactly what the Relativity deniers will continue to > deny. They have their own frame for reference. Happily, the myriad of >laws they labour under do not all apply to me. That is a relief to hear;) === === Subject: A puzzling issue: object with 8 degrees of freedom A friend and I are having a bet. He states that there must be objects or mechanisms with 8 degrees of freedom (not counting translation} which have 3-fold symmetry (at least in some configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners whose angles are not fixed. But such a cubus has - three orientational degrees of freedom - three internal angles which makes a total of only 6 degrees of freedom. A cubus has 3fold symmetry when seen along a diagonal, so that would fit; but 6 are not 8 degrees of freedom. I brought up the idea of a tetrahedral skeleton, (like a methane molecule http://en.wikipedia.org/wiki/Methane ) . It has 8 degrees of freedom, it has 3fold symmetry in some configurations, but we do not see a way to build that in metal or rubber without having more or less than 8 degrees of freedom. On the other hand, I am not able to prove that the puzzle is impossible to solve. Is there another solution? Where can one look for such objects or related theorems? Are there books or sites on these issues? John === Subject: Re: A puzzling issue: object with 8 degrees of freedom > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus .... geometry and robotics. Ken Pledger. === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=PIdpdAoAAABmOZotWTpUEsX0KIi_Gc24 rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John Many answers! Of course a rigid body has only six degrees of freedom. That is why we are thinking about deformable objects or mechanisms. Are there any canonical lists of such mechanisms? We are looking for one with threefold symmetry and 8 degrees of freedom in total. John === Subject: Re: A puzzling issue: object with 8 degrees of freedom >...we are thinking about >deformable objects or mechanisms. >..... >We are looking for one with threefold symmetry and >8 degrees of freedom in total. John Will this one suit your purpose? On the axis of trifold symmetry, a long finger (1) with a central ball joint permitting the one half (1a) to rotate in the axis of symmetry only, At the end of this member, three fingers (2,3,4) attached to it, with pin joints, so they can each rotate in just one plane. At the tip of each of these three members (2,3,4) , a finger joined to each with a pin joint also permitting just one axis of rotation, (labeled 2a,3a,4a). This appears to provide the trifold symmetry you want, in that the mechanism can rotate on the axis of finger (1) and in the colinear axis of finger (1a). Each of three fingers can sweep an angle about the axis of finger (1a) and each of three finger tips (2a,3a,4a) can also sweep an angle with respect to the finger to which they connect. This is only one of numerous way to provide this specification, it seems. Brian W === Subject: Re: A puzzling issue: object with 8 degrees of freedom <117c949m75619f1nl1gnupn9gstmcon4rs@4ax.com> posting-account=PIdpdAoAAABmOZotWTpUEsX0KIi_Gc24 rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) >...we are thinking about >deformable objects or mechanisms. >..... >We are looking for one with threefold symmetry and >8 degrees of freedom in total. >John Will this one suit your purpose? On the axis of trifold symmetry, a long finger (1) with a central > ball joint permitting the one half (1a) to rotate in the axis of > symmetry only, At the end of this member, three fingers (2,3,4) attached to it, with > pin joints, so they can each rotate in just one plane. At the tip of each of these three members (2,3,4) , a finger joined to > each with a pin joint also permitting just one axis of rotation, > (labeled 2a,3a,4a). > This appears to provide the trifold symmetry you want, in that the > mechanism can rotate on the axis of finger (1) and in the > colinear axis of finger (1a). > Each of three fingers can sweep an angle about the axis of finger (1a) > and each of three finger tips (2a,3a,4a) can also sweep an angle > with respect to the finger to which they connect. This is only one of numerous way to provide this specification, it > seems. Brian W Brian, thank you for the proposal. (It almost looks as if it had 4fold symmetry - or am I wrong?) You also mention numerous ways to do this. Can you give a few more? In any case, thank you very much! John === Subject: HANSON! See this one! | | >...we are thinking about | >deformable objects or mechanisms. | >..... | >We are looking for one with threefold symmetry and | >8 degrees of freedom in total. | > | >John | > | > | | Will this one suit your purpose? | | On the axis of trifold symmetry, a long finger (1) with a central | ball joint permitting the one half (1a) to rotate in the axis of | symmetry only, | | At the end of this member, three fingers (2,3,4) attached to it, with | pin joints, so they can each rotate in just one plane. | | At the tip of each of these three members (2,3,4) , a finger joined to | each with a pin joint also permitting just one axis of rotation, | (labeled 2a,3a,4a). | This appears to provide the trifold symmetry you want, in that the | mechanism can rotate on the axis of finger (1) and in the | colinear axis of finger (1a). | Each of three fingers can sweep an angle about the axis of finger (1a) | and each of three finger tips (2a,3a,4a) can also sweep an angle | with respect to the finger to which they connect. | | This is only one of numerous way to provide this specification, it | seems. | | Brian W HAHAHA! http://www.insanesoccer.com/games/files/thefinger.jpg I love it! === Subject: Re: HANSON! See this one! | > | | > | >...we are thinking about deformable objects | > | >or mechanisms. ..... [snipped]..... | > | >We are looking for one with threefold symmetry | > | >and 8 degrees of freedom in total. | > | >John | > | > | > | > | > | Will this one suit your purpose? | > | On the axis of trifold symmetry, a long finger (1) with a central | > | ball joint permitting the one half (1a) to rotate in the axis of | > | symmetry only, | > | At the end of this member, three fingers (2,3,4) attached to it, with | > | pin joints, so they can each rotate in just one plane. | > | | > | At the tip of each of these three members (2,3,4) , a finger joined to | > | each with a pin joint also permitting just one axis of rotation, | > | (labeled 2a,3a,4a). | > | This appears to provide the trifold symmetry you want, in that the | > | mechanism can rotate on the axis of finger (1) and in the | > | colinear axis of finger (1a). | > | Each of three fingers can sweep an angle about the axis of finger (1a) | > | and each of three finger tips (2a,3a,4a) can also sweep an angle | > | with respect to the finger to which they connect. | > | This is only one of numerous way to provide this specification, it | > | seems. | > | Brian W | > | > HAHAHA! | > http://www.insanesoccer.com/games/files/thefinger.jpg | > I love it! | > | ... ahahahaha... Yeah, to you their exchange may sound | funny... ahahahahaha... but John and Brian are simply | having a standard machine shop operator talk. That | is their line and their language. Why they posted that | into sci.physics & sci math that is the funny part.... ahaha... | Both of them are probably laughing louder then you do... | Brian W is definitely having a laugh, John Stanton may have his head up his arse. === Subject: Re: HANSON! See this one! [cut] ahahaha... Is somebody kissing Hanson's ass by any chance? Yep. Once an ass kisser, always an ass kisser. ahahaha... AHAHAHA... ahahaha... Louis Savain Rebel Science News: http://rebelscience.blogspot.com/ === Subject: Re: HANSON! See this one! > Rebel Science News: > http://rebelscience.blogspot.com/ >Shhh... Louis... shhhh ... ahahahaha... >When fancy strikes I will tell you why >your thought after grav .& charge field >gradient velocity is not instantaneous >(for that would lead to infinite regression) >but may turn out to be (on first principles) >a whopping 6*10^71 cm/sec..... which >for practical purposes is instantaneous >and it quantifies Mach's Principle... ahahaha.. >Talk to you later after all the aggrieved Einstein >Dingleberries, in the other thread that you >have started, will have said their prayers... >ahahahaha... ahahahanson Well, for lack of a better word, 'instantaneous' is mostly a manner of speaking since nothing is instantaneous in a universe ruled by cause and effect. Cause must always precede effect by a fundamental interval. What I really want to say is that, as far as gravity and electrostatic fields are concerned, the time between cause and effect does not depend on distance. It's non-local. It's a very short time, though, possibly on the order of Planck time. It's hard to prove experimentally since our instruments cannot measure intervals at that minute scale. However, it should be possible to prove that distance does not affect the measured interval. I'll wait to hear what you have to say about infinite regress. Louis Savain Rebel Science News: http://rebelscience.blogspot.com/ === Subject: Re: A puzzling issue: object with 8 degrees of freedom > Will this one suit your purpose? On the axis of trifold symmetry, a long finger (1) with a central > ball joint permitting the one half (1a) to rotate in the axis of > symmetry only, At the end of this member, three fingers (2,3,4) attached to it, with > pin joints, so they can each rotate in just one plane. At the tip of each of these three members (2,3,4) , a finger joined to > each with a pin joint also permitting just one axis of rotation, > (labeled 2a,3a,4a). > This appears to provide the trifold symmetry you want, in that the > mechanism can rotate on the axis of finger (1) and in the > colinear axis of finger (1a). > Each of three fingers can sweep an angle about the axis of finger (1a) > and each of three finger tips (2a,3a,4a) can also sweep an angle > with respect to the finger to which they connect. This is only one of numerous way to provide this specification, it > seems. So as I stated, re-using the degrees already known is how you would play with the term more than six degrees of freedom. Sadly, You are only using the same 6 degrees of freedom of motions more than once. You have a multiple angles of motion in the same 6 degrees occuring in different places only. Don't ever try and engineer the hair on a shaggy dog. :) -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom > Many answers! Of course a rigid body has only six > degrees of freedom. That is why we are thinking about > deformable objects or mechanisms. Are there any canonical lists of such mechanisms? > We are looking for one with threefold symmetry and > 8 degrees of freedom in total. You are not truly finding 2 extra degrees of freedom You are counting a degree more than once. If you really think that creates multiple degrees of freedom Then a porcupines needles must really blow your mind for degrees of freedom. and boy oh boy Don't even try to think about a forest full of trees and millions (and billions) of branches and leaves etc. :) -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom | > A friend and I are having a bet. He states that there must be | > objects or mechanisms with 8 degrees of freedom | > (not counting translation} which | > have 3-fold symmetry (at least in some | > configurations). But we cannot find any. | > | > He is thinking of objects like a deformable cubus with corners | > whose angles are not fixed. But such a cubus has | > - three orientational degrees of freedom | > - three internal angles | > which makes a total of only 6 degrees of freedom. | > A cubus has 3fold symmetry when seen along | > a diagonal, so that would fit; but 6 are not 8 | > degrees of freedom. | > | > I brought up the idea of a tetrahedral skeleton, | > (like a methane molecule http://en.wikipedia.org/wiki/Methane) . | > It has 8 degrees of freedom, | > it has 3fold symmetry in some configurations, | > but we do not see a way to build that in metal | > or rubber without having more or less than 8 degrees | > of freedom. | > | > On the other hand, I am not able to prove | > that the puzzle is impossible to solve. | > | > Is there another solution? Where can one look for such | > objects or related theorems? Are there books or sites | > on these issues? | > | > | > John | | Many answers! Of course a rigid body has only six | degrees of freedom. That is why we are thinking about | deformable objects or mechanisms. | | Are there any canonical lists of such mechanisms? | We are looking for one with threefold symmetry and | 8 degrees of freedom in total. | | John An object deformed is not symmetric about one of its three axes of symmetry unless the deformation is also symmetric; but that merely returns the 6-DOF of the rigid body. You can't have your cake with a bite out of it. === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=1wPXHwkAAACFV0NiGWX7tZb1o0HYkMjT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; PeoplePal 6.6) w:PACBHO60,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John Just to clarify the meaning of a degree of freedom in my thinking. An independent geometric variable. Angle as a side length appear to be the issue and the basic variable where angle appears dependent was all that need be discussed. | | | here is a right angle and it is clearly independent of side length. A constructed geometric as a whole then allows side to cause angle. IN mechanical interpretation a machine can be described as a geometric function. MY right angle machine above has three apparent degrees of freedom. Two sides and the angle. As soon as a machine is a triangle it has three again, but the angles are side length determined and the meaning as machine itself in abstract appear to be a school idea I am trying to understand. A function of machine applies to any design wheather a robot like device or not. A location of machine parts as function allows a complexity design such as a robot to be discussed, but exact question was apparently unclear to me. A thoughtful man would allow all machine whether a simple constant structure to be a fuunctionally defined structure. A robot type of machine appears the issue. A three fold symmetry mean it has three axises and so all machines with threee axis are functionally equivalent. An inverted triangleoizoid;) was used at the National Insitute of Standards and Technology, NIST as a robot arm/platform. Cables varying the side lengths functionally could cause the tip of the volume to be motionable in an exact functional fashion. It translates left and right and up and down and spins in the center in a circle if required. So all that was required was to allow an actual usage to be stated. I thought it was a crystallike common question and not a mechanical engineeering question. Analogy to crystal was a possible reason for the NIST discovery as a class of robot though. If there is a question concerning the usage of crystal design in robots I would entertain them because there are few machines able to be used in crystal analogy! Here is a small novel machine design based on a crystal geometric. A cubic structure has like maybe ten degree of freedom. And to functionally control side length to cause the function meant the solution would be indeterminate! A rather airfcraft like control function would have to be defined for a cubic to be used as a robot and a test loop in control code would have to prevent nonsolution motion. Making the nIST inverted triangleosoid a truely novel discovery. Maechanical design is very interesting and the basic question here is to either talk of the method of machine analogy or not. === Subject: Re: A puzzling issue: object with 8 degrees of freedom > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. > He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. > I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. > On the other hand, I am not able to prove > that the puzzle is impossible to solve. > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John Just to clarify the meaning of a degree of freedom in my thinking. An independent geometric variable. Angle as a side length appear to be the issue and the basic variable > where angle appears dependent was all that need be discussed. _______ here is a right angle and it is clearly independent of side length. A > constructed geometric as a whole then allows side to cause angle. IN mechanical interpretation a machine can be described as a geometric > function. MY right angle machine above has three apparent degrees of > freedom. Two sides and the angle. As soon as a machine is a triangle > it has three again, but the angles are side length determined and the > meaning as machine itself in abstract appear to be a school idea I am > trying to understand. A function of machine applies to any design wheather a robot like > device or not. A location of machine parts as function allows a > complexity design such as a robot to be discussed, but exact question > was apparently unclear to me. A thoughtful man would allow all machine whether a simple constant > structure to be a fuunctionally defined structure. A robot type of machine appears the issue. A three fold symmetry mean > it has three axises and so all machines with threee axis are > functionally equivalent. An inverted triangleoizoid;) was used at the National Insitute of > Standards and Technology, NIST as a robot arm/platform. Cables > varying the side lengths functionally could cause the tip of the > volume to be motionable in an exact functional fashion. It translates > left and right and up and down and spins in the center in a circle if > required. So all that was required was to allow an actual usage to be stated. I > thought it was a crystallike common question and not a mechanical > engineeering question. Analogy to crystal was a possible reason for > the NIST discovery as a class of robot though. If there is a question concerning the usage of crystal design in > robots I would entertain them because there are few machines able to > be used in crystal analogy! Here is a small novel machine design based on a crystal geometric. A cubic structure has like maybe ten degree of freedom. And to > functionally control side length to cause the function meant the > solution would be indeterminate! A rather airfcraft like control > function would have to be defined for a cubic to be used as a robot > and a test loop in control code would have to prevent nonsolution > motion. Making the nIST inverted triangleosoid a truely novel discovery. Maechanical design is very interesting and the basic question here is > to either talk of the method of machine analogy or not. Hi Doug, The cubic structure still does not have more than 6 degrees of freedom, What you are doing is allowing for the degrees to have a multiple of positions or multiple motions within the normal 6 degrees. That is not extra degrees that is still just an extra motion in the normal 6 degrees of freedom. Any point of the object can have no more than 6 different directions (up, down, left, right, forward, backward) of motion in the 3 planes of 3D space they reside it. Motions that combine lets say up and left, are not an extra degree of freedom. They are a combination of 2 or more of the normal degrees of freedom. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom a truss member has two degree of freedom, namely compress and extension which is the axial force at both ends. a frame member has six degree of freedom, namely translation in the x, y axis plus a rotation at each ends. that means 3 degree of freedom at each end. a cubic would would have three degree of freedom on each faces which is 6 times 3 which is 18 degree of freedom. a material is not measure by it's atomic structure but rather the material property of isotropical or anisotropical which is measure by the modulus of elasticity and the possion ratio.....done > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John > Just to clarify the meaning of a degree of freedom in my thinking. > An independent geometric variable. > Angle as a side length appear to be the issue and the basic variable > where angle appears dependent was all that need be discussed. > _______ > here is a right angle and it is clearly independent of side length. A > constructed geometric as a whole then allows side to cause angle. > IN mechanical interpretation a machine can be described as a geometric > function. MY right angle machine above has three apparent degrees of > freedom. Two sides and the angle. As soon as a machine is a triangle > it has three again, but the angles are side length determined and the > meaning as machine itself in abstract appear to be a school idea I am > trying to understand. > A function of machine applies to any design wheather a robot like > device or not. A location of machine parts as function allows a > complexity design such as a robot to be discussed, but exact question > was apparently unclear to me. > A thoughtful man would allow all machine whether a simple constant > structure to be a fuunctionally defined structure. > A robot type of machine appears the issue. A three fold symmetry mean > it has three axises and so all machines with threee axis are > functionally equivalent. > An inverted triangleoizoid;) was used at the National Insitute of > Standards and Technology, NIST as a robot arm/platform. Cables > varying the side lengths functionally could cause the tip of the > volume to be motionable in an exact functional fashion. It translates > left and right and up and down and spins in the center in a circle if > required. > So all that was required was to allow an actual usage to be stated. I > thought it was a crystallike common question and not a mechanical > engineeering question. Analogy to crystal was a possible reason for > the NIST discovery as a class of robot though. > If there is a question concerning the usage of crystal design in > robots I would entertain them because there are few machines able to > be used in crystal analogy! > Here is a small novel machine design based on a crystal geometric. > A cubic structure has like maybe ten degree of freedom. And to > functionally control side length to cause the function meant the > solution would be indeterminate! A rather airfcraft like control > function would have to be defined for a cubic to be used as a robot > and a test loop in control code would have to prevent nonsolution > motion. > Making the nIST inverted triangleosoid a truely novel discovery. > Maechanical design is very interesting and the basic question here is > to either talk of the method of machine analogy or not. Hi Doug, > The cubic structure still does not have more than 6 degrees of freedom, > What you are doing is allowing for the degrees to have > a multiple of positions or multiple motions within the normal 6 degrees. > That is not extra degrees that is still just an extra motion in the > normal > 6 > degrees of freedom. Any point of the object can have no more than 6 different directions > (up, down, left, right, forward, backward) > of motion in the 3 planes of 3D space they reside it. > Motions that combine lets say up and left, are not an extra > degree of freedom. > They are a combination of 2 or more of the normal degrees of freedom. -- > James M Driscoll Jr > Creator of the Clock Malfunction Theory > Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom >a truss member has two degree of freedom, > namely compress and extension which is > the axial force at both ends. A truss member has at least three degrees of freedom: 1) change in length (+/-) 2) bending off the central axis (up/down, left/right, etc., Euler column) 3) torsion around the central axis David A. Smith === Subject: Re: A puzzling issue: object with 8 degrees of freedom | | >a truss member has two degree of freedom, | > namely compress and extension which is | > the axial force at both ends. | | A truss member has at least three degrees of freedom: | 1) change in length (+/-) A hydraulic ram is free to change in length. It is not a ing truss. A truss has no FREEDOM to change its length, you dork. The six degrees of freedom are x,y,z, pitch, roll, yaw. An aircraft is free to move in any of them. 8-DOF is meaningless drivel. Google 6-DOF. === Subject: Re: A puzzling issue: object with 8 degrees of freedom > a truss member has two degree of freedom, > namely compress and extension which is > the axial force at both ends. A truss member has at least three degrees of freedom: > 1) change in length (+/-) Using only 2 degrees of the 6 known. > 2) bending off the central axis (up/down, left/right, etc., Euler > column) Using 4 degrees of the 6 known. > 3) torsion around the central axis Again Using same 4 degrees of (2) of the 6 known. (up down left right motion of points with a variable of motion for each point) Still only 6 degrees of actual freedom total. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom > a truss member has two degree of freedom, > namely compress and extension which is > the axial force at both ends. > A truss member has at least three degrees of freedom: > 1) change in length (+/-) Using only 2 degrees of the 6 known. No, that is a single degree of freedom. > 2) bending off the central axis (up/down, left/right, > etc., Euler column) Using 4 degrees of the 6 known. Only two. > 3) torsion around the central axis Again Using same 4 degrees of (2) > of the 6 known. (up down left right motion > of points with a variable of motion for each > point) No, this is three degrees of freedom. David A. Smith === Subject: Re: A puzzling issue: object with 8 degrees of freedom /// > A truss member has at least three degrees of freedom: > 1) change in length (+/-) > Using only 2 degrees of the 6 known. No, that is a single degree of freedom. /and so on/ >David A. Smith Dave, I see a problem for you; debating with the folks who have strayed onto an engineering group that actually uses the concept of DoF: it's the one called rassling with pigs.... You WILL get muddy! :-) Better to leave them to campout on sci.physics, sci.maths..... Brian W === Subject: Re: A puzzling issue: object with 8 degrees of freedom /// > A truss member has at least three degrees of freedom: > 1) change in length (+/-) Using only 2 degrees of the 6 known. > No, that is a single degree of freedom. > /and so on/ > David A. Smith Dave, > I see a problem for you; debating with the folks who > have strayed onto an engineering group that actually > uses the concept of DoF: > it's the one called rassling with pigs.... You WILL get muddy! :-) Better to leave them to campout on > sci.physics, sci.maths..... Actually physics does not actually use more than 6 degrees of freedom It is only the math heads that play with such sillyness instead of realizing they are just re-using the same known degrees of freedom already. So it would be best to play with such porcupine needled degrees of freedom that increase with the amount of objects and rubberyness in the math group alone. :) -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom > a truss member has two degree of freedom, > namely compress and extension which is > the axial force at both ends. A truss member has at least three degrees of freedom: > 1) change in length (+/-) > Using only 2 degrees of the 6 known. No, that is a single degree of freedom. No, two directions of motion is 2 degrees of freedom in a single plane of motion. > 2) bending off the central axis (up/down, left/right, > etc., Euler column) > Using 4 degrees of the 6 known. Only two. No, again 4 degrees but now in two planes. I think you are confusing planes with degrees. Each plane has 2 directions of freedom. (2 degrees) > 3) torsion around the central axis > Again Using same 4 degrees of (2) > of the 6 known. (up down left right motion > of points with a variable of motion for each > point) No, this is three degrees of freedom. Nope. It is the same as above. It has 2 directions for for any point in one plane and 2 more directions in the other again, 4 degrees of freedom. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom > a truss member has two degree of freedom, namely compress and > extension which is the axial force at both ends. a frame member has > six degree of freedom, namely translation in the x, y axis plus a > rotation at each ends. that means 3 degree of freedom at each end. a > cubic would would have three degree of freedom on each faces which is > 6 times 3 which is 18 degree of freedom. a material is not measure > by it's atomic structure but rather the material property of > isotropical or anisotropical which is measure by the modulus of > elasticity and the possion ratio.....done Again, The rotation at each end you speak of is still just freedom of motion in the same 6 degrees known. and each end also has a compression factor to allow 6 degrees to still exist at each end. You are merely mixing already known degrees of freedom into extra degrees that are not actually there. You are adding already known degrees of freedom as extra degrees. A cube, has 6 degrees of freedom only, Any single point on the cube or inside the cube also only has 6 degrees of freedom. You can combine any of the degrees for different motion in such free 3D space. But it still only moves with 6 degrees of freedom. Just because it has ends does not give it extra degrees by adding the same degrees. Each end can move in the same 6 degrees the other end can move in. There is no addition of degrees occuring. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom A friend and I are having a bet. He states that there must be >objects or mechanisms with 8 degrees of freedom >(not counting translation} which >have 3-fold symmetry (at least in some >configurations). But we cannot find any. /// >John Better not take the other end of the bet. A rotational degree of freedom occurs in one object capable of rotating with respect to another. An object with several rotatable links can provide several degrees of rotary freedom for each successive link with respect to the base attachment. An object like the Manx three-legged object, can rotate three ways at each ankle with respect to its limb. Brian W === Subject: Re: A puzzling issue: object with 8 degrees of freedom A friend and I are having a bet. He states that there > must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus > with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane molecule > http://en.wikipedia.org/wiki/Methane ) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for > such > objects or related theorems? Are there books or sites > on these issues? If I understood you correctly, the degrees of freedom of which you speak are equal to the cardinal points of the dimensionality. A time-dependent 3 dimensional object (6 degrees of freedom in your context) has two more degrees (6 + 2) where time is a simple parameter of reversible direction. This would, of course, beg treatment in the complex plane, which is inherently 2-dimensional. Tom > John > === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze Filter 1.2.0.72; .NET CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; WWTClient2),gzip(gfe),gzip(gfe) > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? An object with movable appendages, such as the human body, has multiple degrees of freedom. Dave === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=1wPXHwkAAACFV0NiGWX7tZb1o0HYkMjT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; PeoplePal 6.6) w:PACBHO60,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John A pentagram has five sides. A vertex angle makes 5 a symmetry. Making one degree of freedom and one symmetry. A line length or side length makes a five degree of freedom change, each side may be independent of the other. Allowing a legnth as a cause to ratio of side to side then making a ratio symmetry. And a mirror of set of sides allows a ratio of areas. Draw a line between vertexes and mirror. Making the third degree symmetry. And two degrees of freedom for there are only two axis? NO there are three axis, making 9 degree of freedom. SO use a square. A square is a cubic and all cubic exhibit this majic property. Gold as a cubic crystal system allows a functional method of set to be developed. 3 symmetries and 8 degrees of freedom allows a functional set to be designed. D(3) Length(4) Mirror ratio(2) Wait the square has only 7 degree of freedom, sorry! I went through several shapes and found this one. * A triangloid with a certain number of sides. It haa No mirror property because the axis appears a side! SO the angle vertex makes a ratio of side length to side length For all equal sides, two vertexs exist. One degree for each. Allowing the dies to equal the rest of the degrees of freedom. And the third mirror symmetry exists only as a NON-degree of freedom effect. === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=1wPXHwkAAACFV0NiGWX7tZb1o0HYkMjT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; PeoplePal 6.6) w:PACBHO60,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. > He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. > I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. > On the other hand, I am not able to prove > that the puzzle is impossible to solve. > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John > A pentagram has five sides. æ [snip crap] Uncle Al counts 10 external sides. æIdiot. http://www.electricwitch.com/pentagram2.gif -- > Uncle Alhttp://www.mazepath.com/uncleal/ > æ(Toxic URL! Unsafe for children and most mammals)http://www.mazepath.com/uncleal/lajos.htm#a2- Hide quoted text - - Show quoted text - The pentagram was a postulated shape, I then tried a square, then a triangloziod. THe last was OK. * The formal name escapes me now. === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=1wPXHwkAAACFV0NiGWX7tZb1o0HYkMjT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; PeoplePal 6.6) w:PACBHO60,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. > He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. > I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. > On the other hand, I am not able to prove > that the puzzle is impossible to solve. > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John A pentagram has five sides. æA vertex angle makes 5 a symmetry. > Making one degree of freedom and one symmetry. A line length or side length makes a five degree of freedom change, > each side may be independent of the other. Allowing a legnth as a > cause to ratio of side to side then making a ratio symmetry. And a mirror of set of sides allows a ratio of areas. Draw a line > between vertexes and mirror. > Making the third degree symmetry. æAnd two degrees of freedom for > there are only two axis? æNO there are three axis, making 9 degree of > freedom. SO use a square. A square is a cubic and all cubic exhibit this majic property. æGold > as a cubic crystal system allows a functional method of set to be > developed. 3 symmetries and 8 degrees of freedom allows a functional > set to be designed. D(3) > Length(4) > Mirror ratio(2) Wait the square has only 7 degree of freedom, sorry! I went through several shapes and found this one. æ æ æ* A triangloid with a certain number of sides. æIt haa No mirror > property because the axis appears a side! æSO the angle vertex makes a > ratio of side length to side length For all equal sides, two vertexs > exist. æOne degree for each. Allowing the dies to equal the rest of the degrees of freedom. And the third mirror symmetry exists only as a NON-degree of freedom > effect.- Hide quoted text - - Show quoted text - I forgot to mention. A base line or axis drawn through the base to mirror CAN NOT because the Volume of the mirror appears nonexistent. You can not mirror a volume with a line in other words, except as given. A top vertex line only appears to have the property of symmetric formal applied volume, but it appears ZERO. If the base was square | | | | | | | | axis An axis trough the base side can not make a volume mirrior as with the top vertex because the DEGREE of Freedom of the top vertex was a third symmetrical form. It depends as a symmetry on base and side legnth, while the base verticies depende only on square side length. === Subject: Re: A puzzling issue: object with 8 degrees of freedom > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane molecule http://en.wikipedia.org/wiki/Methane ) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > Hi John, The shape or makeup of an object does not change the freedom of it's motion. Freedom of motion has 6 directions, up- down, forward- backward,left-right. Those are the 6 so called degrees that I would call planes of motion instead. 6 maximum planes of motion only. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Aug 2, 6:55 am, Spaceman A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. > He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. > I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. > On the other hand, I am not able to prove > that the puzzle is impossible to solve. > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > Hi John, > The shape or makeup of an object does not change the freedom > of it's motion. > Freedom of motion has 6 directions, up- down, forward- backward,left-right. > Those are the 6 so called degrees that I would call planes of motion > instead. No: under the conventions that mathematicians and physicists use (and those *are* the relevant ones, after all, in this math newsgroup) you have described only three degrees of freedom, not six. These three degrees are East-West, up-down and in-out. This just says that there are three lines along which the motion can be projected, or three coordinates needed to describe velocity. For a generally-shaped so- called rigid body there can be two more degrees of freedom, associated with rotational angles and the like (i.e., the object's orientation). For non-rigid bodies there can be additional degrees of freedom, associated with vibrational modes, internal angles, etc. You need to be careful to count these correctly in order to obtain correct figures for specific heats in polyatomic gasses when doing statistical thermodynamics. Anyway, the OP is, presumably, dealing with only orientation and internal-structure degrees of freedom. I'm still not sure about the exact answer to the OP's question. R.G. Vickson > 6 maximum planes of motion only. -- > James M Driscoll Jr > Creator of the Clock Malfunction Theory > Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom | On Aug 2, 6:55 am, Spaceman | > A friend and I are having a bet. He states that there must be | > objects or mechanisms with 8 degrees of freedom | > (not counting translation} which | > have 3-fold symmetry (at least in some | > configurations). But we cannot find any. | > | > He is thinking of objects like a deformable cubus with corners | > whose angles are not fixed. But such a cubus has | > - three orientational degrees of freedom | > - three internal angles | > which makes a total of only 6 degrees of freedom. | > A cubus has 3fold symmetry when seen along | > a diagonal, so that would fit; but 6 are not 8 | > degrees of freedom. | > | > I brought up the idea of a tetrahedral skeleton, | > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . | > It has 8 degrees of freedom, | > it has 3fold symmetry in some configurations, | > but we do not see a way to build that in metal | > or rubber without having more or less than 8 degrees | > of freedom. | > | > On the other hand, I am not able to prove | > that the puzzle is impossible to solve. | > | > Is there another solution? Where can one look for such | > objects or related theorems? Are there books or sites | > on these issues? | > | > | > Hi John, | > The shape or makeup of an object does not change the freedom | > of it's motion. | > Freedom of motion has 6 directions, up- down, forward- backward,left-right. | > Those are the 6 so called degrees that I would call planes of motion | > instead. | | No: under the conventions that mathematicians and physicists use (and | those *are* the relevant ones, after all, in this math newsgroup) | you have described only three degrees of freedom, not six. These three | degrees are East-West, up-down and in-out. This just says that there | are three lines along which the motion can be projected, or three | coordinates needed to describe velocity. For a generally-shaped so- | called rigid body there can be two more degrees of freedom, associated | with rotational angles and the like (i.e., the object's orientation). | For non-rigid bodies there can be additional degrees of freedom, | associated with vibrational modes, internal angles, etc. You need to | be careful to count these correctly in order to obtain correct figures | for specific heats in polyatomic gasses when doing statistical | thermodynamics. | | Anyway, the OP is, presumably, dealing with only orientation and | internal-structure degrees of freedom. I'm still not sure about the | exact answer to the OP's question. | | R.G. Vickson 6-DOF x, y, z, pitch, roll, yaw. 6-DOF platforms: http://www.inmotionsimulation.com/images/6-dof-2.jpg http://www.inmotionsimulation.com/images/6-dof-1.jpg http://www.ckas.com.au/CKAS%20V4%206DOF%20Motion%20Platform.jpg 16783 === Subject: Question about Apostol Text posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Two questions about Tom Apostol's 2 vol set on calculus. 1) I am considering going to grad school for applied math (starting at the MS level) and I am wondering if Apostol is a good book to review (it's been about 8 years since I got my BS). I've noticed that it appears to be the source book that Rensselaer Poly applied math bases its prelims on. (Obviously, there is no algebra or topology, so I'm looking at Apostol as a way to review the essential facts about real analysis, matrices, vector analysis and diff eqs) 2) When I read the various reviews of Aopstol on Amazon.com, I get the impression that the editions vary significantly. I read one review that said Apostol treated Fourier analysis, yet in the editions available in my library, I did not see any such material covered. Matt === Subject: Re: Question about Apostol Text posting-account=JpxxPAgAAAAgwzQIYqn4j6syK-YhOmcF Gecko/20070309 Firefox/2.0.0.3,gzip(gfe),gzip(gfe) Two questions about Tom Apostol's 2 vol set on calculus. 1) I am considering going to grad school for applied math (starting at > the MS level) and I am wondering if Apostol is a good book to review > (it's been about 8 years since I got my BS). I've noticed that it > appears to be the source book that Rensselaer Poly applied math bases > its prelims on. (Obviously, there is no algebra or topology, so I'm > looking at Apostol as a way to review the essential facts about real > analysis, matrices, vector analysis and diff eqs) 2) When I read the various reviews of Aopstol on Amazon.com, I get the > impression that the editions vary significantly. I read one review > that said Apostol treated Fourier analysis, yet in the editions > available in my library, I did not see any such material covered. > Matt I have Apostol's Calculus, both volumes, not in front on me, but I don't recall seeing anything on Fourier Analysis. There is another book by Apostol, Mathematical Analysis, and this is where he goes in depth into Fourier Analysis (after covering Lebesgue Integrals). Those who contrast Apostol's Calculus and Rudin's Principles of Analysis are comparing apples and oranges. Apostol's Calculus is just - well - Calculus, very rigorous, fairly comprehensive, in fact both volumes include large sections on Linear Algebra and Differential Equations - and yet it's still Calculus rather than Analysis. I would say this is probably the best Calculus book you can get (I still use it as a reference). Im not sure what you mean by 'Various editions exist'. The latest edition was, if I remember correctly, the 2nd edition. You can get inexpensive 'international edition (softcover)' from eBay. Apostol's Calculus is commonly used in 'Honor Calculus' (Calculus with Theory, Rigorous Calculus or whatever the course name is), like at MIT. Im not sure if it's good for reviewing of Calculus, though. The level of details may be overwhelming. I would recommend to get both volumes anyway, just have it as a reference. If you just want to review some Calculus theorem, without rigor or sufficient depth, something like Stewart will be sufficient. Ridin's Principles of Analysis is not a Calculus book and completely unsuitable for reviewing the Calculus. (Yes, I took two semesters of Analysis, with Baby Rudin, so I'm quite familiar with that book). On a subject of Fourier Analysis, for rigorous exposure you need to be familiar with Lebesgue Integrals first. That's what Apostol does in his 'Mathematical Analysis. Incidently, this is another excellent book by Apostol, but once again - not a Calculus book. Highly recommend, though, as a good Introduction to Analysis and not as terse as Rudin's. To summarize, if you want to review Calculus, get some Calculus book, like Stewart. Apostol's Calculus is great but probably not the best book for reviewing. On the other hand, Apostol's Calculus also contains Linear Algebra and Differential Equations, you can read those sections for review as well. I'm not sure how much of analysis you would need for Applied math and whether you need to have some knowledge of it (on a level of, say, the first 7-8 Chapters of Rudin's Principles of Analysis), or having solid Calculus, Linear Algebra and ODE is all that's required. === Subject: Re: Question about Apostol Text posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > I have Apostol's Calculus, both volumes, not in front on me, but I > don't recall seeing anything on Fourier Analysis. There is another > book by Apostol, Mathematical Analysis, and this is where he goes in > depth into Fourier Analysis (after covering Lebesgue Integrals). > OK, that is where the Fourier analysis is coming from: my mistake in getting the two texts confused. So that answers my question about where Apostol covers Fourier analysis. Matt === Subject: Re: Question about Apostol Text Rudin's 'Principles Of Mathematical Analysis' is easier to carry and better preparation for grad school. === Subject: Re: Question about Apostol Text <25705036.1217602850801.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Rudin's 'Principles Of Mathematical Analysis' is easier to carry and better preparation for grad school. When you say better preparation in what sense do you mean this? Do you think Rudin is better in that it is more rigorous than Apostol? I had Rudin as an undergrad for a 2 semester course in analysis and got A's in both classes, but the reasons why I have shied away from it are: i) does not seem to be as comprehensive as Apostol (as far as the variety of topics it covers). ii) I am also not totally convinced that Rudin covers the material in sufficiently greater depth than Apostol does (with the possible exception of the last few chapters of Rudin that deals with measure and differential forms). BUT, I could be wrong here, so I'm interested in what others think in this regard. iii) Since my interest is in applied math, I was not crazy about a book dealing with the basic theory of calculus that has no figures, motivations, or applications. Matt === Subject: Re: Question about Apostol Text In most applied math programs you probably still wouldn't see many numbers. So in that sense Rudin might be better preparation. Also, if you get used to Rudin's style, he has texts on functional analysis and other topics that are useful. I suppose taste plays a part. === Subject: Re: An exact simplification challenge - 70 (MeijerG) posting-account=htkO2QoAAABig9a_npSTZGp0tWMcSMXY Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] Here's what I entered in Maple: s:=convert(c,StandardFunctions); e:=convert(s,Ei); simplify(e); Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this can be simplified further. One could convert it into 1-argument Ei calls (see FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer expression. -- Thomas Richard Maple Support Scientific Computers GmbH http://www.scientific.de === Subject: Re: An exact simplification challenge - 70 (MeijerG) posting-account=uD9kfgoAAABaqjCF8ol-EFTFI3g2PjhE Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] Here's what I entered in Maple: s:=convert(c,StandardFunctions); > e:=convert(s,Ei); > simplify(e); Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this > can be simplified further. > One could convert it into 1-argument Ei calls (see > FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer > expression. > Or, in terms of Si,Ci: 1/2*Pi^(1/2)*(Pi^2-4*Pi*Si(1)+4*Ci(1)^2+4*Si(1)^2) Mate === Subject: Re: An exact simplification challenge - 70 (MeijerG) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] > Here's what I entered in Maple: > s:=convert(c,StandardFunctions); > e:=convert(s,Ei); > simplify(e); > Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this > can be simplified further. > One could convert it into 1-argument Ei calls (see > FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer > expression. Or, in terms of Si,Ci: 1/2*Pi^(1/2)*(Pi^2-4*Pi*Si(1)+4*Ci(1)^2+4*Si(1)^2) > Mate For me it is a bit senseless to ask for simplifications for arbitrary MeijerG, since that class is certainly too large. Already for the hypergeometric type there would be enough, where Maple and other CAS will fail to 'simplify' missing special identities. For example convert 3F1 to MeijerG and try to return ... (which gives identities ...) === Subject: Re: An exact simplification challenge - 70 (MeijerG) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] > Here's what I entered in Maple: > s:=convert(c,StandardFunctions); > e:=convert(s,Ei); > simplify(e); > Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this > can be simplified further. > One could convert it into 1-argument Ei calls (see > FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer > expression. > Or, in terms of Si,Ci: > 1/2*Pi^(1/2)*(Pi^2-4*Pi*Si(1)+4*Ci(1)^2+4*Si(1)^2) > Mate > For me it is a bit senseless to ask for simplifications for > arbitrary MeijerG, since that class is certainly too large. > Already for the hypergeometric type there would be enough, > where Maple and other CAS will fail to 'simplify' missing > special identities. > For example convert 3F1 to MeijerG and try to return ... > (which gives identities ...) > AV> For me it is a bit senseless to ask for simplifications > AV> for arbitrary MeijerG, since that class is certainly too > AV> large. For arbitrary MeijerG, yes. Still, what we post (and have in store) is not arbitrary. Computer algebra systems will deliver to the customers more > powers/flexibility with a better implementation of approaches > to MeijerG (simplification). For example, http://www.planetquantum.com I agree with Mate (essentially: for internal use as mighty tool), that does not say 'do not care for MeijerG'. Well, seeing no systematics (except the one already said) your choices are arbitrary - it makes not that much difference from which chapters in what books or pages they are taken. Even it would be nice if CAS know the vast variety of all the identities - we know: they don't. So why to repeat it. They even do not know all 'useful' identities or transformations for hypergeometric functions. So why repeat it. Of course occasionally one can guess which in your 'tasks' has may be considered as variable and look up books etc. But it is a mess to type those formulae (besides the notational burdens). For example your task 69 reduces to a duplication formula, but no reasonable person would like to type that in hoping to come out with a simple result. BTW: Kelly Roach (=your link) uses exp before the usual MeijerG (at least in older papers). AFAIK in former days he did some work for/with Maple. Certainly a strong Mathematician - and those may 'always' beat existing CAS, giving enhancements. === Subject: Re: An exact simplification challenge - 70 (MeijerG) posting-account=uD9kfgoAAABaqjCF8ol-EFTFI3g2PjhE Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > MeijerG[{{0}, {1}}, {{0, 0, 0, 1/2}, {}}, 1/4] > Here's what I entered in Maple: > s:=convert(c,StandardFunctions); > e:=convert(s,Ei); > simplify(e); > Which returns 2*sqrt(Pi)*Ei(1, I)*Ei(1, -I). I don't know whether this > can be simplified further. > One could convert it into 1-argument Ei calls (see > FunctionAdvisor(identities,Ei(1,z))[1]), but that's a longer > expression. > Or, in terms of Si,Ci: > 1/2*Pi^(1/2)*(Pi^2-4*Pi*Si(1)+4*Ci(1)^2+4*Si(1)^2) > Mate > For me it is a bit senseless to ask for simplifications for > arbitrary MeijerG, since that class is certainly too large. > Already for the hypergeometric type there would be enough, > where Maple and other CAS will fail to 'simplify' missing > special identities. > For example convert 3F1 to MeijerG and try to return ... > (which gives identities ...) AV> For me it is a bit senseless to ask for simplifications > AV> for arbitrary MeijerG, since that class is certainly too > AV> large. For arbitrary MeijerG, yes. Still, what we post (and have in store) is not arbitrary. > MeijerG should be used mainly (if not exclusively) for internal tasks and not for challenges. There are so many _beautiful_ math problems ... === Subject: Re: -- approach to solving a second order differential equation (nonseries? <26983706.1217593241699.JavaMail.jakarta@nitrogen.mathforum.org>, > How might I go about solving the following second > order > differential equation xy'' + y' + xy = 0 > Is there a solution other than a series solution > that will help me solve this equation ? > http://en.wikipedia.org/wiki/Bessel_function Best wishes > Torsten. Bessel function J_0 is one solution. === Subject: Re: zeta stuff Do you know of an expression for the sum side of Product[1/(1-(1/Composite[n]^2))]= Sum = 12/Pi^2 ? n=2,3,4,... === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Note that I refer to a extra-set-theoretical reality here. If you like, > I consider a proof to be a linguistic entity . And your /sequences/ are > just mathematical means to analyze this entities. This would justify > the identification of a one-line proof with the formula making up > this proof.) > [...] I'm using 'proof' here in such a 'technical sense' and not in its > other non-technical sense as 'an argument or discourse that provides > convincing grounds for believing a proposition or an argument or > discourse that provides convincing grounds for believing that there > exists a formal mathematical proof of a certain formula'. >I was in fact using proof vs axiom mostly in the non-technical >sense. What I meant to say was exactly that stating a result as an >axiom cannot provide convincing grounds in the face of an argument >that states the opposite result from the basic properties of the >natural numbers plus induction. Possibly so. That has no relevance here, since you have not > given any such argument: The things you say are almost > always incomprehesibly garbled, and when we try to extract > what you really mean we find an error . Pathetic as usual. In spite of your reiterated bull and personal insults, your math is now provably wrong even by an absolute beginner like me! Go to hell, retard, the game is over. -LV >-LV > Same with me. The realistic view of a proof (as a certain linguistic entity) > does not contradict the notion of proof in this formal sense (due to Frege). > Anyway, I enjoyed our little exchange of ideas/thoughts. > B. > -- > For every line of Cantor's list it is true that this line does not > æcontain the diagonal number. æNevertheless the diagonal number may > æbe in the infinite list. (WM, sci.logic) David C. Ullrich Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result > Note that I refer to a extra-set-theoretical reality here. If you > like, > I consider a proof to be a _linguistic entity_. And your /sequences/ > are > just mathematical means to analyze this entities. This would > _justify_ > the identification of a one-line proof with the formula making up > this proof.) > [...] I'm using 'proof' here in such a 'technical sense' and not in > its > other non-technical sense as 'an argument or discourse that provides > convincing grounds for believing a proposition or an argument or > discourse that provides convincing grounds for believing that there > exists a formal mathematical proof of a certain formula'. >I was in fact using proof vs axiom mostly in the non-technical >sense. What I meant to say was exactly that stating a result as an >axiom cannot provide convincing grounds in the face of an argument >that states the opposite result from the basic properties of the >natural numbers plus induction. > Possibly so. That has no relevance here, since you have not > _given_ any such argument: The things you say are almost > always incomprehesibly garbled, and when we try to extract > what you really mean we find an _error_. > Pathetic as usual. In spite of your reiterated bull and personal insults, You don't realize how funny it is for _you_ to be complaining about personal insults? The other day I said something about the math, not about you, and your reply was OFF. Here I say something about the math, not about you, and you say Go to hell, retard. Exactly what personal insults are you talking about? >your math > is now provably wrong even by an absolute beginner like me! Really? I haven't seen you say _anything_ in reply to the point I made several times. Well, nothing about the math - saying OFF doesn't really count as proving what I said is wrong. By the way, when you call it _my_ math that's very funny. It's the same as the math of every mathematician on the planet. > Go to hell, retard, the game is over. Keep it up - this is the best way to convince people you're right. > -LV >-LV > Same with me. The realistic view of a proof (as a certain linguistic > entity) > does not contradict the notion of proof in this formal sense (due to > Frege). > Anyway, I enjoyed our little exchange of ideas/thoughts. > B. > -- > For every line of Cantor's list it is true that this line does not > æcontain the diagonal number. æNevertheless the diagonal number may > æbe in the infinite list. (WM, sci.logic) > David C. Ullrich > Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) -- David C. Ullrich === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > your math > is now provably wrong even by an absolute beginner like me! So far, the only suggestion of something from you that might be a proof is to take as an axiom some principle or another that contradicts mere intuitionistic logic combined with the axiom schema of separation. That is fine onto itself; we freely admit that ZFC is contradicted by any such principle. However, that leaves us with not an inkling as to how you would derive the mathematical theorems for calculus, or even for arithmetic, or even for any interesting mathematics at all. MoeBlee === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > your math > is now provably wrong even by an absolute beginner like me! So far, the only suggestion of something from you that might be a > proof is to take as an axiom some principle or another that > contradicts mere intuitionistic logic combined with the axiom schema > of separation. That is fine onto itself; we freely admit that ZFC is > contradicted by any such principle. However, that leaves us with not > an inkling as to how you would derive the mathematical theorems for > calculus, or even for arithmetic, or even for any interesting > mathematics at all. MoeBlee Yes, but you can understand that that job cannot be done in this context. There is just too much noise. -LV === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result > your math > is now provably wrong even by an absolute beginner like me! > So far, the only suggestion of something from you that might be a > proof is to take as an axiom some principle or another that > contradicts mere intuitionistic logic combined with the axiom schema > of separation. That is fine onto itself; we freely admit that ZFC is > contradicted by any such principle. However, that leaves us with not > an inkling as to how you would derive the mathematical theorems for > calculus, or even for arithmetic, or even for any interesting > mathematics at all. > MoeBlee Yes, but you can understand that that job cannot be done in this > context. There is just too much noise. Not a very good excuse. I have no problem posting correct mathematics here in spite of the noise. All you have to do is post a correct proof of what you assert - there's no way that the other posts can prevent you from doing that. Of course, the fact that what you're trying to prove is _false_ is going to make it hard. > -LV -- David C. Ullrich === Subject: Re: card(P(N)) = card(N), rev 2: a preliminary result posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > your math > is now provably wrong even by an absolute beginner like me! > So far, the only suggestion of something from you that might be a > proof is to take as an axiom some principle or another that > contradicts mere intuitionistic logic combined with the axiom schema > of separation. That is fine onto itself; we freely admit that ZFC is > contradicted by any such principle. However, that leaves us with not > an inkling as to how you would derive the mathematical theorems for > calculus, or even for arithmetic, or even for any interesting > mathematics at all. > Yes, but you can understand that that job cannot be done in this > context. There is just too much noise. That classical mathematics goes on its merry way is no impediment for you to devise whatever theory you like. Indeed, you could learn from classical mathematics (and many other alternatives to classical mathematics that are part of the literature of the subject) how theories are put together, even if then your twist is to put together a quite different theory. MoeBlee === Subject: Alternate proof of Th7.13 in Rudin's PMA? posting-account=dUaYtQoAAABjEW06yhX-7UQnMbSgN7mB rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Hi folks, I'm self-studying my way through baby Rudin. I've reached theorem 7.13 and have come up with a proof. However, it doesn't use one of the givens so I suspect that it's incorrect. I was hoping someone here could find the flaw in my reasoning. (Please let me know if this is the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and a) {f_n} is a sequence of continuous functions on K, b) {f_n} converges pointwise to a continuous function f on K, c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, we can assign to each point x an integer M(x) such that |f_{n>=M(x)} (x) - f(x)| < epsilon. We have to show that an M can be found that satisfies this condition for all x in K. Due to the continuity of the f_n and of f, we can define a neighbourhood N(x) around each point x such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). These neighbourhoods form an open cover of K. Since K is compact, a finite number of the N(x) covers K, centered at x_i = {x_1,x_2,...x_P}. We can then set M = max M(x_i). Then |f_{n>=M}(x)-f(x)} < epsilon for all x in K, and uniform convergence is proved. As you can see, I haven't used (c) anywhere. Where's my mistake? Sina === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? >I'm self-studying my way through baby Rudin. I've reached theorem 7.13 >and have come up with a proof. However, it doesn't use one of the >givens so I suspect that it's incorrect. I was hoping someone here >could find the flaw in my reasoning. (Please let me know if this is >the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and >a) {f_n} is a sequence of continuous functions on K, >b) {f_n} converges pointwise to a continuous function f on K, >c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... >Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, >we can assign to each point x an integer M(x) such that |f_{n>=M(x)} >(x) - f(x)| < epsilon. We have to show that an M can be found that >satisfies this condition for all x in K. Due to the continuity of the >f_n and of f, we can define a neighbourhood N(x) around each point x >such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). These >neighbourhoods form an open cover of K. Since K is compact, a finite >number of the N(x) covers K, centered at x_i = {x_1,x_2,...x_P}. We >can then set M = max M(x_i). Then |f_{n>=M}(x)-f(x)} < epsilon for >all x in K, and uniform convergence is proved. As you can see, I haven't used (c) anywhere. Where's my mistake? Consider the following counterexample without c. For x in [0,1] and n >= 1, let f_n(x) = 4nx(1-x)/(1+(n-1)x)^2. Note that for all x in [0,1], lim f (x) = 0 n->oo n Therefore, the f_n satisfy a and b, yet 1 f ( --- ) = 1 n n+1 Thus, the f_n do not converge uniformly. Rob Johnson take out the trash before replying === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? > Hi folks, I'm self-studying my way through baby Rudin. I've reached theorem 7.13 > and have come up with a proof. However, it doesn't use one of the > givens so I suspect that it's incorrect. I was hoping someone here > could find the flaw in my reasoning. (Please let me know if this is > the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and > a) {f_n} is a sequence of continuous functions on K, > b) {f_n} converges pointwise to a continuous function f on K, > c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... > Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, > we can assign to each point x an integer M(x) such that |f_{n>=M(x)} > (x) - f(x)| < epsilon. We have to show that an M can be found that > satisfies this condition for all x in K. Due to the continuity of the > f_n and of f, we can define a neighbourhood N(x) around each point x > such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). The mistake is here. For each n >= M(x), there is a neighborhood U_n(x) that works for f_n - f, but the intersection of all these U_n(x)'s is what you need, and there's no reason for the intersection to be open. > These > neighbourhoods form an open cover of K. Since K is compact, a finite > number of the N(x) covers K, centered at x_i = {x_1,x_2,...x_P}. We > can then set M = max M(x_i). Then |f_{n>=M}(x)-f(x)} < epsilon for > all x in K, and uniform convergence is proved. As you can see, I haven't used (c) anywhere. Where's my mistake? > Sina === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? , Hi folks, I'm self-studying my way through baby Rudin. I've reached theorem 7.13 > and have come up with a proof. However, it doesn't use one of the > givens so I suspect that it's incorrect. I was hoping someone here > could find the flaw in my reasoning. (Please let me know if this is > the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and > a) {f_n} is a sequence of continuous functions on K, > b) {f_n} converges pointwise to a continuous function f on K, > c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... > Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, > we can assign to each point x an integer M(x) such that |f_{n>=M(x)} > (x) - f(x)| < epsilon. We have to show that an M can be found that > satisfies this condition for all x in K. Due to the continuity of the > f_n and of f, we can define a neighbourhood N(x) around each point x > such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). The mistake is here. For each n >= M(x), there is a neighborhood > U_n(x) that works for f_n - f, but the intersection of all these > U_n(x)'s is what you need, and there's no reason for the intersection > to be open. Is this acceptable? M_n = max|f - f_n| >= 0 exists, because f and f_n are continuous on a compact set. M_n -> 0, because f_n -> f everywhere on K. M_n >= M_{n+1} by c). For eps > 0 exists N such that for all n > N, we have M_n < eps. This last statement is equivalent to the desired result. -- Michael Press === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? > , > Hi folks, I'm self-studying my way through baby Rudin. I've reached theorem 7.13 > and have come up with a proof. However, it doesn't use one of the > givens so I suspect that it's incorrect. I was hoping someone here > could find the flaw in my reasoning. (Please let me know if this is > the wrong group to be posting questions like this). Here's the theorem: Suppose K is compact, and > a) {f_n} is a sequence of continuous functions on K, > b) {f_n} converges pointwise to a continuous function f on K, > c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... > Then f_n => f uniformly on K. My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, > we can assign to each point x an integer M(x) such that |f_{n>=M(x)} > (x) - f(x)| < epsilon. We have to show that an M can be found that > satisfies this condition for all x in K. Due to the continuity of the > f_n and of f, we can define a neighbourhood N(x) around each point x > such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). The mistake is here. For each n >= M(x), there is a neighborhood > U_n(x) that works for f_n - f, but the intersection of all these > U_n(x)'s is what you need, and there's no reason for the intersection > to be open. Is this acceptable? M_n = max|f - f_n| >= 0 exists, because > f and f_n are continuous on a compact set. M_n -> 0, because f_n -> f everywhere on K. Two problems with this. First, you could set M_n = sup|f - f_n|, which exists without any continuity assumptions. Second, M_n -> 0 is equivalent to f_n -> f uniformly (exercise: prove this). So you are obtaining uniform convergence without using (c), whoops. > M_n >= M_{n+1} by c). For eps > 0 exists N such that for all n > N, we have M_n < eps. > This last statement is equivalent to the desired result. === Subject: Re: Alternate proof of Th7.13 in Rudin's PMA? >, > Hi folks, > I'm self-studying my way through baby Rudin. I've reached theorem 7.13 > and have come up with a proof. However, it doesn't use one of the > givens so I suspect that it's incorrect. I was hoping someone here > could find the flaw in my reasoning. (Please let me know if this is > the wrong group to be posting questions like this). > Here's the theorem: Suppose K is compact, and > a) {f_n} is a sequence of continuous functions on K, > b) {f_n} converges pointwise to a continuous function f on K, > c) f_n(x) >= f_{n+1}(x) for all x in K, n = 1,2,3,... > Then f_n => f uniformly on K. > My proof: Pick some epsilon > 0. Since {f_n} converges pointwise to f, > we can assign to each point x an integer M(x) such that |f_{n>=M(x)} > (x) - f(x)| < epsilon. We have to show that an M can be found that > satisfies this condition for all x in K. Due to the continuity of the > f_n and of f, we can define a neighbourhood N(x) around each point x > such that |f_{n>=M(x)} (y) - f(y)| < epsilon for all y in N(x). > The mistake is here. For each n >= M(x), there is a neighborhood > U_n(x) that works for f_n - f, but the intersection of all these > U_n(x)'s is what you need, and there's no reason for the intersection > to be open. Is this acceptable? M_n = max|f - f_n| >= 0 exists, because >f and f_n are continuous on a compact set. M_n -> 0, because f_n -> f everywhere on K. M_n >= M_{n+1} by c). For eps > 0 exists N such that for all n > N, we have M_n < eps. >This last statement is equivalent to the desired result. Just because f_n -> f everywhere does not mean that M_n -> 0. See the counterexample that I gave earlier. That is, 4nx(1-x) f (x) = ------------ n (1+(n-1)x)^2 As before, for all x in [0,1], lim f (x) = 0 n->oo n Yet, the hump just moves toward 0 1 f ( --- ) = 1 n n+1 So M_n = 1 for all n. Another counterexample is n n f (x) = 4 x (1-x ) n Here we get that the hump moves toward 1 -1/n f ( 2 ) = 1 and again, M_n = 1 for all n. Rob Johnson take out the trash before replying === Subject: Re: Graph Theory: Cutting a Graph into Two in an Artificial Chemistry I don't have time to do this properly, but the general question for general graphs has the feel of NP-Complete. However, your situation will have many many more constraints on the graph. For example, you won't have 6000 vertices, all mutually connected. It feels to me like the graphs you are interested in will have only limited sorts of minimal cycles, and each atom will have small degree. In that case there are going to be clever branch-and-bound searches that should work. I don't, however, know of anything off the shelf. === Subject: Re: Graph Theory: Cutting a Graph into Two in an Artificial Chemistry <6fer50Fbb9lnU1@mid.individual.net> posting-account=oTDIagkAAACTxHurtPutBWvNQS8ZCNO9 Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Jul 31, 3:03 pm, Chris Gordon-Smith It's unclear what you mean by cut - there is more > than one definition, and for each definition, someone > thinks it's obviously the right one. Let me put the question a different way. Here is a brute force way to find > what I am looking for:- 1) I start with a graph that has a single connected component (such that > any node can be reached from any other node by traversing a series of > edges) 2) I then divide the graph's nodes into two mutually exclusive groups and > remove all of the edges between nodes in different groups, leaving the > other edges in place. If I do this and the result is exactly two connected > components, then I count that as a valid way that my original graph > (molecule) could split. If the result is more than two components, then I > discard the result as invalid. I would count a single node on its own as a > connected component. 3) I repeat (2) until I have tried all of the possible ways of dividing the > nodes of the original graph into two mutually exclusive groups 4) The set of all 'valid' splits from (2) is the answer I want. Is there an efficient way to get this set of 'valid' splits? > Since no one else has responded, no solution, but some thoughts... First, it sounds like you don't want to just count these splits, you want to obtain a list of pairs of disjoint sub-graphs which cover the original set of nodes. Second, your problem will become quite a bit more difficult if you want only a list of non-isomorphic such pairs. For eample, for ethane, one could argue that there are only two different such pairings: (CH3, CH3) and (CH3CH2, H); even though your algorithm above would naively count 7 distinct such pairings. And determining whether two graphs are isomorphic is (generally) not so easy. At any rate, there are relatively quick algorithms for determining whether two nodes a and b are members of the same component of a graph (for a starting point see: http://en.wikipedia.org/wiki/Disjoint-set_data_structure ). This suggests that you can try removing edges E_i = (n_a, n_b) one at a time, then check for connectivity of n_a and n_b. The trick would be to bound your searches so that once you have determined that some set of edges (E_1, E_2, .., E_k) causes a split, you don't re-examine sets of edge removals which contain that subset a second time; e.g., by considering the directed graph of all subsets of edges, and then traversing /that/ tree, stopping at any depth that produces a split. This approach can probably be modified so that if you know that some set of edges are are essentially the same (e.g., the C-H bonds in ethane), then you don't revisit removing a single C-H edge 6 times. It would probably be useful to know the size and complexity of the graphs you are working with. Trivially, if your graphs are trees, then the number of such splits is the number of edges. So if they are sufficiently small and generally tree-like except for a few cycles, it may be that this will affect the average expected complexity (as opposed to the worst-case complexity); and perhaps even a brute force search will be sufficient to proceed. Anyway, post if you find a nice algorithm for this... === Subject: Re: Graph Theory: Cutting a Graph into Two in an Artificial Chemistry <25198735.1217459003945.JavaMail.jakarta@nitrogen.mathforum.org> <6fer50Fbb9lnU1@mid.individual.net > It's unclear what you mean by cut - there is more than one > definition, and for each definition, someone thinks it's obviously > the right one. > Let me put the question a different way. Here is a brute force way to > find what I am looking for:- > 1) I start with a graph that has a single connected component (such > that any node can be reached from any other node by traversing a series > of edges) > 2) I then divide the graph's nodes into two mutually exclusive groups > and remove all of the edges between nodes in different groups, leaving > the other edges in place. If I do this and the result is exactly two > connected components, then I count that as a valid way that my original > graph (molecule) could split. If the result is more than two > components, then I discard the result as invalid. I would count a > single node on its own as a connected component. > 3) I repeat (2) until I have tried all of the possible ways of > dividing the nodes of the original graph into two mutually exclusive > groups > 4) The set of all 'valid' splits from (2) is the answer I want. > Is there an efficient way to get this set of 'valid' splits? > Since no one else has responded, no solution, but some thoughts... First, it sounds like you don't want to just count these splits, you > want to obtain a list of pairs of disjoint sub-graphs which cover the > original set of nodes. Second, your problem will become quite a bit more difficult if you want > only a list of non-isomorphic such pairs. For eample, for ethane, one > could argue that there are only two different such pairings: (CH3, > CH3) and (CH3CH2, H); even though your algorithm above would naively > count 7 distinct such pairings. And determining whether two graphs are > isomorphic is (generally) not so easy. It may be OK to count all of the isomorphic cases, because each results from the breaking of a different bond and I want my model to reflect the likelihood of each outcome. Other things being equal, if one bond is broken at random then the likeliest outcome is CH3CH2. > At any rate, there are relatively quick algorithms for determining > whether two nodes a and b are members of the same component of a graph > (for a starting point see: http://en.wikipedia.org/wiki/Disjoint-set_data_structure > may well end up using this. > ). This suggests that you can try removing edges E_i = (n_a, n_b) one at a > time, then check for connectivity of n_a and n_b. The trick would be to > bound your searches so that once you have determined that some set of > edges (E_1, E_2, .., E_k) causes a split, you don't re-examine sets of > edge removals which contain that subset a second time; e.g., by > considering the directed graph of all subsets of edges, and then > traversing /that/ tree, stopping at any depth that produces a split. This approach can probably be modified so that if you know that some set > of edges are are essentially the same (e.g., the C-H bonds in ethane), > then you don't revisit removing a single C-H edge 6 times. It would probably be useful to know the size and complexity of the > graphs you are working with. Trivially, if your graphs are trees, then > the number of such splits is the number of edges. So if they are > sufficiently small and generally tree-like except for a few cycles, it > may be that this will affect the average expected complexity (as opposed > to the worst-case complexity); and perhaps even a brute force search > will be sufficient to proceed. I don't want to limit the graphs to being trees. Also, the graphs could get quite large. In real chemistry molecules can have hundreds or even thousands of atoms. I don't know whether this would make the searches too large. If so, one possibility might be to limit the search so that I find a 'representative' set of splits rather than all of the possible splits. Chris www.simsoup.info Anyway, post if you find a nice algorithm for this... > === Subject: Information that can free you financially posting-account=ca3sVwoAAABbBgOuHm48TMnvyz3CC7Ze Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) Improve Your Financial Situation NOW! No matter what your personal financial goals may be ($4K to $20K a month or more), we have the training, resources, and amazing team support to help you achieve these goals, quickly and easily! http://www.mygoldplan.com/ebusaf/ === Subject: Re: help with deducing general eqn of limacon?! posting-account=x9DlGAoAAAAvZTRvRYG8JjmPJCeyCza7 SV1),gzip(gfe),gzip(gfe) æFormulation is easier with respect to the invariant origin. æTo derive further from center of rolling circle, you want to add > æextra (p,q) or ( c cos(gam, c sin(gam) ), where gam is angle of > tracer ptæending crank to the horizontal. æLets assume the rolling circle has radius n times that of fixed > æcircle, so has angle subtended at its center is n times less, as > arc length is same. gam = t + n t - pi. æhttp://i36.tinypic.com/5p0qq1.jpg æ(picture modified from from epicycloids wikipedia where n < 1 ) æSo the first and second coordinate increments with respect to > originæas fixed circle on cranks or vector lengths (c + c n) and cn > ærespectively to generate epicycloid are: æc (1+ n) { cos(t), sin(t) } æ+ c { cos(gam), sin(gam) } æIn Mathematica notation æParametricPlot[ (n+1)c {Cos[t],Sin[t]} - n c{Cos[(1/n+1)t],Sin[(1/n > æ+1)t] }, {t,0, æ2 n Pi }, AspectRatio->Automatic] æConversion into polar coordinates: radius = Sqrt[1 + 2*n*(1 + n) - > æ2*n*(1 + n)*Cos[t/n]] and æpolar angle = ArcTan[ U Cos[t] - Cos[U], > æU Sin[t] - Sin[U] ] ; U = (1 + 1/n)*t Hi Narasimham, I am very grateful for your help, but how can I derive the simpler formula r = b + a cos .9b for the limacon?? Sorry, but I'm not getting this! Michael === Subject: Re: the adjoint Any teacher who, when asked What does > this mean?, answers by parroting the > book definition, is a CRUMMY TEACHER. > Is that what you think about my previous writings? Perhaps it would do you good if you were to learn some book definitions yourself. It also helps your intuition, you know. Ignored from now on. Sebastiaan. === Subject: rumpled surfaces? posting-account=twLK8gkAAADX2eq14ORonqtyAuVoEcNf Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) In familiar Euclidean plane geometry, a set of angles can be placed edge-to-edge, with their vertices on a common point. When the sum of the angle measures equals exactly 360 degrees, the plane region about the point is filled. If the sum is positive and less than 360 degrees, the angles can still be placed edge-to-edge, and form a vertex of a solid figure. For example, the three right angles at the corner of a cube sum to 270 degrees. If the sum is greater than 360 degrees, the surface becomes rumpled (for lack of the correct term). Try fitting together ten 60- degree angles made of paper, and you'll see what I mean. The surface described in the paragraph above is rumpled about one point. Is it possible for a surface to be rumpled about every point? If so, what special geometries or topologies are required? What is the correct terminology for this concept, so that I can look it up for myself? Ted Shoemaker === Subject: Re: rumpled surfaces? > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? > Yes. A saddle shape has everywhere negative curvature and is hence is rumpled everywhere. > What is the correct terminology for this concept, so that I can look > it up for myself? > Google negative curvature surface, or hyperbolic geometry. Many/most of the links assume the surface has negative curvature everywhere, and not just at specific points. Ted Shoemaker === Subject: Re: rumpled surfaces? posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? What is the correct terminology for this concept, so that I can look > it up for myself? > Ted Shoemaker It has to do with double curvature of the originating smooth surface, whether normal curvature in two principal perpendicular directions is on the same side or on opposite sides. If a smooth surface of positive Gauss curvature(sphere, ellipsoid,paraboloid etc., like a football) is now discretized by triangulation, say more than 4 triangles meet at such a vertex, the sum of angles is less than 180 degrees.It is a synclastic surface, belongs to elliptic geometry. If a smooth surface of negative Gauss curvature(pseudosphere,hyperbolic paraboloid, catenoid etc.,like a Pringles potato chip) is now discretized by triangulation, say more than 4 triangles meet at a vertex, the sum of angles is more than 180 degrees.It is an anticlastic surface, belongs to hyperbolic geometry, all the points are saddle points. The relationship between rotation around a contour and the solid angle ( called integral curvature in steridians ) subtended by normals is simply stated in Gauss-Bonnet theorem.It can be stated for smooth as well as discretized/ triangulated/tesselated surfaces.. Hope it helps. Narasimham === Subject: Re: rumpled surfaces? posting-account=twLK8gkAAADX2eq14ORonqtyAuVoEcNf Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) triangle is less than 180 degrees. Yes, but that's not what I'm looking for. Just to be sure, I did look up Gaussian curvature and a few related topics; I couldn't see that they were the topic I wanted. Either I'm not understanding you, or you're not understanding me. Let's try again. CASE 1 Take a sheet of paper, and cut it into angles (I didn't say triangles) of various measures. Assemble the angles so that their vertices share a common point, and the angles are adjacent but not overlapping each other. If you do this on a plane, you can exactly fill the space around a point with 360 degrees of angles. If you do this on a sphere, you STILL get 360 degrees. If you doubt that, look at the north pole on a globe, and start counting longitude lines. CASE 2 Assemble your angles as before, except use a total of less than 360 degrees. Tape the adjacent edges together. Now tape the two outside edges together. You will have formed part of a cone or a prism. Noneuclidean geometry not required. (Not yet, anyway.) CASE 3 Assemble your angles again, this time making the angle measure total more than 360 degrees. Tape the adjacent edges together. Tape the two edges at the extremes together. You now have, for lack of the right term, a wrinkled skirt, with a single vertex. THE QUESTIONS, RESTATED 1. Is it possible to have a surface such that every point is surrounded by exactly 360 degrees of surface? (YES. Examples are the plane, sphere, etc.) 2. Is it possible to have a surface such that every point is the vertex of a wrinkled skirt? 3. Is it possible to have a surface such that every point is surrounded by less than 360 degrees of surface? 4. What branch of math deals with this? 5. What is the correct terminology for me to look up? Ted Shoemaker > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. > If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. > If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. > The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? > What is the correct terminology for this concept, so that I can look > it up for myself? > Ted Shoemaker It has to do with double curvature of the originating smooth surface, > whether normal curvature in two principal perpendicular directions is > on the same side or on opposite sides. If a smooth surface of positive Gauss curvature(sphere, > ellipsoid,paraboloid etc., like a football) is now discretized by > triangulation, say more than 4 triangles meet at such a vertex, the > sum of angles is less than 180 degrees.It is a synclastic surface, > belongs to elliptic geometry. If a smooth surface of negative Gauss > curvature(pseudosphere,hyperbolic paraboloid, catenoid etc.,like a > Pringles potato chip) is now discretized by triangulation, say more > than 4 triangles meet at a vertex, the sum of angles is more than 180 > degrees.It is an anticlastic surface, belongs to hyperbolic geometry, > all the points are saddle points. The relationship between rotation around a contour and the solid angle > ( called integral curvature in steridians ) subtended by normals is > simply stated in Gauss-Bonnet theorem.It can be stated for smooth as > well as discretized/ triangulated/tesselated surfaces.. Hope it helps. Narasimham === Subject: Re: rumpled surfaces? posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > triangle is less than 180 degrees. Yes, but that's not what I'm > looking for. Just to be sure, I did look up Gaussian curvature and a > few related topics; I couldn't see that they were the topic I wanted. Either I'm not understanding you, or you're not understanding me. > Let's try again. CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. If you doubt that, look > at the north pole on a globe, and start counting longitude lines. CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. Noneuclidean geometry not required. (Not yet, anyway.) CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? > 4. What branch of math deals with this? > 5. What is the correct terminology for me to look up? > Ted Shoemaker > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. > If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. > If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. > The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? > What is the correct terminology for this concept, so that I can look > it up for myself? > Ted Shoemaker > It has to do with double curvature of the originating smooth surface, > whether normal curvature in two principal perpendicular directions is > on the same side or on opposite sides. > If a smooth surface of positive Gauss curvature(sphere, > ellipsoid,paraboloid etc., like a football) is now discretized by > triangulation, say more than 4 triangles meet at such a vertex, the > sum of angles is less than 180 degrees.It is a synclastic surface, > belongs to elliptic geometry. > If a smooth surface of negative Gauss > curvature(pseudosphere,hyperbolic paraboloid, catenoid etc.,like a > Pringles potato chip) is now discretized by triangulation, say more > than 4 triangles meet at a vertex, the sum of angles is more than 180 > degrees.It is an anticlastic surface, belongs to hyperbolic geometry, > all the points are saddle points. > The relationship between rotation around a contour and the solid angle > ( called integral curvature in steridians ) subtended by normals is > simply stated in Gauss-Bonnet theorem.It can be stated for smooth as > well as discretized/ triangulated/tesselated surfaces.. Hope it helps. > Narasimham For a smooth surface the angle sum between tangents at any point is locally always 360 degrees. For polyhedrons all the three cases are possible.So at a polyhedral vertex,three cases: On the plane, or any point of developable surface (exclude cone/ pyramid vertex singularity etc) .. sum = 360, K = 0. On elliptic convex or concave surfaces of discretized elliptic geometry, cone vertex, pyramid vertex, .. sum < 360, K > 0. On warped surfaces discretized saddle points of hyperbolic geometry,.. sum > 360, K < 0. Mathematica imaging may be helpful in seeing polyhedral vertices. But there is a hole, to avoid crowded convergent lines: << RealTime3D` Pringle =u { Cos[v] , Sin[v], Sin[2 v]/2} PR=ParametricPlot3D[Pringle,{u, 0, 1},{v,0,2 Pi},PlotPoints->{6,30} ] cone = u { Cos[v] , Sin[v], 1} CO=ParametricPlot3D[cone,{u, 0, 1},{v,0,2 Pi},PlotPoints->{6,30} ] disc = u { Cos[v] , Sin[v], 0} DI=ParametricPlot3D[disc,{u, 0, 1},{v,0,2 Pi},PlotPoints->{6,30} ] Show[PR,CO,DI] The depiction of surfaces using geodesic polar coordinates in differential geometry I think would be very useful in your further studies. Narasimham === Subject: Re: rumpled surfaces? posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) A smooth surface can be discretized and a discretized surface(rumpled or many faceted like the exterior of a diamond)can be smoothed. After discretization of a smooth surface lot of edges(E), vertices (V) and faces (F) develop, and their numbers obey the Euler law: V + F = E + 2. The sum of angles at a spherical vertex is less than 360 degrees. (Sorry, not 180 degrees as I typed wrongly).Please look at a model and faces e.g., of an icosahedron or dodecahedron or tetrahedron from among the set of ideal Platonic solids. The total sums up respectively to 60 X 5 = 300, 108 X 3 = 324 and 60 X 3 = 180 all less than 360 degrees. > triangle is less than 180 degrees. No no, by spherical triangle is meant a curvilinear triangle whose sides are geodesic arcs of great circles, the sum of three internal triangles is more than 180 degrees: Neither I referred to it nor is your query about it. You are asking about situation at any vertex of a polyhedron. Compared to a smooth differentiable surface, for a polyhedron normal curvature has sudden jumps at the edges. As the number of edges goes to infinity and area of faces created by a certain subdivision of curves passing through the vertices tends to zero, the polyhedron can be made to tend to a smooth surface, but the two are quite different. > Yes, but that's not what I'm > looking for. Just to be sure, I did look up Gaussian curvature and a > few related topics; I couldn't see that they were the topic I wanted. After some more study you will see that discretization of positive and negative Gaussian curvature surfaces is your topic. > Either I'm not understanding you, or you're not understanding me. > Let's try again. May be the former, but I shall help in explanations. > CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. > If you doubt that, look at the north pole on a globe, and start counting longitude lines. No, you are not distinguishing between a polyhedron and a smooth sphere. The sum of angles is < 360 degrees in the former, if number of faces -> Infinity, then the sum -> 360 degrees . > CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. you meant a pyramid. > CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. This warped surface that you call a wrinkled skirt can be described as discretized hyperbolic paraboloid with two hilly humps and two descending valleys for the two legs of a horse rider sitting on the horse saddle. Or even a monkey saddle , where there are three humps to go in between them are monkey's two legs and a tail. The monkey sits on it, bottom contacting a central (singular, but don't bother for now) point. > THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) plane OK, but not sphere. You can include cylinder, cone. > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? Yes. Look at the surface of a helicoid for instance.It has a simple parametrization. x= u cos(v), y= u sin(v), z = v. Two lines cut at any point, angle sum is 360 for this smooth surface. But do not make the same error as you did at the sphere north pole, because when you join the vertices forming a polytope, sum of angles at any vertex is more than 360 degrees. > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? Yes, e.g., at any vertex of Platonic solid, or any convex surface like ellipsoid suitably triangulated (by Veronoi, Delaunay etc. on convex hulls)and when the vertices are joined not by lines on the surface but by straight lines through the air. > 4. What branch of math deals with this? Differential geometry. Beware of going too deep into topology at this stage. > 5. What is the correct terminology for me to look up? Gauss curvature,Gauss-Bonnet theorem linking differential geometry and topology, parametrization of surfaces,triangulation,discretization. I suggest you build models of Platonic solids and also some from hyperbolic geometry like hyperbolic paraboloid or a catenoid or a helicoid using cardboard cuttings that may provide insight not just for what is happening at each point but how the entire surface is building up.I also suggest going through the book by David Hilbert and Cohn Vossen: Geometry and Imagination. Narasimham > Ted Shoemaker > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. > If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. > If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. > The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? > What is the correct terminology for this concept, so that I can look > it up for myself? > Ted Shoemaker > It has to do with double curvature of the originating smooth surface, > whether normal curvature in two principal perpendicular directions is > on the same side or on opposite sides. > If a smooth surface of positive Gauss curvature(sphere, > ellipsoid,paraboloid etc., like a football) is now discretized by > triangulation, say more than 4 triangles meet at such a vertex, the > sum of angles is less than 180 degrees.It is a synclastic surface, > belongs to elliptic geometry. > If a smooth surface of negative Gauss > curvature(pseudosphere,hyperbolic paraboloid, catenoid etc.,like a > Pringles potato chip) is now discretized by triangulation, say more > than 4 triangles meet at a vertex, the sum of angles is more than 180 > degrees.It is an anticlastic surface, belongs to hyperbolic geometry, > all the points are saddle points. > The relationship between rotation around a contour and the solid angle > ( called integral curvature in steridians ) subtended by normals is > simply stated in Gauss-Bonnet theorem.It can be stated for smooth as > well as discretized/ triangulated/tesselated surfaces.. Hope it helps. > Narasimham === Subject: Re: rumpled surfaces? May be the former, but I shall help in explanations. > CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. > If you doubt that, look at the north pole on a globe, and start counting > longitude lines. No, you are not distinguishing between a polyhedron and a smooth > sphere. The sum of angles is < 360 degrees in the former, if number > of faces -> Infinity, then the sum -> 360 degrees . > The sum of the angles around any point on a sphere is 360. What he says is completely true. He isn't asking about polyhedra in his question, and nor do I think its relevant to his problem. > CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. you meant a pyramid. No, I think he means a cone. His assembly process will result in a circle (if all the strips are the same length) pit a pie shaped degment removed - roll this up as he describes, and a cone is formed. > CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. This warped surface that you call a wrinkled skirt can be described as > discretized hyperbolic paraboloid with two hilly humps and two > descending valleys for the two legs of a horse rider sitting on the > horse saddle. Or even a monkey saddle , where there are three humps > to go in between them are monkey's two legs and a tail. The monkey > sits on it, bottom contacting a central (singular, but don't bother > for now) point. > Simpler than that. His wrinled shirt is a surface where the sum of the internal angles of a triangle is less than 360. Any hyperbolic surface does that. > THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) plane OK, but not sphere. You can include cylinder, cone. > plane OK, sphere OK, endless cylinder OK, cone not OK. I score you 2/4, which is no better than chance. > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? Yes. Look at the surface of a helicoid for instance.It has a simple > parametrization. x= u cos(v), y= u sin(v), z = v. Two lines cut at any > point, angle sum is 360 for this smooth surface. But do not make the > same error as you did at the sphere north pole, because when you join > the vertices forming a polytope, sum of angles at any vertex is more > than 360 degrees. > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? Yes, e.g., at any vertex of Platonic solid, or any convex surface like > ellipsoid suitably triangulated (by Veronoi, Delaunay etc. on convex > hulls)and when the vertices are joined not by lines on the surface but > by straight lines through the air. > He said every point. On Platonic surfaces, the ONLY points where the sum of the angles is less 360 are the finite number of points at the vertices. > 4. What branch of math deals with this? Differential geometry. Beware of going too deep into topology at this > stage. > 5. What is the correct terminology for me to look up? Gauss curvature,Gauss-Bonnet theorem linking differential geometry and > topology, parametrization of surfaces,triangulation,discretization. I suggest you build models of Platonic solids and also some from > hyperbolic geometry like hyperbolic paraboloid or a catenoid or a > helicoid using cardboard cuttings that may provide insight not just > for what is happening at each point but how the entire surface is > building up.I also suggest going through the book by David Hilbert and > Cohn Vossen: Geometry and Imagination. > I suggested he just Google hyperbolic geometry, which seems to be all he is really talking about. === Subject: Re: rumpled surfaces? <4895a49d$0$1025$afc38c87@news.optusnet.com.au> posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > May be the former, but I shall help in explanations. > CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. > If you doubt that, look at the north pole on a globe, and start counting > longitude lines. > No, you are not distinguishing between a polyhedron and a smooth > sphere. The sum of angles is < 360 degrees in the former, if number > of faces -> Infinity, then the sum -> 360 degrees . The sum of the angles around any point on a sphere is 360. What he says is > completely true. Indeed it is, no disagreement about the tangent plane situation where angle sum is 360 degrees. > He isn't asking about polyhedra in his question, and nor do I think its > relevant to his problem. Did you read OP's first post where he talks about the example 3 squares, 3 edges of a cube? If you assemble straight edges at a point, the point must necessarily be a polyhedron vertex. (The total can be more,equal or less than 360 degrees). A flat development for a curved surface of non-zero K is impossible by Gauss Egregium theorem and a faithful assembly in the tangent plane at the vertex where the straight adjacent edges are brought together all along contacting is also consequently impossible. > CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. > you meant a pyramid. No, I think he means a cone. His assembly process will result in a circle > (if all the strips are the same length) pit a pie shaped degment removed - > roll this up as he describes, and a cone is formed. Yes he meant a cone no doubt about it, next I suggested he meant a pyramid instead of a prism. > CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. > This warped surface that you call a wrinkled skirt can be described as > discretized hyperbolic paraboloid with two hilly humps and two > descending valleys for the two legs of a horse rider sitting on the > horse saddle. Or even a monkey saddle , where there are three humps > to go in between them are monkey's two legs and a tail. The monkey > sits on it, bottom contacting a central (singular, but don't bother > for now) point. Simpler than that. His wrinled shirt is a surface where the sum of the > internal angles of a triangle is less than 360. Any hyperbolic surface does > that. You perhaps mean sum of internal angles of a quadrilateral of 4 geodesic arcs is less than 360 degrees. I imagined a ballerina's mini skirt shape in spin dance. Does not matter, Gauss curvature K should be negative in any example given. > THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) > plane OK, but not sphere. You can include cylinder, cone. plane OK, sphere OK, endless cylinder OK, cone not OK. I mentioned cone the cylinder meaning all points of a cone except at the vertex which is a point of singularity.So cone OK except at cone vertex that should be mentioned which I hope OP would now begin to understand. Every point of a zero K surface is developable and is surrounded by exactly 360 degrees. There are other cases of zero K surfaces like the developable helicoid. > I score you 2/4, which is no better than chance. :) ! > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? > Yes. Look at the surface of a helicoid for instance.It has a simple > parametrization. x= u cos(v), y= u sin(v), z = v. Two lines cut at any > point, angle sum is 360 for this smooth surface. But do not make the > same error as you did at the sphere north pole, because when you join > the vertices forming a polytope, sum of angles at any vertex is more > than 360 degrees. > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? > Yes, e.g., at any vertex of Platonic solid, or any convex surface like > ellipsoid suitably triangulated (by Veronoi, Delaunay etc. on convex > hulls)and when the vertices are joined not by lines on the surface but > by straight lines through the air. He said every point. On Platonic surfaces, the ONLY points where the sum > of the angles is less 360 are the finite number of points at the vertices. Specifically mentioning it first for a vertex only, I proceeded to convex surface as I think that going from known to unknown would be more instructive for a student... without splitting hairs too much. > 4. What branch of math deals with this? > Differential geometry. Beware of going too deep into topology at this > stage. > 5. What is the correct terminology for me to look up? > Gauss curvature,Gauss-Bonnet theorem linking differential geometry and > topology, parametrization of surfaces,triangulation,discretization. > I suggest you build models of Platonic solids and also some from > hyperbolic geometry like hyperbolic paraboloid or a catenoid or a > helicoid using cardboard cuttings that may provide insight not just > for what is happening at each point but how the entire surface is > building up.I also suggest going through the book by David Hilbert and > Cohn Vossen: Geometry and Imagination. I suggested he just Google hyperbolic geometry, which seems to be all he > is really talking about. Differential and Riemannian geometries are more comprehensive for his guidance for non-Euclidean geometries. They include elliptic,hyperbolic and flat parabolic geometries as special cases.Pure hyperbolic geometry will not help him when angle sum is less than 360 degrees. There is also a classical book by Felix Klein on Non-Euclidean geometry (Vorlesungen ueber..) in German, IIRC another by HSM Coxeter etc. Narasimham === Subject: Re: rumpled surfaces? > triangle is less than 180 degrees. Yes, but that's not what I'm > looking for. Just to be sure, I did look up Gaussian curvature and a > few related topics; I couldn't see that they were the topic I wanted. Either I'm not understanding you, or you're not understanding me. > Let's try again. CASE 1 > Take a sheet of paper, and cut it into angles (I didn't say > triangles) of various measures. Assemble the angles so that their > vertices share a common point, and the angles are adjacent but not > overlapping each other. If you do this on a plane, you can exactly > fill the space around a point with 360 degrees of angles. If you do > this on a sphere, you STILL get 360 degrees. If you doubt that, look > at the north pole on a globe, and start counting longitude lines. CASE 2 > Assemble your angles as before, except use a total of less than 360 > degrees. Tape the adjacent edges together. Now tape the two > outside edges together. You will have formed part of a cone or a > prism. Noneuclidean geometry not required. (Not yet, anyway.) CASE 3 > Assemble your angles again, this time making the angle measure total > more than 360 degrees. Tape the adjacent edges together. Tape the > two edges at the extremes together. You now have, for lack of the > right term, a wrinkled skirt, with a single vertex. THE QUESTIONS, RESTATED > 1. Is it possible to have a surface such that every point is > surrounded by exactly 360 degrees of surface? > (YES. Examples are the plane, sphere, etc.) Yes. Examples are the plane, sphere etc. Indeed, any surface that has a curvature defined at every point has this characteristic. As you zoom in on the point, the surface looks locally more and more flat (eg a lake looks flat, even though its surface is actually part of a sphere the size of the earth), so the sum of angles is always 360. > 2. Is it possible to have a surface such that every point is the > vertex of a wrinkled skirt? Yes, a hyperbolic surface. I note that you have introduced a new and undefined term - vertex. I don't actually care what a vertex is in this example, other than it is a point, as every point on a hyperbolic plane has this characteristic you have described. So I don't care which points are also verices. > 3. Is it possible to have a surface such that every point is > surrounded by less than 360 degrees of surface? No, or at least not one which is everywhere differentiable (but see note below). > 4. What branch of math deals with this? Non-euclidean geometry, for starters. Topology for desert. > 5. What is the correct terminology for me to look up? > Non-euclidean geometry. Hyperbolic surfaces. Ted Shoemaker All everywhere differentiable surfaces are locally Euclidean (but see note below), so the sum of the angles around a point is always 360 degrees. Your cone fails because this is not true at the point of the cone, and this is the only place it fails. Zoom in as much as you like, but the area around the point never looks flat. Your use of the word vertex would seem to highlight your confusion. I am assuming that a vertex is a point on the surface which is not differentiable. This is a different issue to whether a triangle on the surface has interior angles adding to 180. As long as the triangle on a cone does not include the vertex, its interior angle will add to 180 degrees; a cone has zero curvature except at the vertex, where the curvature is undefined (or infinite, if you prefer). Note: This is well outside my field of expertise, and I am only guessing that the requirement is that the surface is everywhere differentiable; I think this is a necessary condition but may not be sufficient. There may be pathological functions that are everywhere differentiable but still have points with undefined curvature. === Subject: Re: rumpled surfaces? > In familiar Euclidean plane geometry, a set of angles can be placed > edge-to-edge, with their vertices on a common point. When the sum of > the angle measures equals exactly 360 degrees, the plane region about > the point is filled. If the sum is positive and less than 360 degrees, the angles can still > be placed edge-to-edge, and form a vertex of a solid figure. For > example, the three right angles at the corner of a cube sum to 270 > degrees. If the sum is greater than 360 degrees, the surface becomes > rumpled (for lack of the correct term). Try fitting together ten 60- > degree angles made of paper, and you'll see what I mean. The surface described in the paragraph above is rumpled about one > point. Is it possible for a surface to be rumpled about every > point? If so, what special geometries or topologies are required? What is the correct terminology for this concept, so that I can look > it up for myself? I think you're talking about a surface of negative Gaussian curvature. Look up: Gaussian curvature (e.g. ) -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: ? general norm soln for a model with errors > Hi: Most commonly seen approximated soln to a system of linear eqns A*x = b is the least-squares soln. One can find the solutions either when the system matrix A without and with uncertainty. But that assumes 2-norm being used. How does one solve the same problem when the matrix A is with uncertainty by using more general > norm(s)? > By using more general optimization algorithms. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: ? general norm soln for a model with errors posting-account=H-IscAoAAABkDNrURGSxo9jPN3MJ3a8A 1.0.3705; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) On Aug 1, 2:55æpm, Robert Israel > Hi: > æMost commonly seen approximated soln to a system of linear eqns > A*x = b is the least-squares soln. One can find the solutions either > when the system matrix A without and with uncertainty. > æBut that assumes 2-norm being used. How does one solve the same > problem when the matrix A is with uncertainty by using more general > norm(s)? By using more general optimization algorithms. > -- > Robert Israel æ æ æ æ æ æ æisr...@math.MyUniversitysInitials.ca > Department of Mathematics æ æ æ æhttp://www.math.ubc.ca/~israel > University of British Columbia æ æ æ æ æ æVancouver, BC, Canada I think I should make it more precise: can one have some more explicit formula to the soln of this kind of problem? I tried pure numerical approach for the case simply using the 2-norm and this pure numerical approach suffers convergent problem. When the 2-norm is used, there is explicit soln to this model-with-error problem: solving SVD of the extended system matrix and then take the (n+1) right singular vector, assuming full-ranked system. So I can compare numerical soln with explicit soln; numerical soln does not always provide a convergent soln. For exmaple, using MatLab built-in optimization routine with the following objective function. function y = funcTLS(x, AExtend) xExtend = [-1.0; x]; y = AExtend*xExtend; The above does not always give comparable soln with that of explicit form; depending on IC used. So let's narrow down my question: are there explicit soln for model- with-error problem using a norm that is not 2-norm or are there more reliable numerical approach to solve this? === Subject: Re: ? general norm soln for a model with errors posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > On Aug 1, 2:55 pm, Robert Israel > Hi: > Most commonly seen approximated soln to a system of linear eqns > A*x = b is the least-squares soln. One can find the solutions either > when the system matrix A without and with uncertainty. > But that assumes 2-norm being used. How does one solve the same > problem when the matrix A is with uncertainty by using more general > norm(s)? > By using more general optimization algorithms. > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada I think I should make it more precise: can one have some more explicit > formula to the soln of this kind of problem? I tried pure numerical approach for the case simply using the 2-norm > and this pure numerical approach suffers convergent problem. When the 2-norm is > used, there is explicit soln to this model-with-error problem: solving SVD > of the extended system matrix and then take the (n+1) right singular vector, > assuming full-ranked system. So I can compare numerical soln with explicit > soln; numerical soln does not always provide a convergent soln. For exmaple, using MatLab built-in optimization routine with the > following objective function. function y = funcTLS(x, AExtend) xExtend = [-1.0; x]; > y = AExtend*xExtend; The above does not always give comparable soln with that of explicit > form; depending on IC used. So let's narrow down my question: are there explicit soln for model- > with-error problem using a norm that is not 2-norm or are there more reliable > numerical approach to solve this? Are you saying that you have a system of equations A*x = b in which the elements of A and b are not known exactly? If so, you might benefit by looking at the literature on chance-constrained programming (now rather old) or the much more recent (and reportedly much more effective) approach of robust optimization. Reportedly, this technique is starting to have important impacts in areas such as bridge and building design, machine design, financial planning and the like. Of course, your problem does not look immediately like an optimization problem, but you could consider it as the problem of maximizing 0*x, subject to A*x = b, and so turn it into an optimization. Good luck. R.G. Vickson === Subject: Binary Sequences posting-account=dGiPYgkAAABSJ3xUlNLViQdT0h489hR6 AppleWebKit/523.10.3 (KHTML, like Gecko) Version/3.0.4 Safari/523.10,gzip(gfe),gzip(gfe) I have recently posted these three related sequences to the Encyclopedia Of Integer Sequences: (I don't know if they have appeared yet.) %S A143220 1,1,0,1,0,1,1,0,0,0,1,1,1,1,0,1,0,1,0,0,1,1,1,0 %N A143220 a(0)=1. For n >=1, a(n) = 1 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 0 otherwise. %e A143220 The binary representation of 21 is 10101. This occurs in the concatenation of terms a(0) through a(20) like so: 1101011000111(10101)001. So a(21) = 1. %Y A143220 A118268,A143221,A143222 %O A143220 0 %K A143220 ,base,more,nonn, %S A143221 1,0,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0,0 %N A143221 a(0)=1. For n >=1, a(n) = 0 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 1 otherwise. %e A143221 The binary representation of 21 is 10101. This occurs in the concatenation of terms a(0) through a(20) like so: 10010111(10101)00010010. So a(21) = 0. %Y A143221 A143220,A143222 %O A143221 0 %K A143221 ,base,more,nonn, %S A143222 0,1,1,0,1,0,0,1,1,0,0,1,0,0,1,1,1,1,0,0,0,1 %N A143222 a(0)=0. For n >=1, a(n) = 0 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 1 otherwise. %e A143222 The binary representation of 20 is 10100. This occurs in the concatenation of terms a(0) through a(19) like so: 01(10100)1100100111100. So a(20) = 0. %Y A143222 A143220,A143221 %O A143222 0 %K A143222 ,base,more,nonn, (Every term of the sequence where a(0)=0 and a(n)=1 if n does occur and a(n)=0 otherwise is trivially 0.) Let a = sum{k=0 to inf} A143220(k)/2^k, b = sum{k=0 to inf} A143221(k)/2^k, and c = sum{k=0 to inf} A143222(k)/2^k. Can a, b, and/or c be connected via any mathematical relations? Are there closed expressions for a, b, or c? Maybe the sequences have closed forms for their generating functions, I wonder. (I am guessing that the GFs with closed forms, if any, would be the ordinary GFs, but I could be wrong.) -- By the way, doing a search on Google Groups for Leroy Quet now brings up ZERO hits, when in fact I have posted THOUSANDS of posts to sci.math. (And I signed each one. So my name appears in each post, and should have been seen by a search.) Does anyone know what the hell is wrong with Goggle's search? Leroy Quet === Subject: Re: Binary Sequences posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) > I have recently posted these three related sequences to the > Encyclopedia Of Integer Sequences: > (I don't know if they have appeared yet.) %S A143220 1,1,0,1,0,1,1,0,0,0,1,1,1,1,0,1,0,1,0,0,1,1,1,0 > %N A143220 a(0)=1. For n >=1, a(n) = 1 if the binary representation of > n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). > a(n) = 0 otherwise. > %e A143220 The binary representation of 21 is 10101. This occurs in > the concatenation of terms a(0) through a(20) like so: > 1101011000111(10101)001. So a(21) = 1. > %Y A143220 A118268,A143221,A143222 > %O A143220 0 > %K A143220 ,base,more,nonn, %S A143221 1,0,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0,0 > %N A143221 a(0)=1. For n >=1, a(n) = 0 if the binary representation of > n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). > a(n) = 1 otherwise. > %e A143221 The binary representation of 21 is 10101. This occurs in > the concatenation of terms a(0) through a(20) like so: > 10010111(10101)00010010. So a(21) = 0. > %Y A143221 A143220,A143222 > %O A143221 0 > %K A143221 ,base,more,nonn, %S A143222 0,1,1,0,1,0,0,1,1,0,0,1,0,0,1,1,1,1,0,0,0,1 > %N A143222 a(0)=0. For n >=1, a(n) = 0 if the binary representation of > n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). > a(n) = 1 otherwise. > %e A143222 The binary representation of 20 is 10100. This occurs in > the concatenation of terms a(0) through a(19) like so: > 01(10100)1100100111100. So a(20) = 0. > %Y A143222 A143220,A143221 > %O A143222 0 > %K A143222 ,base,more,nonn, (Every term of the sequence where a(0)=0 and a(n)=1 if n does occur > and a(n)=0 otherwise is trivially 0.) Let a = sum{k=0 to inf} A143220(k)/2^k, > b = æsum{k=0 to inf} A143221(k)/2^k, and > c = æsum{k=0 to inf} A143222(k)/2^k. Can a, b, and/or c be connected via any mathematical relations? > Are there closed expressions for a, b, or c? Maybe the sequences have closed forms for their generating functions, > I wonder. > (I am guessing that the GFs with closed forms, if any, would be the > ordinary GFs, but I could be wrong.) -- By the way, doing a search on Google Groups for Leroy Quet now > brings up ZERO hits, when in fact I have posted THOUSANDS of posts to > sci.math. (And I signed each one. So my name appears in each post, and > should have been seen by a search.) > Does anyone know what the hell is wrong with Goggle's search? I've noticed curious bevaviour also. Just a couple minutes ago and it couldn't find it. I did find it eventually by looking for different keywords. In looking for leroy quet, I got 4 hits. Looking for quet returned 6000, so I would guess they are still all there. Here's one of the summaries. I've put square brackets [] around the bolded words that mark keyword matches. But note only 2 of the 3 occurences of quet are bolded (searching for Quet gave me exactly the same result, quet bolded, Quet not bolded). There definitely appears to be something wrong with the matching of keywords to content. That sucks. Imposter? Re:Sum Of Product Involving Group: rec.puzzles A leroy [quet], not this Leroy Quet, has posted something in my name, or so it seems. A few possibilities; 1) There is another leroy [quet]. 2) Someone is trying to mess with my mind. (Who? A long while back, someone else DID post something in my name. Have they returned?) 3) Looking at the content, I am starting ... Leroy Quet === Subject: moderator analysis posting-account=r5Mu_AoAAAA4Or1qD_TiZuru2F3PYCEL Gecko/20080702 Firefox/2.0.0.16 eMusic DLM/4.0_1.0.0.1,gzip(gfe),gzip(gfe) Hi everyone, I am analyzing the retention of subjects across the 3 timepoints of a study. I would like to determine whether the level of a variable (say varA) at baseline moderates continuation in the study as measured by presence at time 1 and time2. I am not sure how to best run that analysis. I was thinking I should get the correlations between varA, PresentAtTime1, and PresentAtTime2 where PresentAtTime1 is an indicator variable. I was also thinking I should run some logistic models. I am confused about those. PresentAtTime1= logistic(varA)? PresentAtTime2= logistic(PresentAtTime1+PresentAtTime1*varA)? I would appreciate some help or some pointers. === Subject: Re: How long did it take him to get home? I currently am reading Me, Myself and Them by Kurt > Snyder, which is > a well-written, interesting first person account of > paranoid > schizophrenia. At one point he is driving home and > developes the > delusion that the CIA is somehow controlling his > route home. To thwart > semi-random route > home, instead of letting them control me. I would > only make a turn in > the direction of my house if the last digit of the > minutes on the > digital clock read 1, 3, 5, 7. Otherwise, I would > continue straight > ahead. As you can probably guess, it took me a *very* > long time to get > home. (pgs. 79-80). Under certain simplifying assumptions, this can be > modeled by a Markov > chain. Take as states ordered triples (i,j,d) where > (i,j) is a point > with integer coordinates and d is in {N,S,E,W}. Home > states are of the > form (0,0,d). State transitions are hard to spell out > in detail > (especially if u-turns are not permitted). Home > states are absorbing. > Assuming u-turns it would be like this. With > probability p = 0.6 you > would move in the same direction. For example > (1,-3,S) would lead to > (1,-4,S). With probability 1-p you would go to a > neighboring point in > the plane (and adjust d accordingly) in such a way > that the taxicab > distance is decreased. If there is a tie between two > adjacent states > in this situation make the choice randomly. Note that > decreasing the > distance home might imply going in the direction you > are already > heading, so in some states the probability of going > straight might be > 1 rather than 0.6 (in which case you make a bee-line > home). He didn't > explicitly say that in the passage quoted above, but > it seems like a > natural interpretation. I don't know enough about > Markov chains to > know for sure, but it seems clear enough that with > probability one he > *does* get home with the expected time heavily > dependent on the > initial state. It is easy to see that, for any > initial state, > infinitely many states are inaccessible These > includine those on the > coordinate axis heading in the wrong direction > (unless the initial > state is on the same axis and also heading in the > wrong direction). > Are there other inaccessible states? Has anybody studied such semi-random walks? At the > very least it > would make for some fun computer simulations. -scattered p.s. The author also describes how when he was in > college he had > delusions of grandeur: I believed that I was going > to discover some > fabulous new mathematical principle that would > transform the way we > view the universe. I told no one about these > thoughts. I started > looking for clues to this mathematical theory in math > books I found at > the library. I actually learned very little about > math though, because > I couldn't focus on any material for any significant > length of > time ... but I still thought that one day I would get > a flash of > inspiration and become famous. (pgs. 1,2) I suspect > that most of the > cranks who post here are *not* schizophrenic, but > passages like that > just quoted make you wonder. OT, but if you liked that book (haven't read it, though I think I might like to)you might want to read another first person account of descent into schizophrenia-- Mark Vonnegut, The Eden Express. It takes place in the 1960s, when Vonnegut (son of Kurt Vonnegut, from whom he inherited the condition)was young. Mark Vonnegut, after treatment, became a successful medical doctor. Tom === Subject: Re: How long did it take him to get home? >p.s. The author also describes how when he was in college he had >delusions of grandeur: I believed that I was going to discover some >fabulous new mathematical principle that would transform the way we >view the universe. I told no one about these thoughts. I started >looking for clues to this mathematical theory in math books I found at >the library. I actually learned very little about math though, because >I couldn't focus on any material for any significant length of >time ... but I still thought that one day I would get a flash of >inspiration and become famous. (pgs. 1,2) That feels awfully familiar! I think I'd find this book a good read. (I've never actually been schizophrenic, but I'm a long way off from being normal). >I suspect that most of the >cranks who post here are *not* schizophrenic, but passages like that >just quoted make you wonder. Psychology is not an empirical science (notwithstanding the undoubted existence of an empirical science that calls itself psychology), so it's hardly surprising if its categories are as fluid and ambiguous as its subject matter. -- Angus Rodgers Contains mild peril === Subject: Efficient method to find roots of equations. Hi everyone! I currently encounter a root finding problem. f_1(x1)=f_2(x2)=...=f_N(xN) g_1(x1)+g_2(x2)+...+g_N(xN)=B === Subject: Re: Efficient method to find roots of equations. posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Hi everyone! I currently encounter a root finding problem. f 1(x1)=f 2(x2)=...=f N(xN) g 1(x1)+g 2(x2)+...+g N(xN)=B I know there is and only is one vector X=[x1,x1,...,xN] satisfying these equations. There is N equations and N variables. I wonder whether the Newton[CapitalEth]Raphson can be used to find the roots of these equations? If there there is one equation ,the Newton[CapitalEth]Raphson works, but does it work for multiple equations? Yes, of course, and it is one of the underlying methods of multivariate optimization. For multiple-equation Newton-Raphson, see, eg., www.haoli.org/nr/bookf/f9-6.ps or Section 7 of http://en.wikipedia.org/wiki/Newton's method or http://trond.hjorteland.com/thesis/node28.html . R.G. Vickson === Subject: Re: Efficient method to find roots of equations. === Subject: Re: Partially Persistent Tree Implementation Face: iVBORw0KGgoAAAANSUhEUgAAADAAAAAwAQMAAABtzGvEAAAABlBMVEUAAAD///+l2Z/dAAAA oElEQVR4nK3OsRHCMAwF0O8YQufUNIQRGIAja9CxSA55AxZgFO4coMgYrEDDQZWPIlNAjwq9 033pbOBPtbXuB6PKNBn5gZkhGa86Z4x2wE67O+06WxGD/HCOGR0deY3f9Ijwwt7rNGNf6Oac l/GuZTF1wFGKiYYHKSFAkjIo1b6sCYS1sVmFhhhahKQssRjRT90ITWUk6vvK3RsPGs+M1RuR mV+hO/VvFAAAAABJRU5ErkJggg== > I'm trying to implement a partially persistent tree structure. I plan > to cache a portion of the frequently accessed nodes in the memory > while the full tree will reside on the disk and changes committed to > memory/on-disk nodes will get propogated to their on-disk/memory > correspondents. Before going into further implementation details, I > want to check the literature for anything had done in the past > subject? I've got no literature, but once I implemented a quad-tree as a memory-mappable data structure. Since the number of children in a quad-tree is fixed (to 4), there's no need for pointers or offsets, you can compute directly the offset of a node from the position in the tree. However, for random trees, you can still do it, simply using offsets instead of pointers, so the data can be mapped at any address. Well I guess the only literature you need is mmap(2). -- __Pascal Bourguignon__ http://www.informatimago.com/ Indentation! -- I will show you how to indent when I indent your skull! === Subject: solution manual posting-account=N_qD3woAAACvn11QjAmpGm8Ieo5qY02Y 2.0.50727),gzip(gfe),gzip(gfe) Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ A first course in probability - Sheldon M. Ross - 7 ed Adaptive Control - Karl J. Astrom - 2 ed Advanced Macroeconomics - Jeffrey Rohaly Advanced Microeconomic Theory - Geoffrey Jehle Advanced Modern Engineering Mathematics - Glyn James - 3 ed Algebra- Baldor An introduction to numerical analysis - E. Suli, F. Mayers Analytical Mechanics - Fowles and Cassiday - 7 ed Antenna theory - Constantine Balanis - 2 ed Applied Numerical Analysis - Curtis F. Gerald, Patrick O. Wheatley - 7 ed Applied Numerical Methods - Steven Chapra Applied Probability models with optimization applications - Sheldon M. Ross Applied strength of materials - Robert L. Mott - 4 ed Artificial Intelligence - Stuart J. Russell y Peter Norvig - 2 ed Automatic control systems - Kuo and Golnaraghi - 8 ed Basic Engineering Circuit Analysis - David Irwin - 8 ed Calculo several variables - Hallet, Gleason McCallum - 4 ed Calculus - George B. Thomas - 11 ed - Vol 1 Calculus - George B. Thomas - 11 ed - Vol 2 Calculus - George Thomas Vol.2 Calculus - James Stewart - 5 ed Calculus - Jerrold Marsden, Alan Weinstein - vol 1 Calculus - Leithold - 7 ed Calculus - Purcell - 9 ed Calculus several variables - James Stewart - 4 Ed Calculus - Wards y Penney 4 ed Calculus 1 variable - Hallet, Gleason McCallum - 4 ed Calculus 1 variable - James Stewart - 4 Ed Calculus one and several variables - Salas Hille Etgen - 8 ed Calculus several variables - Neta B Calculus.Early.Transcendentals - Edition.James.Stewart - 5 ed vol1 and 2 Chemical and Engineering Thermodynamics - Stanley Sandler - 3 ed Communication systems engineering - John G.Proakis - 2 ed Computer Networking - Kurose, W. Ross - 3 ed Computer Networks - Andrew Tanenbaum - 4 ed Control Systems Engineering - Norman Nice Design and analysis of experiments - Douglas C. Montgomery - 6 ed Design of Machinery - Robert Norton - 3 ed Device Electronics for Integrated Circuits - Richard S. Muller, Theodore I. Kamins - 3 ed Differential equations - Dennis G Zill - 7 ed Differential equations linear algebra - Jerry Farlow - 2 ed Digital Comunications - Bernard Sklar - 2 ed Digital Comunications - John G. Proakis - 4 ed Digital image processing - Rafael C. Gonzalez, Richard E. Woods - 2 ed Digital Signal Processing - John G. Proakis - 3 ed Digital signal processing - Sanjit K. Mitra Discrete time signal processing - Alan V. Oppenheim Dynamics - Bedford Fowler - 4 ed Dynamics - Bedford Fowler - 5 ed Dynamics - Hibbeler - 11 ed Economics econometric analysis - William H. Greene - 5 ed Electric Circuits - Nilsson - 7 ed Electric machinery - Fitzgerald , Kingsley, Uman - 6 ed Electric Machinery Fundamentals - Stephen Chapman - 4 ed Elementary mechanics and Thermodynamics - Jhon W. Norbury Elementary Principles of Chemical Processes - Richard Felder y Ronald Rousseau Engineering Mechanics, Statics - R. C. Hibbeler - 10 ed Engineering Circuit Analysis - William H. Hayt - 6 ed Engineering electromagnetics - Hayt - 6 ed Engineering fluid mechanics - Clayton T. Crowe - 6 ed Engineering fluid mechanics - Crowe, Elger, Robertson - 7 ed Engineering mathematics - John Bird - 4 ed Engineering Mechanics, Statics - Hibbeler - 11 ed Feedback Control Dynamic Systems - Franklin Powel Emami - 4 ed Field and Wave Electromagnetics - David K. Cheng - 2 ed Field Theory Electromagneticos - Alexander Sadiku Fluid mechanics - Frank M. White - 6 ed Fluid mechanics, Thermodynamics of turbomachinery - 5 ed Fourier and laplace transforms Fracture mechanics fundamentals and applications - T.L. Anderson - 2 ed Fundamentals of Aerodynamics - John D. Anderson - 3 ed Fundamentals of Applied Electromagnetics - Fawwaz T. Ulaby - 5 ed Fundamentals of engineering electromagnetics - David K. Cheng Fundamentals of engineering thermodynamics - Moran M.J, Shapiro H.N - 5 ed Fundamentals of fluid mechanics - Bruce R. Munson - 4 ed Fundamentals of Physics - Halliday Resnick vol 1 - 7 ed Fundamentals of Physics - Halliday Resnick vol 2 - 7 ed Fundamentals of thermodynamics - Sonntag, Bognakke, Van Wyler - 6 ed Fundamentals.of.Electric.Circuits - C.K.Alexander, M.N.O.Sadiku - 2 ed Heat transfer - Fundamentals of heat and mass transfer - Frank P. Incropera, David Dewitt heat transfer, fundamentals of heat and mass transfer - P. Incropera, D. P. Dewitt Introduction to algorithms - Thomas H. Cormen Charles E. Leiserson - 2 ed Introduction to Electric Circuits - R. C. Dorf y J. A. Svoboda - 6 ed Introduction to electrodynamics - David J. Griffiths - 3 ed Introduction to linear algebra - Gilbert Strang - 3 ed Introduction to mechanics of Fluidos - Robert Fox, Alan McDonald, y Philip J. Pritchard - 6 ed Introduction to probability- Dimitri P. Bertsekas and John N. Tsitsiklis Introduction to Queueing theory - Robert B. Cooper - 2 ed Introductions to chemical engineering Thermodynamics - J. M. Smith, H. C. Van Ness, M. M Abbott - 6 ed Introductions to chemical engineering Thermodynamics - J. M. Smith, H. C. Van Ness, M. M Abbott - 7 ed Introductory quantum optics - C. C. Gerry and P. L. Knight Linear Algebra - Jim Hefferon Linear Algebra and its Applications - David C. Lay - 3rd ed Linear circuit analysis - R. A DeCarlo, Pen Min Lin - 2 ed Materials science and engineering - W.D. Callister - 6 ed Mathematical Analysis - Apostol Mechanical engineering - Shigleys - 8 ed Mechanical Engineering Design - S Mischke, R Budynas - 7 ed Mechanics of Fluids - Bernard Massey - 8 ed Mechanics of materials - Beer Johnston and Dewolf - 3 ed Mechanics of materials - Gere - 6 ed Mechanics of materials - Hibbeler - 4 ed Mechanics of materials - Hibbeler - 4 ed Microeconomic analysis - Hal R. Varian - 3 ed Microelectonics - Millman Microelectronic Circuits - Adel S. Sedra - 4 ed Microelectronic Circuits - Kenneth C. Sedra, Kc Smith - 4 ed Modern Control Engineering - Problems B - Katsuhiko Ogata - 3 ed Modern control system - Richard Dorf y Robert Bishop - 9 ed Modern digital and analog communications systems - B. P. Lathi Organic Chemistry - Carey - 5 ed Organic Chemistry - Hornback - 2 ed Physical chemistry - Peter Atkins, Julio de Paula - 7 ed Physical chemistry - Peter W. Atkins - 7 ed Physics - Paul A. Tipler - 5 ed Physics - Sears, Zemansky, Young, Freedman vol1 - 11 ed Physics - Sears, Zemansky, Young, Freedman vol2 - 11 ed Physics by Resnick Halliday Krane vol 2 - 5 ed Physics for scientists and engineers - Raymond Serway - vol 1 - 5 ed Physics for scientists and engineers - Raymond Serway - vol 1 - 6 ed Physics for scientists and engineers - Raymond Serway - vol 2 - 5 ed Physics for scientists and engineers - Raymond Serway - vol 2 - 6 ed Physics: Principles with Applications - Douglas Giancoli - 6 ed Power System Analysis - John J. Grainger, William D. Stevenson Principles and applications of electrical engineering - Giorgio Rizzoni Principles of electronic materials and devices - S. O. Kasap - 2 ed Probability and statistics for engineers and scientists - Anthony Hayter - 3 ed Probability and statistics for engineers and scientists - Jay L. Devore - 6 ed Probability and statistics for engineers and scientists - Walpole, Myers - 8 ed Probability, Random Variables and Stochastic Processes Solutions - Athanasios Papoulis.- 4 ed Process system analysis and control - Donald R. Coughanowr Quantum Mechanics - Yung-Kuo Lim Science and engineering of materials - Donald R. Askeland - 4 ed Signals and systems - Simon Haykin - 2 ed Signals and systems - Michael J. Roberts Signals and systems - Oppenheim - Willsky - 2 ed Solid state electronic devices - B. G. Streetman, B. Sanjay Solid state physics - Charles Kittel - 8 ed Statics - Meriam Structural analysis - hibbeler - 5 ed System dynamics- Katsuhiko Ogata - 3 ed The econometrics of financial markets - Craig MacKinlay, Andrew W. Lo & John Y. Campbell Thermodynamics an engieneering approach - Yunus Cengel - 5 ed Transport Phenomena - R. Byron Bird, Warren E. Stewart - 2 ed Unit operations of chemical engineering - Warren McCabe, Juan C. Smith, Peter Harriott - 6 ed Vector Mechanics for Engineers: Dynamic - Ferdinand P. Beer - 6 ed Vector Mechanics for Engineers: Dynamics - Ferdinand P. Beer - 7 ed Vector Mechanics for Engineers: Statics - Ferdinand P. Beer - 6 ed Vector Mechanics for Engineers: Statics - Ferdinand P. Beer - 7 ed Wireless Communications - Theodore Rappaport - 2 ed Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ Visit http://www.solutionsmanual.es.tl/ === Subject: Mathieu Functions and Differential Equation Solution posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Hi All, I used Mathematica to solve the following DEQ: x''[t] + (2 Pi)^2(1 + (1/10) Cos[(2 Pi/100) t]) x[t] = 0 The result uses even and odd Mathieu functions. The results are not matching what the paper I am reading shows. Can someone out there use some package and provide a solution to the DEQ to make sure I am getting the correct result (maybe there is an error in the paper - or I could be missing something)? ~A === Subject: Re: Mathieu Functions and Differential Equation Solution > Hi All, I used Mathematica to solve the following DEQ: x''[t] + (2 Pi)^2(1 + (1/10) Cos[(2 Pi/100) t]) x[t] = 0 The result uses even and odd Mathieu functions. The results are not matching what the paper I am reading shows. Can someone out there use some package and provide a solution to the > DEQ to make sure I am getting the correct result (maybe there is an > error in the paper - or I could be missing something)? Maple 12 says: x(t) = _C1*MathieuC(40000,-2000,1/100*Pi*t)+_C2*MathieuS(40000,-2000,1/100*Pi*t) -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Mathieu Functions and Differential Equation Solution posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) On Aug 1, 5:40æpm, Robert Israel > Hi All, > I used Mathematica to solve the following DEQ: > x''[t] + (2 Pi)^2(1 + (1/10) Cos[(2 Pi/100) t]) x[t] = 0 > The result uses even and odd Mathieu functions. > The results are not matching what the paper I am reading shows. > Can someone out there use some package and provide a solution to the > DEQ to make sure I am getting the correct result (maybe there is an > error in the paper - or I could be missing something)? Maple 12 says: x(t) = C1*MathieuC(40000,-2000,1/100*Pi*t)+ C2*MathieuS(40000,-2000,1/100*Pi*t) > -- > Robert Israel æ æ æ æ æ æ æisr...@math.MyUniversitysInitials.ca > Department of Mathematics æ æ æ æhttp://www.math.ubc.ca/~israel > University of British Columbia æ æ æ æ æ æVancouver, BC, Canada If you don't mind - are you able to duplicate Figures 1 through 3 in this paper? http://epsppd.epfl.ch/Roma/pdf/P2 091.pdf ~A === Subject: Re: the Science of Math-DR SALIL PANDE Importance: low >I have always found Math to be a fantastic venture. My father was a >Mathematician and it drew me into the exact sciences. As a doctor I >use math everyday and for some strange reason I find the concepts >behind our most complex theories in math and science fascinating. >Now, I don't want to be wishy-washy here, I just wanted to set up a >post so that people who care about what I care about know how I feel. >Nobody else I tell these things to cares, since my siblings were >always on the more rebellious side. >I think this is a great forum, all of you take care. as my work load >goes down a bit I look forward to being a regular contributor to this >site. >Blessings, >Dr. Salil Pande Welcome! I've never been able to talk about maths either, not even (that's the > problem) to other mathematicians. I'm trying to get into the habit of > talking about it online, in the hope that I might eventually even be > able to literally talk about it when an occasion arises to do so. Maths is indeed fascinating. You forgot your anagram, Angus... Dr. Salil Pande ~ plans a riddle Phil -- -- Microsoft voice recognition live demonstration === Subject: Re: the Science of Math-DR SALIL PANDE indeed, math is quite an interesting field. there are so many possibilities and ways to integrate concepts into our lives. i hope you join our conversations in the near future. - Lana Todorovich === Subject: Prime function(related to zeta) Anything known about the function defined by prod(1/(1 - 1/(p_k - x)^s))? obviously f(x,s) = 0 when x is positive and prime and if x = 0 then it reduces to zeta(s). I'm mainly interested in it for negative values as its better behaved. There seems to be some relationship between it and the zeta function. (I do have a relationship between it and the zeta function but its not closed form) === Subject: Re: the problem with Cantor > When you consider computable reals 0<=cr<1 > what variety do you get in the decimal expansions? > And so another nonstandard mathematician |-|erc, has > appeared out of lurkdom to refute Cantor. > I've only glanced at this thread, but it appears that > |-|erc's argument is another one based on the > assumption (that doesn't hold in ZFC) that if for every > natural number n, phi(n) holds, then we must have > phi(N) holding as well. *************************************************************** [Tonico] > I can't be sure (pretty confussing with all those non-defined terms > and stuff), but at the bottom line it seems |-|erc tries somehow to > turn over the argument used in Cantor's Diagonal proof and he says: OK, I can write down all the computable numbers and that way I get > ALL the possible real numbers, [quote: Computable reals displays EVERY > type of decimal expansion imaginable, there is nothing it misses.]. > Proof? Very simple: tell me which number would I miss doing this! Oh, > but if you can point such a number then it is computable, and thus I > would have written it...taraaaaan! Well, there seems to exist a rather huge logical flaw up there: how > can you know a priori whether you've written down all the conmputable > real numbers? > Or even better, and turning over the tortilla once again: what if > you present me the list (because it will be a list...right??) of ALL > computable real numbers, and then I use Cantor's Diagonal argument and > pinpoint a real number which is NOT in that list? Because believe me: > in any list of real numbers there will be a number I can pinpoint and > prove it is NOT in that list... Of course, it could be |-|erc didn't actually mean the above... [Herc] > Well spotted flipside of my claim - the diag argument. The list will contain every finite prefix of the anti-diagonal. When the diag argument was invented we imagine something like this. 123 > 456 > 789 The diagonal is 159 > The antidiagonal is 261 > Voila - 261 is not on the list. But, if the list is the computable reals, > 2 is on the list > 26 is on the list > 261 is on the list Every finite prefix of the antidiagonal, UP TO OO LENGTH is on the list. This gives me grave doubts of claims that the sequence is missing. > In a way the sequence is there, every finite prefix to infinite length means > you don't miss a digit. Sorry to have to tell you but you're all fools, > only when Artificial Intelligence arrives in 20 years will you be corrected. :)~ Herc ********************************************************** Yeah...that sounded like :you're wrong, you suck, I'm right I win. Wait till jesus (muhammed, buddha, your-favourite-god) arrives...just wait! Then you'll see who's the fool! Well, some fundies already mention NOW the year 2012, after many failed dates in the past, and now you've mentioned A.I. in 20 years more (which I can't fully understand what has to do with this issue...). We shall see, I supose, and no further debate is worth. Tonio You ignored my argument and focused on the little emoticon ending. No wonder you want to withdraw now, your superinfinity is looking pretty weak. You can't even form a unique sequence of digits using diagonalisation on my set can you? http://www.freewebs.com/namesort/linux.html Tell me in your own words what this site does. There's 3 buttons numbered 1 2 3. Press 1 2 3 2 3 2 3 and describe what happens. Then I'll present my proof. Herc === Subject: Re: the problem with Cantor > When you consider computable reals 0<=cr<1 > what variety do you get in the decimal expansions? > And so another nonstandard mathematician |-|erc, has > appeared out of lurkdom to refute Cantor. > I've only glanced at this thread, but it appears that > |-|erc's argument is another one based on the > assumption (that doesn't hold in ZFC) that if for every > natural number n, phi(n) holds, then we must have > phi(N) holding as well. *************************************************************** [Tonico] > I can't be sure (pretty confussing with all those non-defined terms > and stuff), but at the bottom line it seems |-|erc tries somehow to > turn over the argument used in Cantor's Diagonal proof and he says: OK, I can write down all the computable numbers and that way I get > ALL the possible real numbers, [quote: Computable reals displays EVERY > type of decimal expansion imaginable, there is nothing it misses.]. > Proof? Very simple: tell me which number would I miss doing this! Oh, > but if you can point such a number then it is computable, and thus I > would have written it...taraaaaan! Well, there seems to exist a rather huge logical flaw up there: how > can you know a priori whether you've written down all the conmputable > real numbers? Who says they must all be written down in order to know that they are all included in a set? > Or even better, and turning over the tortilla once again: what if > you present me the list (because it will be a list...right??) of ALL > computable real numbers, and then I use Cantor's Diagonal argument and > pinpoint a real number which is NOT in that list? Because believe me: > in any list of real numbers there will be a number I can pinpoint and > prove it is NOT in that list... Of course, it could be |-|erc didn't actually mean the above... > Is Huck really claiming to be able to show that there is a number in in a list and not in that same list? He seems to be challenging WM's position as King Kook. > [Herc] > Well spotted flipside of my claim - the diag argument. The list will contain every finite prefix of the anti-diagonal. When the diag argument was invented we imagine something like this. 123 > 456 > 789 The diagonal is 159 > The antidiagonal is 261 > Voila - 261 is not on the list. But, if the list is the computable reals, > 2 is on the list > 26 is on the list > 261 is on the list Every finite prefix of the antidiagonal, UP TO OO LENGTH is on the list. Up to but not including oo, which means every member of the list has a sufficiently long but finite prefix differing from an equally long prefix of the diagonal. This gives me grave doubts of claims that the sequence is missing. Doubt all you want, but until you can support your doubts with better evidence than you so far have been able to do, they will not convince anyone else. > In a way the sequence is there, every finite prefix to infinite length means > you don't miss a digit. Sorry to have to tell you but you're all fools But not so foolish as to be convinced by such foolish arguments as you have been presenting. , > only when Artificial Intelligence arrives in 20 years will you be corrected. > :)~ Herc === Subject: Re: the problem with Cantor [VIRGIN] You dumb , I can still make a new number [HERC] I don't care about your new number, give me a new sequence of digits. Herc === Subject: Re: the problem with Cantor posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; FunWebProducts; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > [VIRGIN] > You dumb , I can still make a new number [HERC] > I don't care about your new number, give me a new sequence of digits. Herc *************************************************************** I answer here you prior post since for some rather misterious and completely unreachable for me reason that post doesn't accept any answer... 1) Well, let's play along: I entered your link, pressed 1232323 and got a left list of some 60 numbers, then an apaprently randomly generated numbers, and a second list which apparently contains the randly gen. number in its main diagoal entries...so? I don't understand what is this supposed to show. 2) What is superinfinity? And why is it mine? 3) I did not ignore your argument: I did address it in my first post, misunderstood. You ignored this addressing of mine. 4) no wonder I want to withdraw now....from what? A debate? If it indeed is a debate, what path is left for a debate after you write Sorry to have to tell you but you're all fools, only when Artificial Intelligence arrives in 20 years will you be corrected.? It seems not only you've utterly made up your mind about this, but you've also decided we all are fools, you only are wise, we're wrong, you're right...and we all shall see this in 20 years more! Well, you've wrapped it all tight and dandy: YOU closed the door. Tonio === Subject: Re: the problem with Cantor posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > 1) Well, let's play along: I entered your link, pressed 1232323 and > got a left list of some 60 numbers, then an apaprently randomly > generated numbers, and a second list which apparently contains the > randly gen. number in its main diagoal entries...so? I don't > understand what is this supposed to show. I believe I know what Herc is trying to accomplish. Herc is making the following claim (quoted verbatim): Claim: Let L be a list of numbers in [0,1] such that for each d in {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal representation of L_i is d} is infinite. Let D be a random variable uniform on [0,1]. Then almost surely there is a permutation L' of L such that D is the diagonal of L'. But there are two big problems here: 1) The claim most likely _fails_ in ZFC -- even for finite lists! 2) Even if the claim does hold in ZFC, it has absolutely _no_ bearing on the validity of Cantor in ZFC. Let's deal with 1) first. Herc's link is an applet consisting of three steps: 1. Populate list -- clicking this causes a list of sixty real numbers between zero and one, apparently in ternary since only the digits 0,1,2, appear, each with sixty digits. 2. Randomise diagonal -- clicking this causes a 61st real number to appear. 3. Generate list -- clicking this causes a reordering of the first list to appear such that the real number chosen in the second step appears on the diagonal. On the face of it, this should always work. We know that there are 60! (60 factorial) permutations of the original list and only 3^60 possible diagonals -- and factorials increase much faster than exponentiation. (60! ~ 8*10^81, but 3^60 ~ 4*10^28.) So there ought to be more than enough permutations for all the possible diagonals. But I decided to try a little trick. I clicked on the first button to populate the list -- but rather than clicking on the second button, I manually typed in my own diagonal. What I did was look at the first row, which happened to be: 0.021100001022012201100101102022102002102112020210111021110120 Then I used one of the usual tricks for generating an antidiagonal (in Cantor's proof) from a diagonal to make an antirow. I replaced the digits 1 and 2 with 0 and the digit 0 with 1, to obtain the number: 0.100011110100100010011010010100010110010000101001000100001001 which I typed into the diagonal box. Then I clicked on the third button. Sure enough, only 59 numbers appeared rather than 60. And of course, the number that was present on the first list but missing from the second list was the first number. It would be _impossible_ for the first number to appear, since it differs from the diagonal in _every_ digit! And it's easy to see that the same problem would happen for _any_ list, whether finite or infinite, whether saturated or unsaturated. Of course, Herc's claim is that almost surely such a permutation exists. But what is almost surely? Does Herc mean that the set of all real numbers for which a permutation fails to exist has Lebesgue measure zero? If so, then Herc may be right. So far, we've only shown the antirows to be counterexamples -- where an antirow is defined to be a number differing from every digit from a number in a row. Suppose the first row contains the real number 1/2 -- which is 0.111... in ternary. Then an antirow would be any number consisting of only 0 and 2 in ternary -- and the set of all such numbers is the famous Cantor middle-third set, which has Lebesgue measure zero. Similarly, the set of all possible antirows for a given row also has Lebesgue measure zero. And since this is a _list_ of reals, there's exactly one real in this list for every natural, so there are only countably many such reals and their corresponding sets of antirows. So the set of all antirows for the entire list is the countable union of null sets, and therefore itself null. Of course, this isn't to say that there doesn't exist a counterexample to Herc's claim that doesn't happen to be an antirow to any row at all. But the set of the only _known_ counterexamples to Herc's claim does have Lebesgue measure zero. But even if this makes Herc's claim true, we reach problem 2) -- it has nothing to do with Cantor at all! Indeed, let's compare my proof above to Cantor's: I show it's impossible for an _antirow_ to be a _diagonal_. Cantor shows it's impossible for an _antidiagonal_ to be a _row_. If we think about this for a moment, we see that in my proof, I only claim to show that there exists numbers which can't be on the diagonal. That a counterexample happens to be an antirow is irrelevant. I only care about the diagonal, not rows or antirows. Similarly, Cantor claims that there exist real numbers which can't be in any row of the list. That a counterexample happens to be an antidiagonal is irrevelevant. Cantor cares only about _rows_, not the diagonal or the antidiagonal. It's possible for a real number to be the diagonal of one permutation but an antidiagonal of another permutation. Herc was apparently hoping that this wouldn't be the case, but we can even consider the example: 168 249 357 So an antidiagonal is 258, but if we reorder the list: 249 357 168 suddenly 258's the diagonal. Now 369's an antidiagonal, but: 357 168 249 now 369's the diagonal. Therefore, Herc's claim, even if it were true, would neither prove nor disprove _anything_ about Cantor. > 2) What is superinfinity? And why is it mine? By superinfinity, most likely Herc (pejoratively) means uncountable infinity. And it's yours (Tonio's), as much as it belongs to every standard mathematician who believes in its existence (i.e., who adheres to a set theory, such as ZFC, which proves the existence of uncountable sets), as opposed to Herc, who doesn't believe in the existence of uncountable sets. === Subject: Re: the problem with Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > But there are two big problems here: > 1) The claim most likely fails in ZFC -- even for finite lists! > 2) Even if the claim does hold in ZFC, it has absolutely no > bearing on the validity of Cantor in ZFC. Yathink? > But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! Really? Fascinating. You know, no one has been able to see that before. Just to think that all this time we all thought Herc had it absolutely correct all the way down the line. > Therefore, Herc's claim, even if it were true, would neither > prove nor disprove anything about Cantor. Please, say it ain't so! How could Herc ever be wrong about such a thing? MoeBlee === Subject: Re: the problem with Cantor posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU 5.1),gzip(gfe),gzip(gfe) > But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! > Really? Fascinating. You know, no one has been able to see that > before. Just to think that all this time we all thought Herc had it > absolutely correct all the way down the line. When I try to defend Herc or any of the other cranks, MoeBlee often criticizes me as being silly for trying to make a rigorous theory out of their jumbled claims. But in this post, I wasn't defending Herc -- I was actually attacking him. Yet MoeBlee still criticized me -- this time writing a sarcastic post about how I'm merely restating the obvious. So if MoeBlee will criticize me for defending Herc, and criticize me for attacking Herc, then is their anything I can say at all about Herc that won't lead to criticism from MoeBlee? Or am I damned if I do and damned if I don't? The reason for my post above is that here, I'm actually telling Herc that he's wrong -- but being more specific as to why he's wrong. It's refreshing to see, however flawed it may be, a different argument against Cantor rather than the same old infinite induction or only count the finite subsets of N arguments. Many of the standard mathematicians didn't try to figure out what Herc was doing or find out why he's wrong -- they simply knew that whatever Herc's doing is probably wrong (and rightly so). I bet some of the standard mathematicians would just wish that I would stop trying to defend Herc and the other cranks and attack them the same way that they themselves do. But why should I join the standard mathematicians in their attack against the cranks, when the likely response would be a sarcastic post similar to the one MoeBlee posted here? At least Nam Nguyen has already pointed out how MoeBlee has (apparently) contradicted himself (and there's no need to point out that cranks often contradict themselves, too). No matter what I post, someone's not going to like it, so there's no point in trying to please everyone when I post. === Subject: Re: the problem with Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! > Really? Fascinating. You know, no one has been able to see that > before. Just to think that all this time we all thought Herc had it > absolutely correct all the way down the line. When I try to defend Herc or any of the other cranks, > MoeBlee often criticizes me as being silly for trying > to make a rigorous theory out of their jumbled claims. But in this post, I wasn't defending Herc -- I was > actually _attacking_ him. Yet MoeBlee still criticized > me -- this time writing a sarcastic post about how I'm > merely restating the obvious. So if MoeBlee will criticize me for _defending_ Herc, > and criticize me for _attacking_ Herc, then is their > _anything_ I can say at _all_ about Herc that won't > lead to criticism from MoeBlee? Or am I damned if I do > and damned if I don't? What motivated my sarcasm was indeed the irony that we get these lectures from you about how the cranks could conceivably be vindicated but then you turn around to post exactly the kind of critique Herc has been getting for years already. I don't find fault - in itself - in ticking off specifics as to why Herc provides no challenge to the Cantor proofs. What I found worth sarcasm was the context in which you did that. > The reason for my post above is that here, I'm actually > telling Herc that he's wrong -- but being more specific > as to _why_ he's wrong. It's refreshing to see, however > flawed it may be, a different argument against Cantor > rather than the same old infinite induction or only > count the finite subsets of N arguments. Many of the > standard mathematicians didn't try to figure out what > Herc was doing or find out _why_ he's wrong -- they > simply knew that whatever Herc's doing is probably > wrong (and rightly so). See, again, that's where you go off making yourself out to be, in whatever sense, more intellectually inquisitive than standard mathematicians. Look, for SEVERAL years, people have been posting point by point, excuciatingly detailed explanations for Herc of his mistakes and fallacies. You may find yet another Herc variation to be refreshing (even if to refute), but, indeed, it is just another bozoid variation. Excuse us if, after reading years worth of that kind of stuff, we are not all as entertained by it as you are. > I bet some of the standard mathematicians would just > wish that I would stop trying to defend Herc and the > other cranks and attack them the same way that they > themselves do. You don't need you to attack (though you're welcome to do it). There are plenty of people who do that anyway. What I wish you'd stop doing is making sweeping and incorrect generalizations about standard mathematicians. > But why should I join the standard mathematicians in > their attack against the cranks, when the likely > response would be a sarcastic post similar to the > one MoeBlee posted here? See earlier in this post. > At least Nam Nguyen has > already pointed out how MoeBlee has (apparently) > contradicted himself No, Nam has not shown any contradiction in my remarks. I addressed that in the post just before yours here. MoeBlee === Subject: Re: the problem with Cantor MoeBlee, I have a hard time understanding how that big brain of yours ticks over. Please help me out by answering 1 question. I'll make it as brief as possible, a yes/no question. Q: Do you realise that the computable reals contain every possible digit sequence up to infinite length? that is all Herc === Subject: Re: the problem with Cantor <5nPlk.25589$IK1.13607@news-server.bigpond.net.au> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > MoeBlee, I have a hard time understanding how that big brain of yours ticks over. Please help me out by answering 1 question. I'll make it as brief as possible, a yes/no question. Q: Do you realise that the computable reals contain every possible digit sequence > æ æ æ up to infinite length? I can only answer your question if you tell me your axioms and rules of inference by which conclusions are drawn, and if you define your terms, back to your primitives. Such terms in your question include computable, real', possible, digit sequence, up to, and infinite length. MoeBlee === Subject: Re: the problem with Cantor > Q: Do you realise that the computable reals contain every possible digit > sequence up to infinite length? > I can only answer your question if you tell me your axioms and rules > of inference by which conclusions are drawn, and if you define your > terms, back to your primitives. Such terms in your question include > computable, real', possible, digit sequence, up to, and > infinite length. > between non-cranks) I'd say the problematic term here is up to. Does this mean: of finite, but unbound, length? Or does it include sequences of infinite length? In the latter case, the statement is wrong. Actually, not any digit sequence (if infinitely long sequences are admissible) are computable. So I'd propose not to realise that anti-fact. :-) B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor <8m3h94tpddo03n718etaa578lf1e15eo5p@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Q: Do you realise that the computable reals contain every possible digit > æ æsequence up to infinite length? > I can only answer your question if you tell me your axioms and rules > of inference by which conclusions are drawn, and if you define your > terms, back to your primitives. Such terms in your question include > computable, real', possible, digit sequence, up to, and > infinite length. between non-cranks) I'd say the problematic term here is up to. Does > this mean: of finite, but unbound, length? Or does it include > sequences of infinite length? In the latter case, the statement is wrong. Actually, not any digit > sequence (if infinitely long sequences are admissible) are computable. I agree. MoeBlee === Subject: Re: the problem with Cantor I agree. > Additional comment: > Q: Do you realise that the computable reals contain every possible digit > æ æsequence up to infinite length? > [...] I'd say the problematic term here is up to. Does > this mean: of finite, but unbound, length? Or does it include > sequences of infinite length? > One of the main problems when discussing mathematical topics with cranks seems to be there tendency to use unspecific terminology in their statements. (I reckon that there's a dependency relation between their usage of such terms and their problems to comprehend certain mathe- matical facts.) As a result very often such claims (expressed by cranks) are not even wrong. :-/ B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours ticks > over. Please help me out by answering 1 question. I'll make it as brief as possible, a yes/no question. Q: Do you realise that the computable reals contain every possible digit > sequence > up to infinite length? Up to but not including. Every finite binary digit sequence is, at least in theory, computable. In order for every infinite sequence to be computable as well, wouldn't have to be able to count the computation schemes required, one per sequence? And that is provably impossible. === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours ticks > over. > Please help me out by answering 1 question. > I'll make it as brief as possible, a yes/no question. > Q: Do you realise that the computable reals contain every possible digit > sequence > up to infinite length? Up to but not including. Every finite binary digit sequence is, at least in theory, computable. In order for every infinite sequence to be computable as well, wouldn't > have to be able to count the computation schemes required, one per > sequence? And that is provably impossible. Let me get this straight. You AGREE that ALL sequences up to oo length are computable, but you still think there's a NEW sequence of digits? How can you believe AS (all sequences) and ~AS at the same time? in the same breath? HOW DO YOU GET A NEW SEQUENCE when all sequences are computed? No finite part of your new sequence is unique. Therefore its not unique. Herc === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours > ticks > over. > Please help me out by answering 1 question. > I'll make it as brief as possible, a yes/no question. > Q: Do you realise that the computable reals contain every possible digit > sequence > up to infinite length? > Up to but not including. > Every finite binary digit sequence is, at least in theory, computable. > In order for every infinite sequence to be computable as well, wouldn't > have to be able to count the computation schemes required, one per > sequence? > And that is provably impossible. > Let me get this straight. You AGREE that ALL sequences up to oo length > are computable, but you still think there's a NEW sequence of digits? What deludes you to think I agree that all sequences are computable? As there are only countably many computation schemes but uncountably many binary sequences, such an agreement would be illogical. === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours > ticks > over. > Please help me out by answering 1 question. > I'll make it as brief as possible, a yes/no question. > Q: Do you realise that the computable reals contain every possible digit > sequence > up to infinite length? > Up to but not including. > Every finite binary digit sequence is, at least in theory, computable. > In order for every infinite sequence to be computable as well, wouldn't > have to be able to count the computation schemes required, one per > sequence? > And that is provably impossible. > > Let me get this straight. You AGREE that ALL sequences up to oo length > are computable, but you still think there's a NEW sequence of digits? What deludes you to think I agree that all sequences are computable? > As there are only countably many computation schemes but uncountably > many binary sequences, such an agreement would be illogical. You agreed above, up to infinite length. Do you agree or not? All sequences up to infinite length are computable. Herc === Subject: Re: the problem with Cantor > MoeBlee, I have a hard time understanding how that big brain of yours > ticks > over. > Please help me out by answering 1 question. > I'll make it as brief as possible, a yes/no question. > Q: Do you realise that the computable reals contain every possible > digit > sequence > up to infinite length? > Up to but not including. > Every finite binary digit sequence is, at least in theory, computable. > In order for every infinite sequence to be computable as well, wouldn't > have to be able to count the computation schemes required, one per > sequence? > And that is provably impossible. > Let me get this straight. You AGREE that ALL sequences up to oo length > are computable, but you still think there's a NEW sequence of digits? > What deludes you to think I agree that all sequences are computable? > As there are only countably many computation schemes but uncountably > many binary sequences, such an agreement would be illogical. You agreed above, up to infinite length. Up to infinite length is ambiguous, so I do not agree to it. All finite sequences are computable is not ambiguous, and to that I agree. Do you agree or not? I agree that all finite sequences are computable but not to up to infinite length. All sequences up to infinite length are computable. That is your mantra, not mine. Herc === Subject: Re: the problem with Cantor > All sequences up to infinite length are computable. All finite sequences are computable, obviously. So what? > Herc -- Alan Smaill === Subject: Re: the problem with Cantor > 1) Well, let's play along: I entered your link, pressed 1232323 and > got a left list of some 60 numbers, then an apaprently randomly > generated numbers, and a second list which apparently contains the > randly gen. number in its main diagoal entries...so? I don't > understand what is this supposed to show. I believe I know what Herc is trying to accomplish. Herc is making the following claim (quoted verbatim): Claim: Let L be a list of numbers in [0,1] such that for each d in > {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal > representation of L_i is d} is infinite. Let D be a random variable > uniform on [0,1]. Then almost surely there is a permutation L' of L > such that D is the diagonal of L'. But there are two big problems here: > 1) The claim most likely _fails_ in ZFC -- even for finite lists! > 2) Even if the claim does hold in ZFC, it has absolutely _no_ > bearing on the validity of Cantor in ZFC. Let's deal with 1) first. Herc's link is an applet consisting > of three steps: 1. Populate list -- clicking this causes a list of sixty real > numbers between zero and one, apparently in ternary since > only the digits 0,1,2, appear, each with sixty digits. > 2. Randomise diagonal -- clicking this causes a 61st real > number to appear. > 3. Generate list -- clicking this causes a reordering of the > first list to appear such that the real number chosen in the > second step appears on the diagonal. On the face of it, this should always work. We know that > there are 60! (60 factorial) permutations of the original list > and only 3^60 possible diagonals -- and factorials increase > much faster than exponentiation. (60! ~ 8*10^81, but > 3^60 ~ 4*10^28.) So there ought to be more than enough > permutations for all the possible diagonals. But I decided to try a little trick. I clicked on the first button > to populate the list -- but rather than clicking on the second > button, I manually typed in my own diagonal. What I did was > look at the first row, which happened to be: 0.021100001022012201100101102022102002102112020210111021110120 Then I used one of the usual tricks for generating an > antidiagonal (in Cantor's proof) from a diagonal to > make an antirow. I replaced the digits 1 and 2 with > 0 and the digit 0 with 1, to obtain the number: 0.100011110100100010011010010100010110010000101001000100001001 which I typed into the diagonal box. Then I clicked on the third button. Sure enough, only 59 > numbers appeared rather than 60. And of course, the > number that was present on the first list but missing > from the second list was the first number. It would be > _impossible_ for the first number to appear, since it > differs from the diagonal in _every_ digit! And it's easy to see that the same problem would > happen for _any_ list, whether finite or infinite, whether > saturated or unsaturated. Of course, Herc's claim is that almost surely such a > permutation exists. But what is almost surely? Does > Herc mean that the set of all real numbers for which a > permutation fails to exist has Lebesgue measure zero? If the new diagonal is a random variable in [0,1], the probability that it can be reordered with that diagonal is 1. SETS of reals with sufficient variety in the expansions don't really have diagonals, only lists do. A set can be ordered to have virtually any diagonal. There's infinite permutations to choose from. If so, then Herc may be right. So far, we've only shown > the antirows to be counterexamples -- where an > antirow is defined to be a number differing from every > digit from a number in a row. Suppose the first row contains the real number 1/2 -- > which is 0.111... in ternary. Then an antirow would be > any number consisting of only 0 and 2 in ternary -- and > the set of all such numbers is the famous Cantor > middle-third set, which has Lebesgue measure zero. Similarly, the set of all possible antirows for a given > row also has Lebesgue measure zero. And since this > is a _list_ of reals, there's exactly one real in this list > for every natural, so there are only countably many > such reals and their corresponding sets of antirows. So the set of all antirows for the entire list is the > countable union of null sets, and therefore itself null. Of course, this isn't to say that there doesn't exist a > counterexample to Herc's claim that doesn't happen to > be an antirow to any row at all. But the set of the only > _known_ counterexamples to Herc's claim does have > Lebesgue measure zero. But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! Indeed, let's compare my proof above to Cantor's: I show it's impossible for an _antirow_ to be a _diagonal_. > Cantor shows it's impossible for an _antidiagonal_ to be a _row_. If we think about this for a moment, we see that in my > proof, I only claim to show that there exists numbers > which can't be on the diagonal. That a counterexample > happens to be an antirow is irrelevant. I only care about > the diagonal, not rows or antirows. Similarly, Cantor claims that there exist real numbers > which can't be in any row of the list. That a counterexample > happens to be an antidiagonal is irrevelevant. Cantor cares > only about _rows_, not the diagonal or the antidiagonal. It's possible for a real number to be the diagonal of one > permutation but an antidiagonal of another permutation. Herc > was apparently hoping that this wouldn't be the case, but > we can even consider the example: 168 > 249 > 357 So an antidiagonal is 258, but if we reorder the list: 249 > 357 > 168 suddenly 258's the diagonal. Now 369's an antidiagonal, but: 357 > 168 > 249 now 369's the diagonal. Therefore, Herc's claim, even if it were true, would neither > prove nor disprove _anything_ about Cantor. > 2) What is superinfinity? And why is it mine? By superinfinity, most likely Herc (pejoratively) means > uncountable infinity. And it's yours (Tonio's), as much as > it belongs to every standard mathematician who believes > in its existence (i.e., who adheres to a set theory, such > as ZFC, which proves the existence of uncountable sets), > as opposed to Herc, who doesn't believe in the existence > of uncountable sets. ZFC doesn't prove the existence of uncountable sets, it finds a new number not on the list and human operaters interpret this as there must be some larger set type than infinity. It's like a young child arguing 0.999.. is different to 1. Every possible digit sequence up to infinite length is on the computable reals list so what new sequence of digits did it find? Herc === Subject: Re: the problem with Cantor ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? Second order ZF does demand the existence of uncountable sets. Finiteness is a second order property but finiteness is so obvious that nobody should have any problem using it as a starting assumption. http://en.wikipedia.org/wiki/Peano_axioms#Nonstandard_models Cantor's theorem can be proven in a variety of different ways. The diagonal proof is one way but Cantor's theorem can also be proven using membership. Below is a proof that was posted on this newsgroup a while back, using membership, and it holds for any set: Cantor's Theorem: For any set S, |S| < |P(S)|. If (a) |S|<=|P(S)| and (b) |S|=/=|P(S)| are both true then Cantor's theorem is proven. Letting q be the one to one function q:S-->P(S) such that s-->{s}, it is easy to see that (a) is true. To see that (b) is true, assume the contrary. That is, assume there is a bijection r:S-->P(S). Let T be the set: T = {s in S : s not in r(s)} Since r is assumed to be onto, there should be a t in S such that (c) T=r(t). If t is in T then, by definition, t is not in r(t) and this contradicts (c). So it must be that t is not in T but then (c) means that t is in T and this is another contradiction. Thus, r is not onto and therefore not a bijection. QED. k === Subject: Re: the problem with Cantor > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? Second order ZF does demand the existence of uncountable sets. Finiteness > is a second order property but finiteness is so obvious that nobody should > have any problem using it as a starting assumption. http://en.wikipedia.org/wiki/Peano_axioms#Nonstandard_models Cantor's theorem can be proven in a variety of different ways. The diagonal > proof is one way but Cantor's theorem can also be proven using membership. > Below is a proof that was posted on this newsgroup a while back, using > membership, and it holds for any set: Cantor's Theorem: For any set S, |S| < |P(S)|. If (a) |S|<=|P(S)| and (b) |S|=/=|P(S)| are both true then Cantor's theorem > is proven. Letting q be the one to one function q:S-->P(S) such that > s-->{s}, it is easy to see that (a) is true. To see that (b) is true, > assume the contrary. That is, assume there is a bijection r:S-->P(S). Let > T be the set: T = {s in S : s not in r(s)} Since r is assumed to be onto, there should be a t in S such that (c) > T=r(t). If t is in T then, by definition, t is not in r(t) and this > contradicts (c). So it must be that t is not in T but then (c) means that t > is in T and this is another contradiction. Thus, r is not onto and > therefore not a bijection. QED. > This one is my favorite! Which box contains the number of all the boxes that don't contain their own number? There is none - therefore - SUPERINFINITY Herc === Subject: Re: the problem with Cantor > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. In ZFC, one has the following: If A bijects with B then Card(A) = Card(B). If A injects into B then Card(A) <= Card(B) If A injects into B but B does not inject into A then Card(A) < Card(b) It is trivial in ZFC that any set injects into its power set. It is less trivial but still true in ZFC that no power set of a set injects back into that set. Let N be the set of naturals in ZFC and R be the set of lower Dedekind cut sets. One can show that in ZFC, Card(R) = Card(P(N)) and Card(N) < Card(P(N)). > It's like a > young child arguing 0.999.. is different to 1. It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? First prove that a list of all computable sequences is itself computable. If that can be proven at all, which is not obvious to me, then the new sequence must be an uncomputable sequence. And once we have a computable sequence of all computable sequences, we an easily create at least as many uncomputable sequences as there are computable ones. === Subject: Re: the problem with Cantor > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. In ZFC, one has the following: If A bijects with B then Card(A) = Card(B). > If A injects into B then Card(A) <= Card(B) > If A injects into B but B does not inject into A then Card(A) < Card(b) It is trivial in ZFC that any set injects into its power set. > It is less trivial but still true in ZFC that no power set of a set > injects back into that set. Let N be the set of naturals in ZFC and R be the set of lower Dedekind > cut sets. One can show that in ZFC, Card(R) = Card(P(N)) and Card(N) < > Card(P(N)). > It's like a > young child arguing 0.999.. is different to 1. It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? First prove that a list of all computable sequences is itself computable. If that can be proven at all, which is not obvious to me, then the new > sequence must be an uncomputable sequence. And once we have a computable sequence of all computable sequences, we > an easily create at least as many uncomputable sequences as there are > computable ones. Surely a TM could enumerate subsets. The output of a TM is a series of 1s and 0s. e.g. 000011111000111100011000000 Let the 0's be dividers between unary outputs. this becomes {5,4,2} Then the enumeration of subsets is merely UTM(n,0) 1 ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. > In ZFC, one has the following: > If A bijects with B then Card(A) = Card(B). > If A injects into B then Card(A) <= Card(B) > If A injects into B but B does not inject into A then Card(A) < Card(b) > It is trivial in ZFC that any set injects into its power set. > It is less trivial but still true in ZFC that no power set of a set > injects back into that set. > Let N be the set of naturals in ZFC and R be the set of lower Dedekind > cut sets. > One can show that in ZFC, Card(R) = Card(P(N)) and Card(N) < > Card(P(N)). > > It's like a > young child arguing 0.999.. is different to 1. > It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? > First prove that a list of all computable sequences is itself computable. > If that can be proven at all, which is not obvious to me, then the new > sequence must be an uncomputable sequence. > And once we have a computable sequence of all computable sequences, we > an easily create at least as many uncomputable sequences as there are > computable ones. > Surely a TM could enumerate subsets. Given a set of naturals, N, either in ZFC, NBG, or some generally similar set theory. let us see a description of any TM which will ennumerate ALL of its subsets. The output of a TM is a series of 1s and 0s. e.g. 000011111000111100011000000 Let the 0's be dividers between unary outputs. this becomes {5,4,2} Then the enumeration of subsets is merely UTM(n,0) 1 ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. > In ZFC, one has the following: > If A bijects with B then Card(A) = Card(B). > If A injects into B then Card(A) <= Card(B) > If A injects into B but B does not inject into A then Card(A) < Card(b) > It is trivial in ZFC that any set injects into its power set. > It is less trivial but still true in ZFC that no power set of a set > injects back into that set. > Let N be the set of naturals in ZFC and R be the set of lower Dedekind > cut sets. > One can show that in ZFC, Card(R) = Card(P(N)) and Card(N) < > Card(P(N)). > It's like a > young child arguing 0.999.. is different to 1. > It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? > First prove that a list of all computable sequences is itself computable. > If that can be proven at all, which is not obvious to me, then the new > sequence must be an uncomputable sequence. > And once we have a computable sequence of all computable sequences, we > an easily create at least as many uncomputable sequences as there are > computable ones. > > Surely a TM could enumerate subsets. Given a set of naturals, N, either in ZFC, NBG, or some generally > similar set theory. let us see a description of any TM which will > ennumerate ALL of its subsets. > The output of a TM is a series of 1s and 0s. > e.g. 000011111000111100011000000 > Let the 0's be dividers between unary outputs. > this becomes {5,4,2} > Then the enumeration of subsets is merely > UTM(n,0) 1 If so which is ennumerated first, the set of all even naturals or the > set of all odd naturals? > This will output every possible subset of N, i.e. the powerset, indexed by N. No it won't. OK, it will produce all the finite subsets of N. We need to input each natural into the UTM, and the UTM responds either YES (1), or NO (0), as to whether that natural is included in the subset. UTM(a,b) = 1 IFF b is a member of the ath subset. That will enumerate ALL subsets, finite and infinite of N. Herc === Subject: Re: the problem with Cantor > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. > In ZFC, one has the following: > If A bijects with B then Card(A) = Card(B). > If A injects into B then Card(A)<= Card(B) > If A injects into B but B does not inject into A then Card(A)< Card(b) > It is trivial in ZFC that any set injects into its power set. > It is less trivial but still true in ZFC that no power set of a set > injects back into that set. > Let N be the set of naturals in ZFC and R be the set of lower Dedekind > cut sets. > One can show that in ZFC, Card(R) = Card(P(N)) and Card(N)< > Card(P(N)). > It's like a > young child arguing 0.999.. is different to 1. > It is a different name for the same number. > Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? > First prove that a list of all computable sequences is itself computable. > If that can be proven at all, which is not obvious to me, then the new > sequence must be an uncomputable sequence. > And once we have a computable sequence of all computable sequences, we > an easily create at least as many uncomputable sequences as there are > computable ones. > Surely a TM could enumerate subsets. > Given a set of naturals, N, either in ZFC, NBG, or some generally > similar set theory. let us see a description of any TM which will > ennumerate ALL of its subsets. > The output of a TM is a series of 1s and 0s. > e.g. 000011111000111100011000000 > Let the 0's be dividers between unary outputs. > this becomes {5,4,2} > Then the enumeration of subsets is merely > UTM(n,0) 1 Does this schema enumerate any of the infinite subsets of N? > If so which is ennumerated first, the set of all even naturals or the > set of all odd naturals? > This will output every possible subset of N, i.e. the powerset, indexed by N. > No it won't. > OK, it will produce all the finite subsets of N. > We need to input each natural into the UTM, and the UTM responds > either YES (1), or NO (0), as to whether that natural is included in the subset. > UTM(a,b) = 1 IFF b is a member of the ath subset. > That will enumerate ALL subsets, finite and infinite of N. If you want to get the set of all prime numbers, then what does 'a' > look like? > Has noone written a TM that determines if its input is prime? It would probably take 100 or so states to program, so 'a' would be a natural about 100 to 1000 digits long. The emulated TM would go through all numbers from d = 2 to b/2 and test if b div d has a remainder. If it finds a number that divides b with no remainder then b is not prime and it outputs 0. If it finds no divisors it outputs 1. UTM(a, 1) = 0 UTM(a, 2) = 1 UTM(a, 3) = 1 UTM(a, 4) = 0 UTM(a, 5) = 1 UTM(a, 6) = 0 ... Shouldn't be that hard to calculate a. Herc === Subject: Re: the problem with Cantor > If you want to get the set of all prime numbers, then what does 'a' > look like? > > Has noone written a TM that determines if its input is prime? It would > probably take 100 or so states to program, so 'a' would be a natural about > 100 to 1000 digits long. The emulated TM would go through all numbers from d = 2 to b/2 and test if > b div d has a remainder. If it finds a number that divides b with no > remainder > then b is not prime and it outputs 0. If it finds no divisors it outputs 1. UTM(a, 1) = 0 > UTM(a, 2) = 1 > UTM(a, 3) = 1 > UTM(a, 4) = 0 > UTM(a, 5) = 1 > UTM(a, 6) = 0 > ... Shouldn't be that hard to calculate a. Herc If you think it is so easy, then calculate it. === Subject: Re: the problem with Cantor UTM(a,b) = 1 IFF b is a member of the ath subset. That is only possible if one can list the subsets of N, so that every subset of N is an ath subset, which has been proved, in ZFC and the like, to be impossible. That will enumerate ALL subsets, finite and infinite of N. Not if there are more subsets that a-indices. === Subject: Re: the problem with Cantor > UTM(a,b) = 1 IFF b is a member of the ath subset. That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. Ahh, the old which box contains the numbers of all the boxes that don't contain their own number proof! Just by asking this question, which is impossible, or posing the theorem, you prove superinfinity. > That will enumerate ALL subsets, finite and infinite of N. Not if there are more subsets that a-indices. I've given you the computer program, what subset can all computer programs miscalculate? Herc === Subject: Re: the problem with Cantor > > UTM(a,b) = 1 IFF b is a member of the ath subset. > That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. Ahh, the old which box contains the numbers of all the boxes that don't > contain their own number proof! Just by asking this question, which is > impossible, or posing the theorem, you prove superinfinity. In ZFC It is rather more directly proved than herkimer admits. > That will enumerate ALL subsets, finite and infinite of N. > Not if there are more subsets that a-indices. I've given you the computer program, what subset can all computer > programs miscalculate? Why should any program miscalculate any of the sets that it can calculate? As for the many it cannot calculate at all, why should it miscalculate them either? === Subject: Re: the problem with Cantor <_DSlk.25641$IK1.251@news-server.bigpond.net.au> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > UTM(a,b) = 1 IFF b is a member of the ath subset. > That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. Ahh, the old which box contains the numbers of all the boxes that don't > contain their own number proof! æJust by asking this question, which is > impossible, or posing the theorem, you prove superinfinity. > ************************************************************** Second time, at least, that you've mentioned that thing about boxes and superinfinity, whatever that is. What part of Cantor's Theorem using the argument found in Russell's paradox isn't clear to you? The theorem is pretty succinct: it says that for any set X, we have |X| < |P(X)| , |.| meaning cardinal of set and P(X) the power set of X. cardinality higher than N's, meaning: there's a set, namely P(N), in which N can be embedded but from which there is no possible embedding into N. What's unclear here? Where are you stuck? Tonio === Subject: Re: the problem with Cantor > UTM(a,b) = 1 IFF b is a member of the ath subset. > That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. Ahh, the old which box contains the numbers of all the boxes that don't > contain their own number proof! Just by asking this question, which is > impossible, or posing the theorem, you prove superinfinity. > ************************************************************** -Second time, at least, that you've mentioned that thing about boxes -and superinfinity, whatever that is. - -What part of Cantor's Theorem using the argument found in Russell's -paradox isn't clear to you? The theorem is pretty succinct: it says -that for any set X, we have |X| < |P(X)| , |.| meaning cardinal of set -and P(X) the power set of X. - -cardinality higher than N's, meaning: there's a set, namely P(N), in -which N can be embedded but from which there is no possible embedding -into N. - -What's unclear here? Where are you stuck? I have given a description of a mapping from N to P(N). > UTM(a,b) = 1 IFF b is a member of the ath subset. This will compute all possible subsets (the powerset), finite and infinite of N. primes - its there odds - its there evens - its there {1,2,3} - its there How on earth can the set_of_all_computer_programs possibly miss some simple subset calculation on natural numbers? I have full trust in the capacity of a Universal Turing Machine to come up with a numerical list of all possible subsets of N. I don't believe in your proof 1 bit. All this proof does... > Cantor's Theorem: For any set S, |S| < |P(S)|. If (a) |S|<=|P(S)| and (b) |S|=/=|P(S)| are both true then Cantor's theorem > is proven. Letting q be the one to one function q:S-->P(S) such that > s-->{s}, it is easy to see that (a) is true. To see that (b) is true, > assume the contrary. That is, assume there is a bijection r:S-->P(S). Let > T be the set: T = {s in S : s not in r(s)} Since r is assumed to be onto, there should be a t in S such that (c) > T=r(t). If t is in T then, by definition, t is not in r(t) and this > contradicts (c). So it must be that t is not in T but then (c) means that t > is in T and this is another contradiction. Thus, r is not onto and > therefore not a bijection. QED is self reference and negate itself. Its just a mathematical tongue twister in disguise. Its exactly analagous to having an infinite sized room full of numbered boxes and asking which box contains the numbers of all the boxes that don't contain their own number?. It's just a non solvable puzzle its not a new subset. It reduces to which box (subset) contains its own number and also doesn't contain its own number?. You people are plugging rubbish into ZFC and you are getting rubbish out of it. Herc === Subject: Re: the problem with Cantor > I have given a description of a mapping from N to P(N). UTM(a,b) = 1 IFF b is a member of the ath subset. Such a UTM assumes a priori the existence of what herkimer wishes to conclude, that there is a surjective mapping from N to P(N). Such arguments are called circular, and are not accepted by the logically fastidious. This will compute all possible subsets (the powerset), finite and infinite of > N. > primes - its there > odds - its there > evens - its there > {1,2,3} - its there Not unless one has, a priori, a surjection from N to P(N). How on earth can the set_of_all_computer_programs possibly miss some > simple subset calculation on natural numbers? By being of smaller cardinality. I have full trust in the capacity of a Universal Turing Machine to come up > with a numerical list of all possible subsets of N. Your faith does not override the limitations of ZFC and other set theories in which no such Turing machines can be constructed. I don't believe in your proof 1 bit. And we do not believe in your circular arguments. You people are plugging rubbish into ZFC and you are getting rubbish out of > it. Whereas herkimer's rubbish has even less justification than ZFC provides. === Subject: Re: the problem with Cantor posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > UTM(a,b) = 1 IFF b is a member of the ath subset. > That is only possible if one can list the subsets of N, so that every > subset of N is an ath subset, which has been proved, in ZFC and the > like, to be impossible. > Ahh, the old which box contains the numbers of all the boxes that don't > contain their own number proof! Just by asking this question, which is > impossible, or posing the theorem, you prove superinfinity. ************************************************************** -Second time, at least, that you've mentioned that thing about boxes > -and superinfinity, whatever that is. > - > -What part of Cantor's Theorem using the argument found in Russell's > -paradox isn't clear to you? The theorem is pretty succinct: it says > -that for any set X, we have |X| < |P(X)| , |.| meaning cardinal of set > -and P(X) the power set of X. > - > -cardinality higher than N's, meaning: there's a set, namely P(N), in > -which N can be embedded but from which there is no possible embedding > -into N. > - > -What's unclear here? Where are you stuck? I have given a description of a mapping from N to P(N). > UTM(a,b) = 1 IFF b is a member of the ath subset. This will compute all possible subsets (the powerset), finite and infinite of N. > primes - its there > odds - its there > evens - its there > {1,2,3} - its there > **************************************************************** Where's that mapping from N to P(N)? All I can see is UTM(a,b) = 1 iff b is a member of a-th subset, and this is NOT a map with N as definition set: it appears to be that b is a natural number and a is some set, perhaps an element of P(N). Moreover: if a indeed is a way of counting elements of P(N), i.e. subsets of N, then we're already wrong: you can't count these guys! Meaning: you can't put the elements of P(N) in a 1-1 correspondence with elements of N, and then to talk of nth subset is meaningless ***************************************************************** > How on earth can the set of all computer programs possibly miss some > simple subset calculation on natural numbers? > ************************************************************* I don't really know, but (1) what is the set of all computer programs? (2) What is a simple subset calculation on natural numbers? You seem to believe that the set of all computer programs, whatever this means, is so powerful and mighty that it can't possible miss something that, apparently, is simple. Why do you believe that? ************************************************************* > I have full trust in the capacity of a Universal Turing Machine to come up > with a numerical list of all possible subsets of N. > *************************************************************** Why? Is this some kind of axiom or is this assumed in UTM's definition? And again: you seem to beleive you can ennumerate all possible subsets of N, but you really can't: this is EXACTLY what Cantor, among oither things, proved. *************************************************************** > I don't believe in your proof 1 bit. æAll this proof does... > Cantor's Theorem: For any set S, |S| < |P(S)|. > If (a) |S|<=|P(S)| and (b) |S|=/=|P(S)| are both true then Cantor's theorem > is proven. æLetting q be the one to one function q:S-->P(S) such that > s-->{s}, it is easy to see that (a) is true. æTo see that (b) is true, > assume the contrary. æThat is, assume there is a bijection r:S-->P(S). æLet > T be the set: > T = {s in S : s not in r(s)} > Since r is assumed to be onto, there should be a t in S such that (c) > T=r(t). æIf t is in T then, by definition, t is not in r(t) and this > contradicts (c). æSo it must be that t is not in T but then (c) means that t > is in T and this is another contradiction. æThus, r is not onto and > therefore not a bijection. QED is self reference and negate itself. æIts just a mathematical tongue twister in disguise. ************************************************************ Do you know the reductio ad absurdum method to prove mathematical statements? Is it clear to you? If it is not, then you have to research a little on it, and if it is then: do you reject this method? If you do, why? And if you don't, then what part of CT's proof you don't like/accept? ************************************************************ > Its exactly analagous to having an infinite sized room full of numbered boxes and > asking which box contains the numbers of all the boxes that don't contain their > own number?. ************************************************************ No, it is not analogous to this...not even close, leave alone exactly analogous. It could be so-so analogous if you'd say: there are some numbered balls in a room, and we have in another room all the possible boxes formed by introducing some (all, none) of the balls of the first room in a box (thus, we have boxes with one ball, with two balls, with no ball at all, with all the balls, etc), and these boxes are numbered too in some way. We're going to show that there are more boxes than balls, and we do this by assuming otherwise ==> there must be a box which contains all the balls which are not contained in the box with the same number as the balls. You see, the above example comes up pretty cumbersome, complex, and it even assumes that we have a huge number of copies of each ball (as many copies as possible boxes containing that specific ball), and after all this work we haven't attained anything better... ************************************************************ æIt's just a non solvable puzzle its not a new subset. æIt reduces to > which box (subset) contains its own number and also doesn't contain its own number?. You people are plugging rubbish into ZFC and you are getting rubbish out of it. > **************************************************************** So far ZFC has yielded pretty nice results. Have you ever thought, and I'm asking this without the slightest intention to offend, of the possibility that you don't REALLY grasp all this basic logic-set-theory stuff? Could it be that your mathematical education is lacking some basic training in these matters? Are you a mathematician? Of course, not being one doesn't automatically disqualifies you to give you opinion, but perhaps there's some stuff you haven't yet studied...? Tonio > Herc === Subject: Re: the problem with Cantor [...] > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? You can define a real number as the output of a program, as in You say that only computable reals exist. The accepted constructions of the real numbers don't mention computability. To a Platonist, the set of reals is something out there and the reals can't be counted with 1, 2, 3 .... , i.e. they form an uncountable set. You say the Platonists are wrong. To convince Platonists that they are wrong, one way would be to get a contradiction from ZFC, such as 1 =/= 1 . Besides arguments like that, it should be very hard to convert a Platonic set theorist. David Bernier === Subject: Re: the problem with Cantor > [...] > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? You can define a real number as the output of a program, as in You say that only computable reals exist. The accepted constructions > of the real numbers don't mention computability. To a Platonist, > the set of reals is something out there and the reals can't > be counted with 1, 2, 3 .... , i.e. they form an uncountable set. > Platonist and ZFC adherent aren't synonymous. > You say the Platonists are wrong. To convince Platonists that > they are wrong, one way would be to get a contradiction > from ZFC, such as 1 =/= 1 . Besides arguments like that, > it should be very hard to convert a Platonic set theorist. David Bernier Platonist set theorist and ZFC adherent aren't synonymous. ZF's universe, in any larger universe, one of which exists else ZF would be complete, and in being so by argument within itself inconsistent, is irregular and contains itself, thus ZF is inconsistent. That's not so say regular set theory isn't the most applicable foundation for finite combinatorics, because it's an excellent and more or less naturally consistent foundation for finite combinatorics. In the infinite, there's much to be considered about that it is the nature of infinity to have what might seem unintuitive (although not necessarily not intuitionistic) properties. the more closely they are measured, infinitesimal, and the universe is larger the more comprehensively it is measure, infinite, in nature, physics. In terms of particularly the unit interval of reals, in terms of real numbers vis-a-vis natural integers, that there is a way to construct them in their natural order from zero through one, while that was the obvious way to do it in terms of countable additivity in analysis, the density of the individua of the continuum leads to ready conflicts with the notion that they can be so arrayed. While that may be so, in terms of modeling the natural/unit equivalency function as a limit of real functions, much as the unit impulse function is modeled as a limit of real functions, leads to then these implicit mathematical objects, real numbers, only and all of which are on the unit interval, thus that the resulting construction of those numbers leads to that, for example, the antidiagonal and nested intervals results don't apply. I think more people should be interested in surpassing what is basically a limitation of adherence to the cumulative hierarchy of transfinite ordinals and their cardinals. Instead, by analyzing the polydimensional points of the real number line, in the penultimate and ultimate real number space(s), new applications will be discovered, in concordance with nature. Ross F. === Subject: Re: the problem with Cantor > [...] > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? You can define a real number as the output of a program, as in You say that only computable reals exist. The accepted constructions > of the real numbers don't mention computability. To a Platonist, > the set of reals is something out there and the reals can't > be counted with 1, 2, 3 .... , i.e. they form an uncountable set. You say the Platonists are wrong. To convince Platonists that > they are wrong, one way would be to get a contradiction > from ZFC, such as 1 =/= 1 . Besides arguments like that, > it should be very hard to convert a Platonic set theorist. David Bernier You just admitted you are all impervious to reason. I still have no comments on the fact that 1/ the computable reals compute all digit expansions to infinite length 2/ the diagonal of a real list with sufficient variety in the expansions is independent of the list Until YOU can address those issues, everyone in the future reading these historic posts No offence David you've done a pretty good job at being open, as have some others. Herc === Subject: Re: the problem with Cantor > [...] > ZFC doesn't prove the existence of uncountable sets, it finds > a new number not on the list and human operaters interpret this > as there must be some larger set type than infinity. It's like a > young child arguing 0.999.. is different to 1. Every possible digit > sequence up to infinite length is on the computable reals list so > what new sequence of digits did it find? > You can define a real number as the output of a program, as in > You say that only computable reals exist. The accepted constructions > of the real numbers don't mention computability. To a Platonist, > the set of reals is something out there and the reals can't > be counted with 1, 2, 3 .... , i.e. they form an uncountable set. > You say the Platonists are wrong. To convince Platonists that > they are wrong, one way would be to get a contradiction > from ZFC, such as 1 =/= 1 . Besides arguments like that, > it should be very hard to convert a Platonic set theorist. > David Bernier You just admitted you are all impervious to reason. On the contrary, he has just said that it is difficult to convince a Platonist that he is wrong without having any proof that that Platonist is wrong. I still have no comments on the fact that > 1/ the computable reals compute all digit expansions to infinite length That presumes falsely that all such expansions are computable. > 2/ the diagonal of a real list with sufficient variety in the expansions is > independent > of the list There are at least as many anti-diagonals not in list as expansion as in it, but each such anti-diagonal is dependent on the list from which it is produced. Until YOU can address those issues, everyone in the future reading > entrenched text book parrots you all are. There will always be those who stubbornly refuse to be persuaded by your own nonproofs of your own beliefs. === Subject: Re: the problem with Cantor > 1) Well, let's play along: I entered your link, pressed 1232323 and > got a left list of some 60 numbers, then an apaprently randomly > generated numbers, and a second list which apparently contains the > randly gen. number in its main diagoal entries...so? I don't > understand what is this supposed to show. I believe I know what Herc is trying to accomplish. Herc is making the following claim (quoted verbatim): Claim: Let L be a list of numbers in [0,1] such that for each d in > {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal > representation of L_i is d} is infinite. Let D be a random variable > uniform on [0,1]. Then almost surely there is a permutation L' of L > such that D is the diagonal of L'. But there are two big problems here: > 1) The claim most likely _fails_ in ZFC -- even for finite lists! > 2) Even if the claim does hold in ZFC, it has absolutely _no_ > bearing on the validity of Cantor in ZFC. Let's deal with 1) first. Herc's link is an applet consisting > of three steps: 1. Populate list -- clicking this causes a list of sixty real > numbers between zero and one, apparently in ternary since > only the digits 0,1,2, appear, each with sixty digits. But the Cantor 'anti-diagonal' construction does not work in ternary any more than it does in binary, as in ternary, one has to use either 0 or 2 in ternary to be different from 1, and allowing either a 0 or a 2 to appearin the anti-diagonal at least theoretically allows the anti-diagonal to be the dual representation of a listed number. With bases of four and larger, the anti-diagonal construction can always avoid using the digits, 0 and the base-minus-one, which are required in dual representations. > 2. Randomise diagonal -- clicking this causes a 61st real > number to appear. > 3. Generate list -- clicking this causes a reordering of the > first list to appear such that the real number chosen in the > second step appears on the diagonal. On the face of it, this should always work. We know that > there are 60! (60 factorial) permutations of the original list > and only 3^60 possible diagonals -- and factorials increase > much faster than exponentiation. (60! ~ 8*10^81, but > 3^60 ~ 4*10^28.) So there ought to be more than enough > permutations for all the possible diagonals. But I decided to try a little trick. I clicked on the first button > to populate the list -- but rather than clicking on the second > button, I manually typed in my own diagonal. What I did was > look at the first row, which happened to be: 0.021100001022012201100101102022102002102112020210111021110120 Then I used one of the usual tricks for generating an > antidiagonal (in Cantor's proof) from a diagonal to > make an antirow. I replaced the digits 1 and 2 with > 0 and the digit 0 with 1, to obtain the number: 0.100011110100100010011010010100010110010000101001000100001001 which I typed into the diagonal box. Then I clicked on the third button. Sure enough, only 59 > numbers appeared rather than 60. And of course, the > number that was present on the first list but missing > from the second list was the first number. It would be > _impossible_ for the first number to appear, since it > differs from the diagonal in _every_ digit! And it's easy to see that the same problem would > happen for _any_ list, whether finite or infinite, whether > saturated or unsaturated. Of course, Herc's claim is that almost surely such a > permutation exists. But what is almost surely? Does > Herc mean that the set of all real numbers for which a > permutation fails to exist has Lebesgue measure zero? If so, then Herc may be right. So far, we've only shown > the antirows to be counterexamples -- where an > antirow is defined to be a number differing from every > digit from a number in a row. Suppose the first row contains the real number 1/2 -- > which is 0.111... in ternary. Then an antirow would be > any number consisting of only 0 and 2 in ternary -- and > the set of all such numbers is the famous Cantor > middle-third set, which has Lebesgue measure zero. Similarly, the set of all possible antirows for a given > row also has Lebesgue measure zero. And since this > is a _list_ of reals, there's exactly one real in this list > for every natural, so there are only countably many > such reals and their corresponding sets of antirows. So the set of all antirows for the entire list is the > countable union of null sets, and therefore itself null. Of course, this isn't to say that there doesn't exist a > counterexample to Herc's claim that doesn't happen to > be an antirow to any row at all. But the set of the only > _known_ counterexamples to Herc's claim does have > Lebesgue measure zero. But even if this makes Herc's claim true, we reach > problem 2) -- it has nothing to do with Cantor at all! Indeed, let's compare my proof above to Cantor's: I show it's impossible for an _antirow_ to be a _diagonal_. > Cantor shows it's impossible for an _antidiagonal_ to be a _row_. If we think about this for a moment, we see that in my > proof, I only claim to show that there exists numbers > which can't be on the diagonal. That a counterexample > happens to be an antirow is irrelevant. I only care about > the diagonal, not rows or antirows. Similarly, Cantor claims that there exist real numbers > which can't be in any row of the list. That a counterexample > happens to be an antidiagonal is irrevelevant. Cantor cares > only about _rows_, not the diagonal or the antidiagonal. It's possible for a real number to be the diagonal of one > permutation but an antidiagonal of another permutation. Herc > was apparently hoping that this wouldn't be the case, but > we can even consider the example: 168 > 249 > 357 So an antidiagonal is 258, but if we reorder the list: 249 > 357 > 168 suddenly 258's the diagonal. Now 369's an antidiagonal, but: 357 > 168 > 249 now 369's the diagonal. Therefore, Herc's claim, even if it were true, would neither > prove nor disprove _anything_ about Cantor. > 2) What is superinfinity? And why is it mine? By superinfinity, most likely Herc (pejoratively) means > uncountable infinity. And it's yours (Tonio's), as much as > it belongs to every standard mathematician who believes > in its existence (i.e., who adheres to a set theory, such > as ZFC, which proves the existence of uncountable sets), > as opposed to Herc, who doesn't believe in the existence > of uncountable sets. === Subject: Re: the problem with Cantor > I don't care Certainly not about truth. === Subject: Re: the problem with Cantor > I don't care Certainly not about truth. The truth is there is nothing larger than infinity. You have failed to give me a new sequence of digits not on the computable reals. Cantor's proof is an exercise in going round in circles. Herc === Subject: Re: the problem with Cantor > > I don't care > Certainly not about truth. The truth is there is nothing larger than infinity. > You have failed to give me a new sequence of digits not on the computable > reals. You have failed to give us any listing of all computable reals. The existence of such a listing would be proof that the computable reals are countable, and the non-existence of any such listing would be evidence off that set's uncountability. > Cantor's proof is an exercise in going round in circles. It is an eample of clearer thinking than Herkimer has yet demonstrated. === Subject: Re: the problem with Cantor > > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. > You have failed to give me a new sequence of digits not on the computable > reals. You have failed to give us any listing of all computable reals. > The existence of such a listing would be proof that the computable reals > are countable, and the non-existence of any such listing would be > evidence off that set's uncountability. > Cantor's proof is an exercise in going round in circles. It is an eample of clearer thinking than Herkimer has yet demonstrated. Use a Universal Turing Machine to compute all the reals. let the jth digit of the ith real = UTM(i,j) mod 10 where UTM has 2 parameters, the TM it is emulating and the input tape to the TM. UTM(TM, input) = output can continue on computing every computable real. Diagonalisation will not produce any new sequence of digits as I have demonstrated. You are far too stupid to admit that and maintain an argument. Herc === Subject: Re: the problem with Cantor > Use a Universal Turing Machine to compute all the reals. Which of the uncountably many incomputable reals will it compute first? === Subject: Re: the problem with Cantor > Use a Universal Turing Machine to compute all the reals. Which of the uncountably many incomputable reals will it compute first? I meant to say, use a UTM to compute all the computable reals, that's what you asked for. Although there are no incomputable reals. Herc === Subject: Re: the problem with Cantor > > Use a Universal Turing Machine to compute all the reals. > Which of the uncountably many incomputable reals will it compute first? I meant to say, use a UTM to compute all the computable reals, that's what > you asked for. Some of those computable reals will take all the time your UTM has. So it may never get past the first one. Although there are no incomputable reals. What exists outside UTMs is not necessarily computable. If Herkimer's universe is limited to what a UTM can produce, his life is pretty lifeless. === Subject: Re: the problem with Cantor > Use a Universal Turing Machine to compute all the reals. > Which of the uncountably many incomputable reals will it compute first? > I meant to say, use a UTM to compute all the computable reals, that's what > you asked for. Some of those computable reals will take all the time your UTM has. So it may never get past the first one. > Although there are no incomputable reals. What exists outside UTMs is not necessarily computable. If Herkimer's universe is limited to what a UTM can produce, his life is > pretty lifeless. brrrrr bing *pretty* it does not compute Herc === Subject: Re: the problem with Cantor > I don't care > Certainly not about truth. The truth is there is nothing larger than infinity. > You have failed to give me a new sequence of digits not on the computable reals. > Cantor's proof is an exercise in going round in circles. Herc Suppose we have a list LF of the numbers that have a terminating decimal expansion: 2.3, 3.14159, 2000000, 0.00000000000056, etc. 1/3 = 0.33333.... = __ 0.3 [ 0.3periodic]. The first 5 decimals of 1/3 are the same as those of 0.33333 , and so on for 5, 7, ... 100, ... 1000 decimals and more. But if we take a fixed number 'c' on LF and compare it to __ 0.3 [ 0.3periodic] , then either __ c < 0.3 or __ c>0.3 . Don't you think the same thing happens with Omega_{Herc} and LF? After all, you're only comparing a finite (but arbitrarily large) number of decimals. In other words, what works for Omega_{Herc} and LF should work for __ 0.3 and LF ... David Bernier === Subject: Re: the problem with Cantor > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. > You have failed to give me a new sequence of digits not on the computable reals. > Cantor's proof is an exercise in going round in circles. > Herc > > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c < 0.3 or > __ > c>0.3 . Don't you think the same thing happens with Omega_{Herc} and LF? After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... David Bernier You've shattered the illusion that Omega is on some list but the problem with Cantors proof still stands. Every finite prefix of possible sequences is computable, so the original diagonalisation technique fails to find a new sequence. 123 456 789 In this simple finite list, the diagonal is 159. The anti-diagonal is 261. It looks like diagonalisation actually finds new sequences of digits. but when you apply it to the set of computable reals it doesn't, the list is already saturated with every possible sequence. That's the claim, that there is a problem, not that Omega is on the list. The computable reals contains every possible sequence up to infinite length, and you tell me with a cloak and digger trick that you can find new sequences. You can't. All the sequences are computed, up to oo length, end of story. Have you taken a look at http://www.freewebs.com/namesort/linux.html It proves that a list of reals that contains every digit oo times in each column can be sorted to have any random diagonal. The diagonal is independent of such lists, e.g. the computable reals. That's another problem with Cantor's proof, the diagonal does not even depend on the list, it's just a random variable. Herc === Subject: Re: the problem with Cantor [...] > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c< 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... > David Bernier You've shattered the illusion that Omega is on some list but the problem > with Cantors proof still stands. Every finite prefix of possible sequences is computable, so the original > diagonalisation technique fails to find a new sequence. 123 > 456 > 789 In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. That's the claim, that there is a problem, not that Omega is on the list. The computable reals contains every possible sequence up to infinite length, > and you tell me with a cloak and digger trick that you can find new sequences. > You can't. All the sequences are computed, up to oo length, end of story. Let's say a Turing machine is decimal if, starting with a tape with k consecutive 1s , where k is a positive integer, it halts and leaves m(k) 1s on the tape, with 0<=m(k)<10 . This makes precise the idea of a procedure in a programming language with input = a positive integer and output = one of the 10 digits. If you want the diagonal and anti-diagonal to be computable and on the list, since diag and anti-diag depend on the ordering of the list, I think it's for you to explain how to order the list of decimal TMs so that diag and anti-diag are *computable* ; also the list of decimal TMs should be complete. How do you know that the entire anti-diagonal is computable? David Bernier > Have you taken a look at > http://www.freewebs.com/namesort/linux.html It proves that a list of reals that contains every digit oo times in each column > can be sorted to have any random diagonal. The diagonal is independent of > such lists, e.g. the computable reals. That's another problem with Cantor's proof, > the diagonal does not even depend on the list, it's just a random variable. Herc === Subject: Re: the problem with Cantor > [...] > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c< 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the problem > with Cantors proof still stands. > Every finite prefix of possible sequences is computable, so the original > diagonalisation technique fails to find a new sequence. > 123 > 456 > 789 > In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. > That's the claim, that there is a problem, not that Omega is on the list. > The computable reals contains every possible sequence up to infinite length, > and you tell me with a cloak and digger trick that you can find new sequences. > You can't. All the sequences are computed, up to oo length, end of story. Let's say a Turing machine is decimal if, starting with > a tape with k consecutive 1s , where k is a positive integer, it > halts and leaves m(k) 1s on the tape, with 0<=m(k)<10 . This makes precise the idea of a procedure in a programming language > with input = a positive integer and output = one of the 10 digits. If you want the diagonal and anti-diagonal to be computable > and on the list, since diag and anti-diag depend on the ordering > of the list, I think it's for you to explain how to order the list > of decimal TMs so that diag and anti-diag are > *computable* ; also the list of decimal TMs should > be complete. How do you know that the entire anti-diagonal is > computable? David Bernier The anti-diagonal is not computable, therefore it doesn't exist. let UTM(real, digit) mod 10 calculate the list of reals. (where real is deceptively an integer) the antidiagonal is UTM(digit, digit) + 1 mod 10 if this is computable, then some TM computes it and some emulated TM also computes it, UTM(ad, digit) mod 10 = UTM(digit, digit) + 1 mod 10 when digit = ad, UTM(ad, ad) mod 10 = UTM(ad, ad) + 1 mod 10 Contradiction! Therefore antidiag is an invalid formula. This is a more sensible conclusion than higher infinities exist as I've basically just rearranged Cantor's proof. One can argue its a valid specification of a sequence but it doesn't actually compute a new sequence of digits. Herc > Have you taken a look at > http://www.freewebs.com/namesort/linux.html > It proves that a list of reals that contains every digit oo times in each column > can be sorted to have any random diagonal. The diagonal is independent of > such lists, e.g. the computable reals. That's another problem with Cantor's proof, > the diagonal does not even depend on the list, it's just a random variable. > Herc > === Subject: Re: the problem with Cantor > [...] > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c< 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the problem > with Cantors proof still stands. Every finite prefix of possible sequences is computable, so the original > diagonalisation technique fails to find a new sequence. 123 > 456 > 789 In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. That's the claim, that there is a problem, not that Omega is on the list. The computable reals contains every possible sequence up to infinite length, > and you tell me with a cloak and digger trick that you can find new sequences. > You can't. All the sequences are computed, up to oo length, end of story. > Let's say a Turing machine is decimal if, starting with > a tape with k consecutive 1s , where k is a positive integer, it > halts and leaves m(k) 1s on the tape, with 0<=m(k)<10 . > This makes precise the idea of a procedure in a programming language > with input = a positive integer and output = one of the 10 digits. > If you want the diagonal and anti-diagonal to be computable > and on the list, since diag and anti-diag depend on the ordering > of the list, I think it's for you to explain how to order the list > of decimal TMs so that diag and anti-diag are > *computable* ; also the list of decimal TMs should > be complete. > How do you know that the entire anti-diagonal is > computable? > David Bernier The anti-diagonal is not computable, therefore it doesn't exist. let UTM(real, digit) mod 10 calculate the list of reals. (where real is deceptively an integer) With the mod 10, we can avoid talking about decimal TMs. For argument 'real', I'd put TM_n, the n'th TM that halts on any input. For 'digit', I'd put k and we get (after rewriting): UTM(TM_n, k)%10 // UTM a fixed universal TM (Indeed, TM_n and k will be presented to the UTM as input or 1s on the tape of the UTM just before it starts). for the k'th decimal of the n'th TM that halts under all inputs, with TMs: TM_1, TM_2 , .... and for k = 1, 2, 3 .... if we are only concerned with reals in [0, 1], and allowing say 0.5000... to appear for several always halting TMs: TM_a, TM_b and so on . If the anti-diag were computable, there would be a contradiction. I think what's going on is that the unsolvability of the halting problem prevents us from writing a program that finds the n'th always-halting TM. The anti-diagonal is not computable, therefore it doesn't exist. There was a long thread starting with: Uncomputable numbers are all in your head. The conclusion I see is that you're not a Platonist. You could abandon the Power set axiom for infinite sets and the axiom of choice. Constructivists don't draw conclusions using the law of the excluded middle. the antidiagonal is UTM(digit, digit) + 1 mod 10 if this is computable, then some TM computes it and some emulated TM also computes it, UTM(ad, digit) mod 10 = UTM(digit, digit) + 1 mod 10 when digit = ad, UTM(ad, ad) mod 10 = UTM(ad, ad) + 1 mod 10 Contradiction! Therefore antidiag is an invalid formula. This is a more sensible conclusion than higher infinities exist as I've basically just > rearranged Cantor's proof. One can argue its a valid specification of a sequence > but it doesn't actually compute a new sequence of digits. Then we can say you're a computist. David Bernier > Herc === Subject: Re: the problem with Cantor > [...] > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: > 2.3, 3.14159, 2000000, 0.00000000000056, etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , and so > on for > 5, 7, ... 100, ... 1000 decimals and more. > But if we take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c< 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) number > of decimals. > In other words, what works for > Omega_{Herc} and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the problem > with Cantors proof still stands. > Every finite prefix of possible sequences is computable, so the original > diagonalisation technique fails to find a new sequence. > 123 > 456 > 789 > In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. > That's the claim, that there is a problem, not that Omega is on the list. > The computable reals contains every possible sequence up to infinite length, > and you tell me with a cloak and digger trick that you can find new sequences. > You can't. All the sequences are computed, up to oo length, end of story. > Let's say a Turing machine is decimal if, starting with > a tape with k consecutive 1s , where k is a positive integer, it > halts and leaves m(k) 1s on the tape, with 0<=m(k)<10 . > This makes precise the idea of a procedure in a programming language > with input = a positive integer and output = one of the 10 digits. > If you want the diagonal and anti-diagonal to be computable > and on the list, since diag and anti-diag depend on the ordering > of the list, I think it's for you to explain how to order the list > of decimal TMs so that diag and anti-diag are > *computable* ; also the list of decimal TMs should > be complete. > How do you know that the entire anti-diagonal is > computable? > David Bernier > The anti-diagonal is not computable, therefore it doesn't exist. > let UTM(real, digit) mod 10 calculate the list of reals. (where real is deceptively an integer) With the mod 10, we can avoid talking about decimal TMs. For > argument 'real', I'd put TM_n, the n'th TM that halts on any input. > For 'digit', I'd put k and we get (after rewriting): UTM(TM_n, k)%10 // UTM a fixed universal TM (Indeed, TM_n and k will be presented to the UTM as > input or 1s on the tape of the UTM just before it starts). for the k'th decimal of the n'th TM that halts under > all inputs, with TMs: TM_1, TM_2 , .... and for k = 1, 2, 3 .... > if we are only concerned with reals in [0, 1], and allowing > say 0.5000... to appear for several always halting TMs: > TM_a, TM_b and so on . If the anti-diag were computable, there would be a contradiction. > I think what's going on is that the unsolvability of the halting > problem prevents us from writing a program that finds > the n'th always-halting TM. The anti-diagonal is not computable, therefore it doesn't exist. There was a long thread starting with: HA I started that one! Read down a few posts... [Herc] A real number has to be enscribed somehow, computers are just a way of automating that Uncomputable numbers are all in your head. The conclusion I see is that you're not a Platonist. > You could abandon the Power set axiom for > infinite sets and the axiom of choice. > Constructivists don't draw conclusions using > the law of the excluded middle. > the antidiagonal is UTM(digit, digit) + 1 mod 10 > if this is computable, then some TM computes it and some emulated TM also computes it, > UTM(ad, digit) mod 10 = UTM(digit, digit) + 1 mod 10 > when digit = ad, > UTM(ad, ad) mod 10 = UTM(ad, ad) + 1 mod 10 > Contradiction! > Therefore antidiag is an invalid formula. > This is a more sensible conclusion than higher infinities exist as I've basically just > rearranged Cantor's proof. One can argue its a valid specification of a sequence > but it doesn't actually compute a new sequence of digits. Then we can say you're a computist. David Bernier I just think numbers have to be enscribed somehow, the only noncomputable things I've seen are self referencing negating tricks, they define themselves to be undoable. MATHS define an object type define a self reference in the object type and negate it find the contradiction claim something whymsical and far fetched I don't fall for any of it. Computable numbers are very very expressive, there's no limit whatsoever. Infinitely long random strings aren't numbers, its just noise. The anti-diagonal is just a facet of lists. Each digit position in the real list contains infinite of each digit, so you can reorder the list and construct any diagonal you want, its just a random variable. Herc === Subject: Re: the problem with Cantor [Herc] > A real number has to be enscribed somehow, computers are just a way > of automating that Numerals may have to be enscribed, numbers don't. === Subject: Re: the problem with Cantor The anti-diagonal is not computable, therefore it doesn't exist. In what axiom system? Unless you can state exactly what assumptions you are making (and you are making quite a few) there is no way to determine what does or does not exist in the system formed by those assumptions. let UTM(real, digit) mod 10 calculate the list of reals. (where real is > deceptively an integer) the antidiagonal is UTM(digit, digit) + 1 mod 10 Not in any standard decimal system. So the rest of Herk's garbage is just that. === Subject: Re: the problem with Cantor > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. You have > failed to give me a new sequence of digits not on the computable > reals. Cantor's proof is an exercise in going round in circles. > Herc > > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: 2.3, 3.14159, 2000000, 0.00000000000056, > etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , > and so on for 5, 7, ... 100, ... 1000 decimals and more. But if we > take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c < 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) > number of decimals. In other words, what works for Omega_{Herc} > and LF should work for > __ > 0.3 and LF ... > David Bernier You've shattered the illusion that Omega is on some list but the > problem with Cantors proof still stands. Every finite prefix of possible sequences is computable, so the > original diagonalisation technique fails to find a new sequence. Then Herkimer must be claiming that the diagonal appears at some finite position in the list. Which position is it in, dumbkopf? 123 456 789 In this simple finite list, the diagonal is 159. The anti-diagonal is 261. For the above simple finite list, there are 729 different base 10 anti-diagonals. It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. Is any list of all computable reals itself computable? And given any arbitrary infinite list of computable reals, is the diagonal for that list computable? Until Herk the Jerk can definitively answer those, one way or the other, there are holes in his arguemnts. > The computable reals contains every possible sequence up to infinite > length, and you tell me with a cloak and digger trick that you can > find new sequences. I don't claim that any such new sequence is necessarily computable, merely that for any list it exists. > You can't. All the sequences are computed, up to oo length, end of > story. There is nothing in Cantor's proof that requires the 'anti-diagonal to be computable, merely existing in that non-physical way that numbers have of existing. Have you taken a look at http://www.freewebs.com/namesort/linux.html It proves that a list of reals that contains every digit oo times in > each column can be sorted to have any random diagonal. It does not prove anything at all about such lists, since it only works with finite lists of finite lengths. > The diagonal is independent of such lists > That's another problem with Cantor's > proof, the diagonal does not even depend on the list, it's just a > random variable. Nonsense. Cantor's original 'diagonal' proof applies only to lists of infinite binary sequences with values in {m,w}. So anything not valid for that original proof is irrelevant. === Subject: Re: the problem with Cantor > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. You have > failed to give me a new sequence of digits not on the computable > reals. Cantor's proof is an exercise in going round in circles. > Herc > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: 2.3, 3.14159, 2000000, 0.00000000000056, > etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , > and so on for 5, 7, ... 100, ... 1000 decimals and more. But if we > take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c < 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) > number of decimals. In other words, what works for Omega_{Herc} > and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the > problem with Cantors proof still stands. > Every finite prefix of possible sequences is computable, so the > original diagonalisation technique fails to find a new sequence. Then Herkimer must be claiming that the diagonal appears at some finite > position in the list. Which position is it in, dumbkopf? You can't correctly define an anti-diagonal. You also can't cite any sequence of numbers that is new to the list. Oh sure it works for 123 456 789 but it doesn't work for the infinite set of computable reals. The computable reals contains all finite sequences of digits up to oo length. Every pattern possible is computable. > 123 456 789 > In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. For the above simple finite list, there are 729 different base 10 > anti-diagonals. If usenet posts had audio you'd hear clapping right now. Did you remember the antidiagonals for the 6 permutations of the list? > It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. Is any list of all computable reals itself computable? And given any > arbitrary infinite list of computable reals, is the diagonal for that > list computable? yes computable things are computable yes there is no contradiction in computing the diagonal somewhere in the list Until Herk the Jerk can definitively answer those, one way or the > other, there are holes in his arguemnts. > The computable reals contains every possible sequence up to infinite > length, and you tell me with a cloak and digger trick that you can > find new sequences. I don't claim that any such new sequence is necessarily computable, > merely that for any list it exists. > You can't. All the sequences are computed, up to oo length, end of > story. There is nothing in Cantor's proof that requires the 'anti-diagonal to > be computable, merely existing in that non-physical way that numbers > have of existing. Numbers exist because they can be computed. If they couldn't be computed you wouldn't know what they were. > Have you taken a look at http://www.freewebs.com/namesort/linux.html > It proves that a list of reals that contains every digit oo times in > each column can be sorted to have any random diagonal. It does not prove anything at all about such lists, since it only works > with finite lists of finite lengths. Oh you had a look. Claim: Let L be a list of numbers in [0,1] such that for each d in {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal representation of L_i is d} is infinite. Let D be a random variable uniform on [0,1]. Then almost surely there is a permutation L' of L such that D is the diagonal of L'. > The diagonal is independent of such lists > That's another problem with Cantor's > proof, the diagonal does not even depend on the list, it's just a > random variable. Nonsense. Did you randomise the diagonal at the website, and produce an equivalent set in the second list with that new diagonal? What does that mean? Cantor's original 'diagonal' proof applies only to lists of infinite > binary sequences with values in {m,w}. So anything not valid for that original proof is irrelevant. Anything valid for an equivalent proof is relevant. This brings up an important point. Virgil, like most of you, believe numbers and mathematical things exist without being computable? Tell me if you subtracted the theories of mathematics that are computable from the total set of theories in mathematics that you believe in, what are you left with? Herc === Subject: Re: the problem with Cantor > > I don't care > Certainly not about truth. > The truth is there is nothing larger than infinity. You have > failed to give me a new sequence of digits not on the computable > reals. Cantor's proof is an exercise in going round in circles. > Herc > Suppose we have a list LF of the numbers that have a terminating > decimal expansion: 2.3, 3.14159, 2000000, 0.00000000000056, > etc. > 1/3 = 0.33333.... = > __ > 0.3 [ 0.3periodic]. > The first 5 decimals of 1/3 are the same as those of 0.33333 , > and so on for 5, 7, ... 100, ... 1000 decimals and more. But if we > take a fixed number 'c' on LF and compare it to > __ > 0.3 [ 0.3periodic] , then either > __ > c < 0.3 or > __ > c>0.3 . > Don't you think the same thing happens with Omega_{Herc} and LF? > After all, you're only comparing a finite (but arbitrarily large) > number of decimals. In other words, what works for Omega_{Herc} > and LF should work for > __ > 0.3 and LF ... > David Bernier > You've shattered the illusion that Omega is on some list but the > problem with Cantors proof still stands. > Every finite prefix of possible sequences is computable, so the > original diagonalisation technique fails to find a new sequence. > Then Herkimer must be claiming that the diagonal appears at some finite > position in the list. Which position is it in, dumbkopf? You can't correctly define an anti-diagonal. Cantor did, at least for any list of binary sequences (which is all he used, the decimal model came later and was done by others. > You also can't cite any > sequence of numbers that is new to the list. I can cite a rule which will produce a sequence not in a given list. but it doesn't work for the infinite set of computable reals. Then the rule must define an incomputable sequnce > The computable > reals > contains all finite sequences of digits up to oo length. Every pattern > possible is > computable. Since there are uncountably more real numbers than computable real numbers, you must explain how that works. > 123 456 789 > In this simple finite list, the diagonal is 159. > The anti-diagonal is 261. > For the above simple finite list, there are 729 different base 10 > anti-diagonals. If usenet posts had audio you'd hear clapping right now. > Did you remember the antidiagonals for the 6 permutations of the list? It looks like diagonalisation actually finds new sequences of digits. > but when you apply it to the set of computable reals it doesn't, the > list is already saturated with every possible sequence. > Is any list of all computable reals itself computable? And given any > arbitrary infinite list of computable reals, is the diagonal for that > list computable? yes computable things are computable There are things that one can speak of which are not, like the set of all Dedekind cuts of the rationals or the set of all equivalence classes of Cauchy sequences mod the null sequences. Either of which makes the set of reals to computable and ssome members of that set also not computable. > yes there is no contradiction in computing the diagonal somewhere in the list Until Herk the Jerk can definitively answer those, one way or the > other, there are holes in his arguemnts. > The computable reals contains every possible sequence up to infinite > length, and you tell me with a cloak and digger trick that you can > find new sequences. > I don't claim that any such new sequence is necessarily computable, > merely that for any list it exists. > You can't. All the sequences are computed, up to oo length, end of > story. There can, and must be be uncomputable sequences, at least unless there are uncountably many computable sequences. > There is nothing in Cantor's proof that requires the 'anti-diagonal to > be computable, merely existing in that non-physical way that numbers > have of existing. Numbers exist because they can be computed. > If they couldn't be computed > you wouldn't know what they were. That's what uncomputable numbers are like, you don't know exactly what they are. > Have you taken a look at http://www.freewebs.com/namesort/linux.html > It proves that a list of reals that contains every digit oo times in > each column can be sorted to have any random diagonal. > It does not prove anything at all about such lists, since it only works > with finite lists of finite lengths. Oh you had a look. Claim: Let L be a list of numbers in [0,1] such that for each d in > {0,...,9} and for all n >= 1, the set {i| n-th digit of the decimal > representation of L_i is d} is infinite. Let D be a random variable > uniform on [0,1]. Then almost surely there is a permutation L' of L > such that D is the diagonal of L'. Almost surely in any infinite set allows for infinitely many exceptions. Almost surely in an uncountably infinite set allows uncountably many exceptions. > > The diagonal is independent of such lists > > That's another problem with Cantor's > proof, the diagonal does not even depend on the list, it's just a > random variable. > Nonsense. Did you randomise the diagonal at the website, and produce an equivalent set > in the second list with that new diagonal? What does that mean? That each list has its own private diagonal(s) which need not work for any other list. Cantor's original 'diagonal' proof applies only to lists of infinite > binary sequences with values in {m,w}. > So anything not valid for that original proof is irrelevant. Anything valid for an equivalent proof is relevant. This brings up an important point. Virgil, like most of you, believe numbers > and mathematical things exist without being computable? I believe that when working with a set of axioms, whatever the axioms require to exist does exist, at least within that axiom system. Tell me if you > subtracted the theories of mathematics that are computable from the total > set of theories in mathematics that you believe in, what are you left with? Most of mathematics, except for, say, finite groups. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > As always, I believe that there can be rigorous theories in > which the claims made by so-called cranks such as Herc > can hold. Not just so-called. > There are two ways to make card(N) = card(R) -- > either by making N bigger or by making R smaller. Since all Peano systems are isomorphic with one another, and since all complete ordered fields are isomorphic with one another, what you're asking for is at least a pretty tall order. Anyway, cranks aren't interested in rigorous theories. MoeBlee Perhaps, in the spirit of this thread, we could make R smaller > by insisting that only computable numbers exist. Then there > would be no Omega or any other numbers missing from the > list, and so Herc's claim that there exist only countably many > reals can still hold. > of === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au Since all Peano systems are isomorphic with one another ... Not sure what you mean here. By the upward LS theorem, there are models of PA of every infinite cardinality. -- hz === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <4893EDC1.4D1756EF@gmail.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Since all Peano systems are isomorphic with one another ... Not sure what you mean here. æBy the upward LS theorem, there are > models of PA of every infinite cardinality. I didn't say all models of PA are isomorphic. I said all Peano systems (also known as 'Dedekind systems') are isomorphic. Df: P is a Peano system <-> ESfz(P = & zeS & f:S->S~{z} & f is 1-1 & Ax((x subset of S & zex & x closed under f) -> x=S)) Th: All Peano systems are isomporhic. Th: Between any two Peano systems there exists exactly one isomorphism. MoeBlee === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au Since all Peano systems are isomorphic with one another ... > Not sure what you mean here. By the upward LS theorem, there are > models of PA of every infinite cardinality. I didn't say all models of PA are isomorphic. I said all Peano systems > (also known as 'Dedekind systems') are isomorphic. Df: > P is a Peano system <- ESfz(P = & > zeS & > f:S->S~{z} & > f is 1-1 & > Ax((x subset of S & zex & x closed under f) -> x=S)) Th: All Peano systems are isomporhic. Th: Between any two Peano systems there exists exactly one > isomorphism. -- hz === Subject: Re: the problem with Cantor > There are two ways to make card(N) = card(R) -- > either by making N bigger or by making R smaller. > [lwal..., designated crank] > Does this asshole have a brain? If we either make N bigger or R smaller, the resulting set(s) would not be N and/or R anymore. *sigh* (Clearly there's a REASON why we write R* and N* in non-standard analysis instead of N and R.) Hence there's NO WAY to make card(N) = card(R) IF N denotes the set of natural numbers, R the set of real numbers, Card cardinality (defined in the spirit of Cantor and Frege) and = identity (in the way WE -i.e. non-cranks- know it). Right, there are two way to make 1 = 2 -- either by making 1 bigger or by making 2 smaller. (Actually, there's a third way: by redefining = as =/=.) Why oh why are mathematical cranks just so one-dimensional? [Consider] the potentially infinite set N. If we add another element, the set remains the same. (W. Mueckenheim) B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <7eh694plfn9etuavv9gqv1mpepq0ukkd01@4ax.com> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > If we either make N bigger or R > smaller, the resulting set(s) would not be N and/or R anymore. > *sigh* (Clearly there's a REASON why we write R* and N* in > non-standard analysis instead of N and R.) Hence there's NO WAY to > make card(N) = card(R) IF N denotes the set of natural numbers, R > the set of real numbers, Card cardinality (defined in the spirit of > Cantor and Frege) and = identity (in the way WE -i.e. non-cranks- know > it). > æ æ æ æ Right, there are two way to make 1 = 2 -- > æ æ æ æ either by making 1 bigger or by making 2 smaller. > æ æ æ æ (Actually, there's a third way: by redefining > æ æ æ æ æ= as =/=.) In this post, Balthasar clearly compares the formulae card(N) = card(R) and 1 = 2. We know that the the perspective of Balthasar -- and most standard mathematicians -- any poster who claims that card(N) = card(R) might as well be claiming that 1 = 2 -- either claim merits the crank label. But I disagree and feel that there are worlds of difference between card(N) = card(R) and 1 = 2, even though (card(N) = card(R)) <-> (1 = 2) is a theorem of ZFC. Formulae involving infinite sets are open to various interpretations, such as classical, constructivism, intuitionism -- whereas those involving finite sets are not subject to these types of interpretations. Thus I consider ~(1 = 2) to be absolutely true in a way that the formula ~(card(N) = card(R)) isn't -- since the former holds for both classical and constructivist mathematicians, unlike the latter where there is some debate as to what N and R actually are . Herc has mentioned computable reals in this thread, as if he were a sort of constructivist who only believes in the existence of computable reals. To a constructivist, Omega simply isn't a real number -- it doesn't even exist . Omega is thus not an element of R. To someone who only philosophically believes in the existence of computable reals, there exist only countably many reals -- and therefore there's a bijection between N and R -- found by enumerating every Turing algorithm that computes a real. Perhaps, just as Balthasar has intimated with his example of *N and *R, a constructivist shouldn't use the symbol R to denote the set of real numbers that he believes exists, but to use some other notation such as Rc -- so that Herc's claim becomes card(N) = card(Rc). Then the symbol R would be reserved for the full, classical set R -- anyone who doesn't believe in the existence of the classical set R would have no right to use the symbol R in a formula. Maybe it's too bad that this isn't the case (cf. the subthread of the WM thread in which Balthasar, among others, discussed pedantry vs. abuse of notation re: definition of card). > Why oh why are mathematical cranks just so one-dimensional? When I first started posting at sci.math about a year ago, I came in fully prepared that someone would call me a crank. By the standard mathematicians' definition of crank, I am indeed a borderline crank, simply because I do not state that Herc and other posters like him are 100% wrong. I may not like the label, but I cannot deny that I satisfy their definition of the term. And I'd rather be labeled a borderline crank than blindly accept ZFC and classical mathematics as the only theories worth discussing and call anyone who posts anything that contradicts ZFC or classical analysis wrong. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > If we either make N bigger or R > smaller, the resulting set(s) would not be N and/or R anymore. > *sigh* (Clearly there's a REASON why we write R* and N* in > non-standard analysis instead of N and R.) Hence there's NO WAY to > make card(N) = card(R) IF N denotes the set of natural numbers, R > the set of real numbers, Card cardinality (defined in the spirit of > Cantor and Frege) and = identity (in the way WE -i.e. non-cranks- know > it). > æ æ æ æ Right, there are two way to make 1 = 2 -- > æ æ æ æ either by making 1 bigger or by making 2 smaller. > æ æ æ æ (Actually, there's a third way: by redefining > æ æ æ æ æ= as =/=.) In this post, Balthasar clearly compares the formulae > card(N) = card(R) and 1 = 2. We know that the > the perspective of Balthasar -- and most standard > mathematicians -- any poster who claims that > card(N) = card(R) might as well be claiming that > 1 = 2 -- either claim merits the crank label. But I disagree and feel that there are worlds of > difference between card(N) = card(R) and 1 = 2, > even though (card(N) = card(R)) <-> (1 = 2) is a > theorem of ZFC. Formulae involving infinite sets are > open to various interpretations, such as classical, > constructivism, intuitionism -- whereas those > involving finite sets are not subject to these > types of interpretations. Thus I consider ~(1 = 2) > to be absolutely true in a way that the formula > ~(card(N) = card(R)) isn't -- since the former > holds for both classical and constructivist > mathematicians, unlike the latter where there is > some debate as to what N and R actually are . The distinction between finitistic and infinitistic statements is well understood by many mathematicians who still work in a classical framework. > Herc has mentioned computable reals in this thread, > as if he were a sort of constructivist who only > believes in the existence of computable reals. You checked with Herc on that, right? > To a constructivist, Omega simply isn't a real > number -- it doesn't even exist . That's fine, but does not refute Baltasar's point. IF N and R are taken to exist in the sense of the set of natural numbers and the set of real numbers, then they don't have the same cardinality. > Omega is thus > not an element of R. Who said it is? > To someone who only > philosophically believes in the existence of > computable reals, there exist only countably > many reals -- and therefore there's a bijection > between N and R -- found by enumerating every > Turing algorithm that computes a real. Then that is just taking 'R' in a DIFFERENT sense. > Perhaps, just as Balthasar has intimated with > his example of *N and *R, a constructivist > shouldn't use the symbol R to denote the set > of real numbers that he believes exists, but to > use some other notation such as Rc -- so that > Herc's claim becomes card(N) = card(Rc). Then > the symbol R would be reserved for the full, > classical set R -- anyone who doesn't believe > in the existence of the classical set R would > have no right to use the symbol R in a formula. It's not a question of rights. It's just a matter of being clear. > And I'd rather be labeled a borderline crank > than blindly accept ZFC and classical mathematics > as the only theories worth discussing And who blindly accepts ZFC and classical mathematics as the only theories worth discussing? You keep referring to such people. And I keep challenging you on it. PLEASE, already, say who these people are. > and call > anyone who posts anything that contradicts ZFC > or classical analysis wrong. Who does that? What are wrong are certain (and many) incorrect claims and misconceptions about ZFC. MoeBlee === Subject: Re: the problem with Cantor > To a constructivist [...]. > That's fine, but does not refute Baltasar's point. IF N and R are > taken to exist in the sense of the set of natural numbers and the set > of real numbers, then they don't have the same cardinality. > Right, there are two ways to make 1 = 2 -- either by making 1 bigger or by making 2 smaller. (Actually, there's a third way: by redefining = to mean =/=.) Since this mirrors exactly one of the statements of this guy. SURE: IF wishes were horses, THEN beggars would ride. (But since...) > Omega is thus _not_ an element of R. > Who said it is? > Well at least I would dare to claim that (Chaitin's) Omega (Omega_U) is a real number. ;-) Omega_U is perhaps the most obvious specific example of an uncomputable number. It is also known to be a transcendental number. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ http://mathworld.wolfram.com/ChaitinsConstant.html B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=UJeUTgkAAADYai-ULU41ORCvNnkXmdRu Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) |But I disagree and feel that there are worlds of |difference between card(N) = card(R) and 1 = 2, |even though (card(N) = card(R)) <-> (1 = 2) is a |theorem of ZFC. Formulae involving infinite sets are |open to various interpretations, such as classical,constructivism, intuitionism -- whereas those |involving finite sets are not subject to these |types of interpretations. They're subject to reinterpretation if you work at it. ;-) |Thus I consider ~(1 = 2) |to be absolutely true in a way that the formula |~(card(N) = card(R)) isn't -- since the former |holds for both classical and constructivist |mathematicians, unlike the latter where there is |some debate as to what N and R actually _are_. I can't think offhand of any seriously intended theory or interpretation that has a continuum R, and doesn't render ~(card(N)=card(R)) valid. |Herc has mentioned computable reals in this thread, |as if he were a sort of constructivist who only |believes in the existence of computable reals. | |To a constructivist, Omega simply _isn't_ a real |number -- it doesn't even _exist_. Well, be a little careful. Errett Bishop in his famous textbook on constructive analysis has an aside about how many of these classical constructions can be treated constructively as instances of what he calls fickle reals. A sequence of rationals r_i has fickle convergence if for each epsilon>0 there exists an N such that there does not exist a sequence i1epsilon for j=1,...,N-1. You could say, for each epsilon>0 there is an upper bound on how many times the sequence moves by epsilon. Nonconstructively, fickle convergence implies convergence, but it doesn't follow constructively. Fickle convergence contradicts the claim that for each N, there are i,j>N for which |r_i-r_j|>epsilon, but doesn't necessarily provide you with an example of an N for which this doesn't happen. Bishop doesn't claim there (and as far as I know never did claim) that the fickle continuum was worth studying very much. It presumably would go something like the studies in nonconstructive mathematics of computable sequences of rationals that converge but not necessarily at a computable rate of convergence. |Omega is thus |_not_ an element of R. Depending on your school of constructive mathematics, this not might get put into quotation marks, or else you might say that its being an element of R implies a standard nonconstructive principle, that every Turing machine computation either terminates or does not terminate. |To someone who only |philosophically believes in the existence of |computable reals, there exist only countably |many reals -- and therefore there's a bijection |between N and R -- found by enumerating every |Turing algorithm that computes a real. No, this is incorrect. Cantor's first proof that there's no surjection from N to R (and hence also no bijection) is perfectly constructive. His second proof suffers only from a technical weakness, in that it assumes each real has a decimal expansion, although if we are given a real, we cannot necessarily compute the decimal expansion. There have been serious mathematicians who apparently were willing to assume that every real is computable, and not just as a quaint hypothesis. Beeson describes the Moscow school of constructive mathematics under Markov as having accepted the computability of every sequence of natural numbers as an axiom, which implies that every real is computable too. The same hypothesis implies however that there's no way of enumerating just the Turing machines that compute reals. The Turing machines can be numbered, and there's a subset of them that compute reals, but there's no computable enumeration of the ones that do. |Perhaps, just as Balthasar has intimated with |his example of *N and *R, a constructivist |shouldn't use the symbol R to denote the set |of real numbers that he believes exists, but to |use some other notation such as Rc -- so that |Herc's claim becomes card(N) = card(Rc). Then |the symbol R would be reserved for the full, |classical set R -- anyone who doesn't believe |in the existence of the classical set R would |have no right to use the symbol R in a formula. I disagree with this, especially since not all constructivists assume that all reals are computable. I propose that anybody who uses the usual definition of real line is entitled to refer to the set satisfying the definition as the real line R. Perhaps you're thinking specifically of people who does assume that the reals are all computable. If a person who doesn't assume that the reals are all computable is working with the set of computable reals, then it's often appropriate for them to use a distinctive notation like R_c. In that case, they really aren't referring to R when they talk about R_c. But to someone who actually believes that R = R_c, talking about R and talking about R_c are just the same thing. It's only appropriate for them to make note of the fact that they're assuming that R = R_c (since this is an unusual assumption) but to ask them to use special notation throughout their work is akin to asking them to write as though they were wrong about what R is like, and actually talking about something else. It's a little like in theological discussion when people try to argue that the God of some people is not the same as the God of others, when they all believe in a supreme being but have different beliefs about what God is like. I would say a change in notation is even less appropriate a suggestion in the case of those constructivists who do not assume that all reals are computable. Logicians have been known to cook up exotic theories of the reals, but generally speaking the theories of the reals all share a common core of constructive principles. The so-called Bishop school more or less works with this shared core of principles, treating all the stuff that doesn't follow from it in a sort of agnostic manner, like you might talk about statements you know are independent of your favorite set theory. People working this way are surely entitled to describe their results as being about the real line, since not only are they valid results about the real line, they don't require any assumptions that aren't normally made, so they're acceptable to as large a group as can reasonably be expected. Those results include in particular ~(card(N)=card(R)). talking about when they talk about the real line are those things that are reals, if the law of excluded middle and the axiom of choice are true. It would be tempting to ask mathematicians to make note of the fact that they're working with this expanded version of the real line, that includes Chaitin's Omega and so on. That battle was lost a long time ago, however. Keith Ramsay === Subject: Re: the problem with Cantor > If we either make N bigger or R > smaller, the resulting set(s) would not be N and/or R anymore. > *sigh* (Clearly there's a REASON why we write R* and N* in > non-standard analysis instead of N and R.) Hence there's NO WAY to > make card(N) = card(R) IF N denotes the set of natural numbers, R > the set of real numbers, Card cardinality (defined in the spirit of > Cantor and Frege) and = identity (in the way WE -i.e. non-cranks- know > it). > æ æ æ æ Right, there are two way to make 1 = 2 -- > æ æ æ æ either by making 1 bigger or by making 2 smaller. > æ æ æ æ (Actually, there's a third way: by redefining > æ æ æ æ æ= as =/=.) In this post, Balthasar clearly compares the formulae > card(N) = card(R) and 1 = 2. We know that the > the perspective of Balthasar -- and most standard > mathematicians -- any poster who claims that > card(N) = card(R) might as well be claiming that > 1 = 2 -- either claim merits the crank label. But I disagree and feel that there are worlds of > difference between card(N) = card(R) and 1 = 2, > even though (card(N) = card(R)) <-> (1 = 2) is a > theorem of ZFC. Formulae involving infinite sets are > open to various interpretations, such as classical, > constructivism, intuitionism -- whereas those > involving finite sets are not subject to these > types of interpretations. Thus I consider ~(1 = 2) > to be absolutely true in a way that the formula > ~(card(N) = card(R)) isn't -- since the former > holds for both classical and constructivist > mathematicians, unlike the latter where there is > some debate as to what N and R actually _are_. Herc has mentioned computable reals in this thread, > as if he were a sort of constructivist who only > believes in the existence of computable reals. To a constructivist, Omega simply _isn't_ a real > number -- it doesn't even _exist_. Omega is thus > _not_ an element of R. To someone who only > philosophically believes in the existence of > computable reals, there exist only countably > many reals -- and therefore there's a bijection > between N and R -- found by enumerating every > Turing algorithm that computes a real. I am of the opinion that there is more than one Turing algorithm to compute any computable real, for some computable reals there are very likely countably many such algorithms. So that constructability of such a bijection is, at best, problematical. Perhaps, just as Balthasar has intimated with > his example of *N and *R, a constructivist > shouldn't use the symbol R to denote the set > of real numbers that he believes exists, but to > use some other notation such as Rc -- so that > Herc's claim becomes card(N) = card(Rc). Then > the symbol R would be reserved for the full, > classical set R -- anyone who doesn't believe > in the existence of the classical set R would > have no right to use the symbol R in a formula. Maybe it's too bad that this isn't the case (cf. > the subthread of the WM thread in which > Balthasar, among others, discussed pedantry vs. > abuse of notation re: definition of card). Why oh why are mathematical cranks just so one-dimensional? When I first started posting at sci.math about a > year ago, I came in fully prepared that someone > would call me a crank. By the standard > mathematicians' definition of crank, I am > indeed a borderline crank, simply because I do > not state that Herc and other posters like him > are 100% wrong. I may not like the label, but I > cannot deny that I satisfy their definition of > the term. And I'd rather be labeled a borderline crank > than blindly accept ZFC and classical mathematics > as the only theories worth discussing and call > anyone who posts anything that contradicts ZFC > or classical analysis wrong. What I object to is WM, and others, telling me that I am not allowed to accept ZFC as a theory worth discussing. I have no objections to someone preferring not to accept ZFC or whatever, but I do object to their attempting to control what I am allowed to think about. Particularly in the case of those like WM who are both mathematically and logically incompetent. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=-eQqtQoAAACZVM-kNEsOn3k7GSvoJoS4 CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent Or intentionally and maliciously dishonest. I'm still not sure about WM, but my gut tells me that he knows his position is untenable, and he gets his jollies out of riling up everyone else. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; FunWebProducts; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. ********************************************************** In the case of WM, I honestly think he embraces all four talents: mathematically and logically incompetent, and also intentionally and maliciously dishonest. I've had a hard time swallowing the fact that he's actually teaching people stuff. A pity. Tonio === Subject: Re: the problem with Cantor What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. Well said! Particularly in the case of those like WM who are both mathematically > and logically incompetent Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. WM has too many published papers supporting his position to be doing it for fun. If those papers were as publicly discredited as they ought to be, it could have serious effects on his job. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; FunWebProducts; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. > Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent > Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** Where has WM publish any of his papers on this stuff? Is there any reachable link where one can acceed to some of them? Thanx Tonio === Subject: Re: the problem with Cantor > > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. > Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent > Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. > WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** Where has WM publish any of his papers on this stuff? Is there any > reachable link where one can acceed to some of them? > Thanx Tonio Try http://arxiv.org/find/math/1/au:+Mueckenheim_W/0/1/0/all/0/1 You might be amused by #8 of the list shown there, in which WM claims to establish that the set of naturals is non-denumerable by reason of there being no bijection between N and itself. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=-eQqtQoAAACZVM-kNEsOn3k7GSvoJoS4 CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** > Where has WM publish any of his papers æon this stuff? Is there any > reachable link where one can acceed to some of them? > Thanx > Tonio Tryhttp://arxiv.org/find/math/1/au:+Mueckenheim W/0/1/0/all/0/1 You might be amused by #8 of the list shown there, in which WM claims to > establish that the set of naturals is non-denumerable by reason of there > being no bijection between N and itself Yeah, but none of those are what I would call mainstream mathematics. You have a crank giving a talk at the meeting of the German Math. Soc. --- happens all the time. (Note: no proceedings.) You have a paper on Physical Constraints of Numbers. I didn't look at it, but he is not alone in feeling unease about this, and the rest is unpublished. He has tenure, I presume, and if he chooses not to subject his ravings to peer review, and his university is happy with his level of output, he is not going to get into trouble. His students give him great reviews. It's clearly not ideal, but there is nothing so blatant that he could lose his job over. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> posting-account=1U85NgoAAABKAgcWPdpT0VtXQLBqKIly SIMBAR={5671DD94-1EBC-4971-8627-EEDFF9B1090D}; Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; IEMB3; IEMB3),gzip(gfe),gzip(gfe) > What I object to is WM, and others, telling me that I am not allowed to > accept ZFC as a theory worth discussing. I have no objections to someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. > Well said! > Particularly in the case of those like WM who are both mathematically > and logically incompetent > Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. > WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** > Where has WM publish any of his papers æon this stuff? Is there any > reachable link where one can acceed to some of them? > Thanx > Tonio Tryhttp://arxiv.org/find/math/1/au:+Mueckenheim W/0/1/0/all/0/1 You might be amused by #8 of the list shown there, in which WM claims to > establish that the set of naturals is non-denumerable by reason of there > being no bijection between N and itself.- Hide quoted text - - Show quoted text - What is your opinion about it Virgil. Zuhair === Subject: Re: the problem with Cantor > > What I object to is WM, and others, telling me that I am not > allowed to > accept ZFC as a theory worth discussing. I have no objections to > someone > preferring not to accept ZFC or whatever, but I do object to their > attempting to control what I am allowed to think about. > Well said! > Particularly in the case of those like WM who are both > mathematically > and logically incompetent > Or intentionally and maliciously dishonest. I'm still not sure about > WM, but my gut tells me that he knows his position is untenable, and > he gets his jollies out of riling up everyone else. > WM has too many published papers supporting his position to be doing it > for fun. > ******************************************************** > Where has WM publish any of his papers æon this stuff? Is there any > reachable link where one can acceed to some of them? > Thanx > Tonio > Tryhttp://arxiv.org/find/math/1/au:+Mueckenheim_W/0/1/0/all/0/1 > You might be amused by #8 of the list shown there, in which WM claims to > establish that the set of naturals is non-denumerable by reason of there > being no bijection between N and itself.- Hide quoted text - > - Show quoted text - What is your opinion about it Virgil. Zuhair WM seems to think that a natural number can have a decimal representation with more than finitely many non-zero digits in it. There are such number systems allowing infinitely many non-zero digits to the left of the radix point, such the p-adics, but even in them, the numerals for naturals don't have infinitely many non-zero digits to the left of the radix point. Also, anyone who claims that there cannot be a bijection of a set with itself, as WM does in this paper, is obviously not in touch with any sort of set theoretical reality or simple logic. === Subject: Re: the problem with Cantor There are two ways to make card(N) = card(R) -- > either by making N bigger or by making R smaller. [lwal..., designated crank] Does this asshole have a brain? > And if so - why doesn't he use it? B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <7eh694plfn9etuavv9gqv1mpepq0ukkd01@4ax.com> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Does this asshole have a brain? > And if so - why doesn't he use it? In Balthasar's opinion, if I were to use my brain, I would see that Herc is 100% wrong and that there's no possible interpretation in which Herc could be right. Of course, I'm fully aware that in that in ZFC, Herc's reasoning is invalid. I already know that the negations of Herc's claims are theorems of ZFC. But Herc already has enough people to tell him that he's wrong. Why should I simply repeat what others are telling him? Indeed, I'd claim that for me to keep repeating Herc, you're wrong! over and over again isn't fully using my brain either. I believe that repeating You're wrong seldom wins anyone to one's point of view. I prefer to say I disagree instead -- though I'm not perfect and have probably called several people wrong -- both standard and crank -- during my first year of posting here. When Herc makes a claim that N and R have the same cardinality, I point out that his claim doesn't hold in ZFC, but perhaps there's some other rigorous theory in which his claim could possibly hold. In other words, I'm trying to find the shortest distance between the claim and something that does hold. Indeed, Balthasar has already mentioned *N and *R. The sets *N and *R do have the same cardinality, namely c. So this is a short path from the claim to something that does hold (though I don't believe that Herc had *N and *R in mind, but this may be similar to WM's ideas, since WM has mentioned Robinson in his posts). Since Herc mentions computable reals, we see how there are only countably many reals that are computable. So I use my brain and realize that to someone who only believes in the existence of computable reals, there are exactly as many (computable) reals as there are naturals. This may not be exactly what Herc has in mind, but it's the shortest distance between his claim that fails in ZFC and something that does hold. It's the difference between insulting someone and constructive criticism. Many standard mathematicians prefer the former -- since it is easier -- while I strive for the latter. I believe that it's better to find a theory in which Herc's claims hold rather than simply repeat how he's wrong. And I have found such a theory -- the theory of computable reals. === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <7eh694plfn9etuavv9gqv1mpepq0ukkd01@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Does this asshole have a brain? > And if so - why doesn't he use it? In Balthasar's opinion, if I were to use my > brain, I would see that Herc is 100% wrong > and that there's no possible interpretation > in which Herc could be right. Of course, I'm fully aware that in that in > ZFC, Herc's reasoning is invalid. I already > know that the negations of Herc's claims are > theorems of ZFC. But Herc already has enough people to tell > him that he's wrong. Why should I simply > repeat what others are telling him? Indeed, > I'd claim that for me to keep repeating Herc, > you're wrong! over and over again isn't > fully using my brain either. I believe that repeating You're wrong > seldom wins anyone to one's point of view. I > prefer to say I disagree instead -- though > I'm not perfect and have probably called > several people wrong -- both standard and > crank -- during my first year of posting here. When Herc makes a claim that N and R have the > same cardinality, I point out that his claim > doesn't hold in ZFC, but perhaps there's some > other rigorous theory in which his claim could > possibly hold. Herc doesn't care about rigorous theories; he has barely a notion of a rigorous thoery. Moreover, it's possible that all kinds of statements are theorems of various rigorous theories. So what? If there is substance here with regard to a rigorous theory, then we'd like to know at least SOMETHING about such a theory and about what it can prove in mathematics. > In other words, I'm trying to > find the shortest distance between the claim > and something that does hold. No, you're not. Rather, you're trying to make yourself out to be some kind of spokesman for fairness and open-mindedness. If what you're interested in is a rigorous theory, then, please, by all means, give us at least a sketch or even a glimmer of a notion of a particular one. > Indeed, Balthasar has already mentioned *N > and *R. The sets *N and *R do have the same > cardinality, namely c. So this is a short > path from the claim to something that does > hold (though I don't believe that Herc > had *N and *R in mind, but this may be > similar to WM's ideas, since WM has mentioned > Robinson in his posts). Just about anyone somewhat informed about these matters understands that if 'N' and 'R' refer to something other than the set of natural numbers and the set of real numbers, then we can say different things about N and R. > Since Herc mentions computable reals, we see > how there are only countably many reals that > are computable. So I use my brain and realize > that to someone who only believes in the > existence of computable reals, there are > exactly as many (computable) reals as there > are naturals. This is well known. > This may not be exactly what Herc has in > mind, but it's the shortest distance between > his claim that fails in ZFC and something that > does hold. So what? > It's the difference between insulting someone > and constructive criticism. Do you have any IDEA how many thousands and thousands of words have been typed in wasted effort trying to communicate with Herc? > Many standard > mathematicians prefer the former Well, usually, after all other efforts have failed. > -- since it > is easier No, since it is all that is LEFT after years and years of trying to reason with someone impervious to reason. > -- while I strive for the latter. I > believe that it's better to find a theory in > which Herc's claims hold rather than simply > repeat how he's wrong. And I have found such > a theory -- the theory of computable reals. WHAT THEORY of computable reals? What SPECIFIC axioms and rules of inference do you have in mind? Anyway, the notion of computable reals is formalizable in classical mathematics and it is a theorem that the set of computable reals is 1-1 with omega. MoeBlee === Subject: Re: the problem with Cantor > When Herc makes a claim that N and R have the > same cardinality, I point out that his claim > doesn't hold in ZFC, but perhaps there's some > other rigorous theory in which his claim could > possibly hold. > [...] it's possible that all kinds of statements > are theorems of various rigorous theories. So what? > Good point! Consider the following theory (formulated in the context of FOPL with identity). (Axiom of difference) 0 =/= 1. Now I can prove the following theorem in this theory: 0 =/= 1. Wow! You see, we NORMALLY would be tempted to claim that it is not the case that 0 =/= 1, but of course there's some other rigorous theory in which his claim could possibly hold - as has just been proved! :-) B. -- For every line of Cantor's list it is true that this line does not contain the diagonal number. Nevertheless the diagonal number may be in the infinite list. (WM, sci.logic) === Subject: Re: the problem with Cantor > Moreover, it's possible that all kinds of statements > are theorems of various rigorous theories. So what? If there is > substance here with regard to a rigorous theory, then we'd like to > know at least SOMETHING about such a theory and about what it can > prove in mathematics. Who in the real world are the ones who would have the right to *define* what substance *must* be? And how would such a definition help or hurt mathematical reasoning in general? > No, you're not. Rather, you're trying to make yourself out to be some > kind of spokesman for fairness and open-mindedness. You sounded as if speaking for fairness and open-mindedness were some kind of a bad thing people should avoid doing? > If what you're > interested in is a rigorous theory, then, please, by all means, give > us at least a sketch or even a glimmer of a notion of a particular > one. Didn't you just say a moment ago So what? to a rigorous theory, why now encouraging one to talk about such a thing? MoeBlee > -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: the problem with Cantor <488ff1d0$0$30460$afc38c87@news.optusnet.com.au> <7_Sjk.24048$IK1.19105@news-server.bigpond.net.au> <7eh694plfn9etuavv9gqv1mpepq0ukkd01@4ax.com> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Moreover, it's possible that all kinds of statements > are theorems of various rigorous theories. So what? If there is > substance here with regard to a rigorous theory, then we'd like to > know at least SOMETHING about such a theory and about what it can > prove in mathematics. Who in the real world are the ones who would have the right to *define* > what substance *must* be? It's an informal notion. Anyone is welcome to propose whatever he or she feels is or is not substantive. > And how would such a definition help or hurt > mathematical reasoning in general? It's up to each individual to decide for him or herself whether any given bit of writing has enough substance in it to be of mathematical interest. > No, you're not. Rather, you're trying to make yourself out to be some > kind of spokesman for fairness and open-mindedness. You sounded as if speaking for fairness and open-mindedness were some kind > of a bad thing people should avoid doing? No, I don't say it's a bad thing or that one should avoid doing it. Rather, in the present case, I find that the poster seems more interested in giving the impression that he is doing it (or even feels that he is doing it) than in showing us any formal theories he keeps talking of the possibility of. And also, that in the present case, the poster is misguided as to who is and isn't being fair in certain of these discussions. I consider it quite fair to mention that anyone is free to propose theories and that they can be evaluated publically and individually on whatever basis of evaluation the evaluator mentions, meanwhile being careful to keep straight just what is the case as to questions of what is or is not provable in such theories as ZFC. And I find not so fair or open-minded the continual misinformed and misconceived dogma of cranks that such theories as ZFC are wrong. (And I don't even hold that one may not have reasonable grounds for not preferring ZFC or even thinking that ZFC is fundamentally wrong; except that cranks don't give reasonable grounds, as instead they give dogma.) > If what you're > interested in is a rigorous theory, then, please, by all means, give > us at least a sketch or even a glimmer of a notion of a particular > one. Didn't you just say a moment ago So what? to a rigorous theory, why now > encouraging one to talk about such a thing? I've said so what to the MERE mention that something may be a theorem of some formal theory. That doesn't mean that I am unconcerned about formal theories. MoeBlee === Subject: Re: the problem with Cantor > It's the difference between insulting someone > and constructive criticism. Many standard > mathematicians prefer the former -- since it > is easier -- But please don't mistake 'sci.math regulars' for 'standard mathematicians'. > ... while I strive for the latter. I > believe that it's better to find a theory in > which Herc's claims hold rather than simply > repeat how he's wrong. And I have found such > a theory -- the theory of computable reals. Reactions that suggest that standard mathematics can't make sense of certain ideas may indeed be the main reason why crackpots keep thinking they've something valuable to contribute. Unfortunately, experience learns that referring a crackpot to some relevant parts of respectable mathematics doesn't work. Oddly enough, it also doesn't stop anti-crackpots from continuing their bashing. In short: don't spoil your time; it's hopeless either way. -- Herman Jurjus === Subject: Re: the problem with Cantor > Indeed, Balthasar has already mentioned *N > and *R. The sets *N and *R do have the same > cardinality, namely c. So this is a short > path from the claim to something that does > hold (though I don't believe that Herc > had *N and *R in mind, but this may be > similar to WM's ideas, since WM has mentioned > Robinson in his posts). I think I asked this before: *N and *R are not uniquely defined in Robinson's approach. So what do you mean when you say that *N and *R have the cardinality c? There are certainly non-standard models of PA that are countable. Of course, there are countable models of the reals with their arithmetic properties also, from Lowenheim-Skolem. -- Alan Smaill === Subject: Re: the problem with Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > at best classical constructivism. classical constructivism, funny. And Herc hardly represents any coherent version of constructivism. > despite not accepted @ sci.math not new either. Most mathematicians are not constructivists. But that direct that explanations from a constructivist viewpoint are unwelcome merely for being constructivist. On the contrary, much work in mathematical logic concerns results about the ramifications of constructivism even for classical mathematics. MoeBlee === Subject: Re: the problem with Cantor <2ah49414i6l7cihuf20qabo9ukjlfasppu@4ax.com> posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > In a fully-computable realm, ... Do we have a formal definition of /fully-computable realm/, Moe? Moreover does this idiot ASSUME that the context of our considerations > is a fully-computable realm? - Whatever that may be. Now that the discussion is over and all results are clear, i can take the time for you: Apart from your absolute cluelessness, you are by far the most revolting piece of around. Keep enjoying your conversantion. -LV > ... any formalizable property does express a set [...]. Fascinating. One might think that (at least in a standard realm) > /being not an element of itself/ is a formalizable property: æ æ æ æ x !e x. Still it's hard to see how in a fully-computable realm the Russell-set > can exist, i.e. a set y such that æ æ æ æ Ax(x e y <-> x !e x). Note that this property (in a class theory) does express (determine) > a proper class. (Though in NF, where there are only sets, the predicate > x !e x is not admissible, since it's not stratified.) > It is rather the accepted approach that is problematic in this respect. Crank speak. B. -- For every line of Cantor's list it is true that this line does not > æcontain the diagonal number. æNevertheless the diagonal number may > æbe in the infinite list. (WM, sci.logic) === Subject: first eigenvalue i am still looking for the first eigenvalue of the bilaplacian on the unit ball in R^N with zero Dirichlet boundary conditions. Does anyone know it in terms of known or computable constants? (clearly it is known and is the first zero of some special function or something) craig === Subject: Re: first eigenvalue <21335622.1217629592027.JavaMail.jakarta@nitrogen.mathforum.org>, > i am still looking for the first eigenvalue of the bilaplacian on the unit > ball in R^N with zero Dirichlet boundary conditions. Does anyone know it in terms of known or computable constants? If not does anyone have any suggestion where i look or what i should type in (clearly it is known and is the first zero of some special function or > something) Have you considered posting to sci.math.research? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: old hat? or just not true? No, I think Prof. Edgar's proof was correct. Empirically it's tantalizing. === Subject: ? derivative of max norm posting-account=H-IscAoAAABkDNrURGSxo9jPN3MJ3a8A 1.0.3705; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) The def of max norm of a vector is: max( abs(x_i), i = 1 to N ). But how does one determine its derivative? === Subject: Re: ? derivative of max norm > The def of max norm of a vector is: max( abs(x_i), i = 1 to N ). But how does one determine its derivative? > By using a precise definition of derivative and then thinking about it. Really, how much thought have you put into this? Care to share some ideas you've tried? === Subject: Re: ? derivative of max norm > The def of max norm of a vector is: max( abs(x_i), i = 1 to N ). But how does one determine its derivative? > The partial deriv wrt x_i is 1 if ||x|| = x_i > all other |x_j| -1 if ||x|| = -x_i > all other |x_j| 0 if |x_i| < ||x|| undefined otherwise -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: New Produced Luxury Residence - Classified Preview 2008/11 posting-account=uWkSGAoAAABcv1Qo8K_q913gt5Szmp28 Hotbar 4.5.1.0),gzip(gfe),gzip(gfe) 28 Juli, 05:01 Title: Munifus Landmark Estate » House Manufacturing Worldwide Project Contracting City: Stockholm, London, Moscow, Los Angeles, Washington, New York, Miami, Chicago, Austin, Tokyo, Jordan, Yemen, Oslo, Helsinki, Berlin, Vienna, Bejing, Istanbul, Alger, Gibraltar, Monte Carlo, Paris, Brussels, Amsterdam, Hamburg, Frankfurt, Zurich, Barcelona, Madrid, State: NON-US Description: Munifus Landmark Estate » represents a building-contractor located in many of the most beautiful and highly attractive locations around the world. The following represents a detailed description for one selection of Munifus Luxury Property ». We have chosen to present a European-style residence to fit an extraordinary home residence in architectural richness with panoramic views, a lovely design and custom built home. Munifus House » has since it suit its first master safe home seems a perfect family living and grand entertaining.With plans and permits in place Munifus House » has found to the comfortable architectural style and setting on a peaceful inviting skilfully ground with wealth of charming details. Completely in the perfect featuring Munifus Luxury Property » is been aimed highest degree for quality to a superb living. Spacious and luxurious family living has been a typical decorative Munifus House » impression. Information: Prize Indices Analysis, Insurance, Inspection, Finance. Agent: Mr Roger K. Olsson Int call: +46 (0) 705474830 Address: Mr Roger K. Olsson PL 25 40 Kuttainen SE: 980 16 Karesuando Real estate website: http://www.rapidsellers.com/molsson/ Prospect: http://publishing.yudu.com/Freedom/Ak8i8/DevelopmentProject/ === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 1, 7:16æam, David Formosa (aka ? the Platypus) [...] > | [...] > | > | > existence exists : æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ tautology > | > nonexistence is nonexistent : æ æ æ æ æ æ æ æ æ ætautology > | > indeterminate existence exists indeterminately : tautology > | > | These are not tautologies, these are type errors. [...] > Admittedly, my usage of existence was a bit liberal here. What do you mean by existence? æTypically we use exists to mean has > the property of being within the concrete universe. æBy this > definition abstractions like red, two and existence don't exist as > they are not within the concrete universe. æRather we all we can do is > see examples of objects that have these properties. > However, I > think that the overall picture that is being painted is valid, Could you give me an argument to support your argument that this is a > valid thesis? [...] My very liberal usage of the words (1)existence, (2)nonexistence, and (3) existentially indeterminate, can be thought of or even defined more rigorously as follows: Definition of existence (as I am using it) The sum totality of all things which are said to exist. Definition of nonexistence (as I am using it) The sum totality of all things which are said to be nonexistent. This includes both those things which cannot exist, and those things which could exist but simply do not. I dont think that it is logical to attempt to draw a distinction between those two classes (which are'nt even classes btw). Definition of the existially indeterminate (as I am using it) The sum totality of all things which may or may not exist. I'll post some analysis below to justify this, a worked problem. But to reword things, [1] The sum totality of all things which are said to exist, exists. [2] The sum totality of all things which are said to be nonexistent, clearly do not exist. [3] The sum totality of all things which are existially indeterminate, obviously may or may not exist. That's a bit better I think. One strange implication of [3] is that no matter how much work you do in this area, nothing is provable and the whole thing can simply evaporate. > [1] Existence exists. Somehow, this forms the basis of all known > mathematics and allows logic to function like a classical Newtonian > Clock. No logic I know of depends on this. æMathematics can function quite > well as a formal game that has no existance. > [2] Nonexistence does not exist. There is a paradox here, and the > reason becomes quite obvious. Normally a paradox is an indercator that your axioms have problems. I think that the paradox here is caused because you cannot perform acts of logic (like math) upon the nonexistent. But, because nonexistence is an existential form, if we expect it to have this property of being self referential, then a paradox must result. This also sheds some light on zero and the problems which can be devised, e.g. zero apples is zero oranges etc. > You cannot make statements about the nonexistent. In standard mathmatics one can make statements about the properties of > the empty set, however they all end up being trivally true. Empty set is very much like zero. It creates no problems, but I am pretty sure that there are some things that you are not supposed to do with it to avoid division by zero type issues. What you cant do is calculate the area of a 4-sided triangle. > You cant say that it exists or not, because it does not > exist. One could put that argument as X doesn't exist so therefor X's > existance can't be established, however you have just established > truth value of X's existance in the first part of the sentence. > Nor would it make sense to calculate the area of a 4sided > triangle because it does not exist. 4 sided triangles are logically inconstant, this means that you will > not find any objects that both possess the triangle property and the 4 > sided property. > The presence of paradox implies the reflexivity of the property. Can you please explain this? Some analysis and a worked problem. First, the central motivation for this approach. [1] Length can be graduated by Plancklengths. [2] The precise position of graduations is indeterminate. The graduations can slide back and forth freely. [3] Therefore length is probabilistic, and any length contained in R can exist in space, despite Plancklength. To consider the idea of random length, you would have one of two possible cases : Case1 A random segment is added to a given length, L = |--------------------| + |~ ~ ~| Case2 The randomness is multiplied into the length and the whole thing stretches like a rubberband L' = |~~~~~~~~~~~~~~~~~| For Case2, you must have existential indeterminacy. You must be able to say that points exist with probability a,b,c.... So, thats the motivation. -------------------------------------------------------------------- Examples: Calculate the distance between 2 points (0,0) and (1,1), where the existential potential function is given as z = f(x,y) = 1/2. What this means is that each point in the region in question has a 50:50 probability of existing, and we want the distance from 0,0) to (1,1). Our answer will be an expected length. Ordinarily, the distance is given by SQRT(2)/2. But in this case the expected length is SQRT(2)/4. The math is simple enough, and I could haul out all kinds of integrals but I dont want to muddle it up. The philosophy is the hard part, not the algebra. If you want the area of the unit square where the existential potential function is given as z = f(x,y) = 1/2, the expected area of the unit square is simply 1/2 * 1/2 = 1/4. Again, you could do this all with integrals (which resemble PDFs), but we'll just stick to simple cases for the time being. We can talk about these simple examples, but what I am moving toward is explaining the precession of perihelion of planets. I think that it can be calculated pretty easily with just vector calculus and an appropriate choice of existential potential function for 3 or 4 space. === Subject: Re: New math. Dont read this. > I think that the paradox here is caused because you > cannot perform > acts of logic (like math) upon the nonexistent. Nope, not true. Example: Penrose triangle. > But, > because > nonexistence is an existential form, if we expect it > to have this > property of being self referential, then a paradox > _must_ result. Nope. Self reference is not paradox. It merely limits the number of true statements one can extract from a system. Self reference can lead to paradox (such as Russell's antinomy); that is, however, a result of the inherent limitation. > This also sheds some light on zero and the problems > which can be > devised, e.g. zero apples is zero oranges etc. No problem here. One apple is also equivalent to one orange. As is 23 apples to 23 oranges. And so on. Ordinal identity is independent of object properties. You cannot make statements about the nonexistent. > We do it all the time. Proof by double negation depends on our ability to make statements of that which cannot exist. > In standard mathmatics one can make statements > about the properties of > the empty set, however they all end up being > trivally true. > The term trivial in mathematics (though often overused) refers to conclusions, not assumptions. Zero has a long history of development (See Charles Seife, Zero: The Biography of a Dangerous Idea) and is far from trivial. The empty set is subtler still. Empty set is very much like zero. It creates no > problems, but I am > pretty sure that there are some things that you are > not supposed to do > with it to avoid division by zero type issues. > Such as? > What you cant do is calculate the area of a 4-sided > triangle. > But you can count the legs on a three legged horse. Your point is vacuous. You cant say that it exists or not, because it > does not > exist. > You just DID say that it exists or not. > One could put that argument as X doesn't exist so > therefor X's > existance can't be established, however you have > just established > truth value of X's existance in the first part of > the sentence. > No, you haven't. Tom > Nor would it make sense to calculate the area of > a 4sided > triangle because it does not exist. > 4 sided triangles are logically inconstant, this > means that you will > not find any objects that both possess the triangle > property and the 4 > sided property. > The presence of paradox implies the reflexivity > of the property. > Can you please explain this? > Some analysis and a worked problem. First, the central motivation for this approach. > [1] Length can be graduated by Plancklengths. > [2] The precise position of graduations is > indeterminate. The > graduations can slide back and forth freely. > [3] Therefore length is probabilistic, and _any_ > length contained in R > can exist in space, despite Plancklength. To consider > the idea of > random length, you would have one of two possible > cases : Case1 > A random segment is _added_ to a given length, > L = |--------------------| + |~ ~ ~| Case2 > The randomness is _multiplied_into_ the length and > the whole thing > stretches like a rubberband > L' = |~~~~~~~~~~~~~~~~~| For Case2, you must have existential indeterminacy. > You must be able > to say that points exist with probability a,b,c.... So, thats the motivation. ------------------------------------------------------ > -------------- Examples: Calculate the distance between 2 points (0,0) and > (1,1), where the > existential potential function is given as z = f(x,y) > = 1/2. What this > means is that each point in the region in question > has a 50:50 > probability of existing, and we want the distance > from 0,0) to (1,1). > Our answer will be an expected length. Ordinarily, the distance is given by SQRT(2)/2. But > in this case the > expected length is SQRT(2)/4. The math is simple enough, and I could haul out all > kinds of integrals > but I dont want to muddle it up. The philosophy is > the hard part, not > the algebra. > If you want the area of the unit square where the > existential > potential function is given as z = f(x,y) = 1/2, the > expected area > of the unit square is simply 1/2 * 1/2 = 1/4. Again, > you could do this > all with integrals (which resemble PDFs), but we'll > just stick to > simple cases for the time being. > We can talk about these simple examples, but what I > am moving toward > is explaining the precession of perihelion of > planets. I think that it > can be calculated pretty easily with just vector > calculus and an > appropriate choice of existential potential function > for 3 or 4 > space. > === Subject: Re: New math. Dont read this. [...] > But, > because > nonexistence is an existential form, if we expect it > to have this > property of being self referential, then a paradox > _must_ result. Nope. Self reference is not paradox. It merely > limits the number of true statements one can extract from > a system. Indeed most of the systems that mathmatics deals with (at least thouse powerfull enought to embed natural numbers) can be made into a self referencial system. === Subject: Re: New math. Dont read this. > [...] > But, > because > nonexistence is an existential form, if we expect > it > to have this > property of being self referential, then a > paradox > _must_ result. > Nope. Self reference is not paradox. It merely > limits the number of true statements one can > extract from > a system. Indeed most of the systems that mathmatics deals with > (at least thouse > powerfull enought to embed natural numbers) can be > made into a self > referencial system. > Exactly. Tautology is a useful tool in this context. Tom === Subject: Re: New math. Dont read this. <27358300.1217673383465.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 2, 7:11æpm, David Formosa (aka ? the Platypus) > [...] > But, > because > nonexistence is an existential form, if we expect it > to have this > property of being self referential, then a paradox > must result. > Nope. æSelf reference is not paradox. æIt merely > limits the number of true statements one can extract from > a system. The paradox is not caused by the self reference. It it caused because the self reference is being forced to act upon nonexistence, which is not a valid thing to do from a logical standpoint. The fact that nonexistence is indeed paradoxical, this seems to validate the thesis that it is acting upon itself vis-a-vis self reference. === Subject: Re: New math. Dont read this. > On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > _must_ result. > Nope. Self reference is not paradox. It merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical standpoint. The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. > Do some study into what paradox means. Tom === Subject: Re: New math. Dont read this. <19917341.1217764210902.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > must result. > Nope. æSelf reference is not paradox. æIt merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical standpoint. > The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. Do some study into what paradox means. Tom- Hide quoted text - - Show quoted text - Is an apple an orange, or is an apple not an orange. Perhaps you could give me an answer, either yes or no. Should be a very simple thing to do. Is an apple an orange ? Please check the appropriate box. [ ] yes [ ] no === Subject: Re: New math. Dont read this. > On Aug 3, 6:49 am, T.H. Ray On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > _must_ result. > Nope. Self reference is not paradox. It > merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. > It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical > standpoint. > The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. > Do some study into what paradox means. > Tom- Hide quoted text - > - Show quoted text - > Is an apple an orange, or is an apple not an orange. > Perhaps you could > give me an answer, either yes or no. Should be a very > simple thing to > do. Is an apple an orange ? Please check the appropriate box. [ ] yes > [ ] no > Sorry, your naive conceptions of identity are way off from mathematical modeling or logical tractability. Let's say A and O are equal (i.e., the same)when A=O. Then your zero apples is the same as zero oranges claim is simply trivial, isn't it? No paradox, and nothing even of interest. Tom === Subject: Re: New math. Dont read this. <21685791.1217777047116.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Aug 3, 6:49 am, T.H. Ray On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > must result. > Nope. æSelf reference is not paradox. æIt > merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. > It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical > standpoint. > The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. > Do some study into what paradox means. > Tom- Hide quoted text - > - Show quoted text - > Is an apple an orange, or is an apple not an orange. > Perhaps you could > give me an answer, either yes or no. Should be a very > simple thing to > do. > Is an apple an orange ? > Please check the appropriate box. > [ ] yes > [ ] no Sorry, your naive conceptions of identity are way > off from mathematical modeling or logical tractability. > Let's say A and O are equal (i.e., the same)when A=O. > Then your zero apples is the same as zero oranges > claim is simply trivial, isn't it? æNo paradox, and > nothing even of interest. Tom- Hide quoted text - - Show quoted text - Actually Tom, we need to distinguish between the abstract and the physical. Lets talk math for a second. You could easily refute me by saying that apples are never oranges, because you need to divide by zero in order to yield the statement that apples = oranges. You could easily refute me that way. Even though I would argue that it is in fact the act of multiplying that yields the contradiction. Regardless, that is the abstract case and it is irresolvable in my opinion - and I know you disagree. But lets consider the physical case instead. Imagine your kitchen table. There are no apples, and there are no oranges. You have nothing on the table. So, do you have zero apples on the table ? Yes or No ? That is where the paradox lives. Now, if math were a valid model of reality, you would'nt just invent rules in order to duct tape things together to keep arithmetic from blowing up. Math is supposed to model reality. I dont think that it is valid to invent rules simply because we dont like paradoxes and we want to illegalize them from a particular model. Truth is not a referrendum. === Subject: Re: New math. Dont read this. [...] > Now, if math were a valid model of reality, you would'nt just invent > rules in order to duct tape things together to keep arithmetic from > blowing up. Math is supposed to model reality. Math is supposed to amuse mathimatisions. If it happens to model reality thats a happy coincidence. > I dont think that it is > valid to invent rules simply because we dont like paradoxes and we > want to illegalize them from a particular model. For the nonconstructivest amoung us concluding that a theorm is true from the fact that maths blows up if its false is quite legitlimate reasoning. === Subject: Re: New math. Dont read this. <21685791.1217777047116.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 3, 4:49æpm, David Formosa (aka ? the Platypus) [...] > Now, if math were a valid model of reality, you would'nt just invent > rules in order to duct tape things together to keep arithmetic from > blowing up. Math is supposed to model reality. Math is supposed to amuse mathimatisions. æIf it happens to model > reality thats a happy coincidence. > I dont think that it is > valid to invent rules simply because we dont like paradoxes and we > want to illegalize them from a particular model. For the nonconstructivest amoung us concluding that a theorm is true > from the fact that maths blows up if its false is quite legitlimate > reasoning. Try this on. You can prove that a photon is a wave. You can also prove that a If I have a segment of length and I can demonstrate that it is continuous, but I can also demonatrste that it is discrete.....do you really think that reducto ad absurdum is going to help you out here ? I say that if you can prove it to be a wave, and you can prove it to whether it is one or the other. Is an apple an orange ? Or, is it indeterminate whether you have an apple or an orange ? === Subject: Re: New math. Dont read this. > On Aug 3, 4:49 pm, David Formosa (aka ? the > Platypus) > [...] > Now, if math were a valid model of reality, you > would'nt just invent > rules in order to duct tape things together to > keep arithmetic from > blowing up. Math is supposed to model reality. > Math is supposed to amuse mathimatisions. If it > happens to model > reality thats a happy coincidence. > I dont think that it is > valid to invent rules simply because we dont like > paradoxes and we > want to illegalize them from a particular model. > For the nonconstructivest amoung us concluding that > a theorm is true > from the fact that maths blows up if its false is > quite legitlimate > reasoning. > Try this on. You can prove that a photon is a wave. You can also > prove that a > one can measure simultaneously both the position and Nothing mysterious here. > If I have a segment of length and I can demonstrate > that it is > continuous, but I can also demonatrste that it is > discrete.....do you > really think that reducto ad absurdum is going to > help you out here ? > You can only demonstrate that a length is continuous, to arbitrary accuracy. > I say that if you can prove it to be a wave, and you > can prove it to > _indeterminate_ > whether it is one or the other. > Google quantum superposition. > Is an apple an orange ? Or, is it _indeterminate_ > whether you have an > apple or an orange ? No more than it is indeterminate whether you have a Tom === Subject: Re: New math. Dont read this. Actually Tom, we need to distinguish between the > abstract and the > physical. > We need to distinguish between mathematical logic and nonsense. > Lets talk math for a second. That'd be a switch. >You could easily refute > me by saying that > apples are never oranges, because you need to divide > by zero in order > to yield the statement that apples = oranges. Apples could equal _anything_ if one could divide by zero. Did you not understand that? > You > could easily refute > me that way. Even though I would argue that it is in > fact the act of > multiplying that yields the contradiction. Your argument would, however, be wrong. It is in the fact that there is no multiplicative inverse (i.e., division) that division by zero is forbidden. I explained this. > Regardless, that is the > abstract case and it is irresolvable in my opinion - > and I know you > disagree. > I disagree, because it is meaningless. > But lets consider the physical case instead. > Imagine your kitchen table. There are no apples, and > there are no > oranges. You have nothing on the table. So, do you > have zero apples on > the table ? Yes or No ? > I also have zero platypuses and zero dingbats. > That is where the paradox lives. > Argh. Stop doing violence to the word paradox. > Now, if math were a valid model of reality, you > would'nt just invent > rules in order to duct tape things together to keep > arithmetic from > blowing up. Math is supposed to model reality. I dont > think that it is > valid to invent rules simply because we dont like > paradoxes and we > want to illegalize them from a particular model. > Truth is not a > referrendum. > Sigh. The rules are not arbitrary or whimsical. They are imposed for the sake of the system's self consistency. Tom > === Subject: Re: New math. Dont read this. <7186082.1217788891533.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Actually Tom, we need to distinguish between the > abstract and the > physical. We need to distinguish between mathematical logic > and nonsense. That is exactly it Tom, exactly. We do need to distinguish between the sensical and the nonsensical. My thesis is that there might be regarded a grey area in between where it is indeterminate whether one is performing sense or nonsense. And I say might be ragarded because it is indeterminate. I know that what I am saying sounds ridiculous but I dont think that you guys have refuted me, Im just not convinced that Im wrong, Perhaps I am, and I remain open to that but I just dont see any solid reason why I would be refuted at this point. === Subject: Re: New math. Dont read this. > On Aug 3, 1:41 pm, T.H. Ray Actually Tom, we need to distinguish between the > abstract and the > physical. > We need to distinguish between mathematical logic > and nonsense. > That is exactly it Tom, exactly. We do need to distinguish between the sensical and > the nonsensical. My > thesis is that there might be regarded a grey area > in between where > it is indeterminate whether one is performing sense > or nonsense. And I say might be ragarded because it is > indeterminate. I know that what I am saying sounds ridiculous but I > dont think that > you guys have refuted me, Im just not convinced that > Im wrong, Perhaps > I am, and I remain open to that but I just dont see > any solid reason > why I would be refuted at this point. You fail to understand that to be refuted (i.e., falsified), your proposition must provide the means of falsification. To merely make a claim equivalent to God created the world. Refute that. has nothing to do with logic or mathematics. Tom === Subject: Re: New math. Dont read this. <16634840.1217857338102.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > On Aug 3, 1:41 pm, T.H. Ray Actually Tom, we need to distinguish between the > abstract and the > physical. > We need to distinguish between mathematical logic > and nonsense. > That is exactly it Tom, exactly. > We do need to distinguish between the sensical and > the nonsensical. My > thesis is that there might be regarded a grey area > in between where > it is indeterminate whether one is performing sense > or nonsense. > And I say might be ragarded because it is > indeterminate. > I know that what I am saying sounds ridiculous but I > dont think that > you guys have refuted me, Im just not convinced that > Im wrong, Perhaps > I am, and I remain open to that but I just dont see > any solid reason > why I would be refuted at this point. You fail to understand that to be refuted (i.e., > falsified), your proposition must provide the means of > falsification. æTo merely make a claim equivalent to > God created the world. æRefute that. has nothing to > do with logic or mathematics. Tom- Hide quoted text - - Show quoted text - That's the existential dichotomy doin all the talkin'. === Subject: Re: New math. Dont read this. <21685791.1217777047116.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Aug 3, 6:49 am, T.H. Ray On Aug 2, 7:11 pm, David Formosa (aka ? the > Platypus) > [...] > But, > because > nonexistence is an existential form, if we > expect it > to have this > property of being self referential, then a > paradox > must result. > Nope. æSelf reference is not paradox. æIt > merely > limits the number of true statements one can > extract from > a system. > The paradox is not caused by the self reference. > It > it caused because > the self reference is being forced to act upon > nonexistence, which > is not a valid thing to do from a logical > standpoint. > The fact that nonexistence is indeed paradoxical, > this seems to > validate the thesis that it is acting upon itself > vis-a-vis self > reference. > Do some study into what paradox means. > Tom- Hide quoted text - > - Show quoted text - > Is an apple an orange, or is an apple not an orange. > Perhaps you could > give me an answer, either yes or no. Should be a very > simple thing to > do. > Is an apple an orange ? > Please check the appropriate box. > [ ] yes > [ ] no Sorry, your naive conceptions of identity are way > off from mathematical modeling or logical tractability. > Let's say A and O are equal (i.e., the same)when A=O. > Then your zero apples is the same as zero oranges > claim is simply trivial, isn't it? æNo paradox, and > nothing even of interest. Tom- Hide quoted text - - Show quoted text - What I meant is that zero apples is indistinguishable from zero oranges. Now either apples are oranges, or they are not oranges. But you still have the case that zero apples cannot be distinguished from zero oranges unless you create special rules. We cannot say that apples are oranges. But we can say that zero apples is zero oranges, and I did not divide by zero. I multiplied. === Subject: Re: New math. Dont read this. What I meant is that zero apples is indistinguishable > from zero > oranges. > So what? > Now either apples are oranges, or they are not > oranges. > Again, so what? The ordinal number of apples or oranges says nothing about the properties of apples and oranges. > But you still have the case that zero apples _cannot_ > be distinguished > from zero oranges unless you create special rules. > Huh? You mean, like describing the properties of apples and oranges? Man, you are confused. > We cannot say that apples are oranges. > We can say anything we want. > But we can say that zero apples is zero oranges, > and I did not > divide by zero. I multiplied. > And the result was, no surprise, zero. ???? Tom === Subject: Re: New math. Dont read this. <27358300.1217673383465.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > I think that the paradox here is caused because you > cannot perform > acts of logic (like math) upon the nonexistent. Nope, not true. æExample: æPenrose triangle. Visual illusion. > But, > because > nonexistence is an existential form, if we expect it > to have this > property of being self referential, then a paradox > must result. Nope. æSelf reference is not paradox. æIt merely > limits the number of true statements one can extract from > a system. æSelf reference can lead to paradox (such > as Russell's antinomy); that is, however, a result of > the inherent limitation. Self reference is indeed paradox if you have a situation where (1) It must be self referential, AND (2) It cannot be self referential. Nonexistence satisfies both (1) and (2). > This also sheds some light on zero and the problems > which can be > devised, e.g. zero apples is zero oranges etc. No problem here. One apple is also equivalent > to one orange. æAs is 23 apples to 23 oranges. æAnd > so on. æOrdinal identity is independent of object > properties. Yikes. I'll never send you to the store for anything..... > You cannot make statements about the nonexistent. We do it all the time. æProof by double negation depends > on our ability to make statements of that which > cannot exist. You are in fact making statements and manipulations on the existent, and then deriving by proof that things dont exist. Proof is a decision making process. You are not really causing logic to act upon nonexistence. You cannot do that anymore than you can divide by zero. > In standard mathmatics one can make statements > about the properties of > the empty set, however they all end up being > trivally true. The term trivial in mathematics (though often > overused) refers to conclusions, not assumptions. > Zero has a long history of development (See Charles > Seife, Zero: The Biography of a Dangerous Idea) and > is far from trivial. æThe empty set is subtler still. > Empty set is very much like zero. It creates no > problems, but I am > pretty sure that there are some things that you are > not supposed to do > with it to avoid division by zero type issues. Such as? AOC > What you cant do is calculate the area of a 4-sided > triangle. But you can count the legs on a three legged horse. > Your point is vacuous. If I said that you cannot dovide by zero - is that also vacuous ? In fact, it is indeterminate whether this whole thesis is vacuous. My point is that indeterminacy does not constitute negation. And so, even though the whole thesis is wholly and thoroughly indeterminate, it is indeed not negated any more than it is validated. It is permanently stuck right where it belongs, in a state of indeterminacy. Neither negated, nor validated. Neither negatable, nor validateable. > You cant say that it exists or not, because it > does not > exist. You just DID say that it exists or not. True. I did say it. I said it because I had to, not because I wanted to. === Subject: Re: New math. Dont read this. >[snip the circular nonsense] > > If I said that you cannot dovide by zero - is that > also vacuous ? > Yes. If you knew the _reason_ that division by zero is a prohibited operation, you would know that the rule is chosen to prevent arithmetic from blowing up; i.e., to allow consistent answers to arithmetic problems. In fact, there is mathematics in which division by zero is allowed. It is not indeterminate. Tom > In fact, it is indeterminate whether this whole > thesis is vacuous. My > point is that indeterminacy does not constitute > negation. And so, even > though the whole thesis is wholly and thoroughly > indeterminate, it is > indeed not negated any more than it is validated. It > is permanently > stuck right where it belongs, in a state of > indeterminacy. Neither > negated, nor validated. Neither negatable, nor > validateable. You cant say that it exists or not, because > it > does not > exist. > You just DID say that it exists or not. > True. I did say it. I said it because I had to, not > because I wanted > to. > === Subject: Re: New math. Dont read this. <10325230.1217691464822.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) >[snip the circular nonsense] > If I said that you cannot dovide by zero - is that > also vacuous ? Yes. If you knew the reason that division by zero is a > prohibited operation, you would know that the rule > is chosen to prevent arithmetic from blowing up; i.e., > to allow consistent answers to arithmetic problems. > In fact, there is mathematics in which division by > zero is allowed. æIt is not indeterminate. Tom And you think that you know the reason ? The real reason ? Perhaps there is a deeper reason than just arithmetic ? Or that would be impossible ? The only place where division by zero is allowed is in QM and I dont find it very gratifying. === Subject: Re: New math. Dont read this. > On Aug 2, 10:37 am, T.H. Ray [snip the circular nonsense] > If I said that you cannot dovide by zero - is > that > also vacuous ? > Yes. > If you knew the _reason_ that division by zero is a > prohibited operation, you would know that the rule > is chosen to prevent arithmetic from blowing up; > i.e., > to allow consistent answers to arithmetic problems. > In fact, there is mathematics in which division by > zero is allowed. It is not indeterminate. > Tom > And you think that you know the reason ? The real > reason ? Perhaps > there is a deeper reason than just arithmetic ? Or > that would be > impossible ? > Yes, Huang, I know the real reason. It is not mystical nor difficult to understand. It is simply that multiplication of any term by zero is zero. Therefore, division being the inverse of multiplication, there is no accommodation for division by zero. 1/0 times 0 does not equal 1. If it did, one could prove anything mathematically and numbers would be useless. There would be no logical structure at all. (In Seife's book I reference earlier, he uses division by zero to prove that Winston Churchill is a carrot.) > The only place where division by zero is allowed is > in QM and I dont > find it very gratifying. You need to come up with an example before anyone can determine what you're trying to say here. Tom === Subject: Re: New math. Dont read this. <31376811.1217764074350.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > The only place where division by zero is allowed is > in QM and I dont > find it very gratifying. You need to come up with an example before anyone can > determine what you're trying to say here. Tom- Hide quoted text - > I already did but the conversation seems to have shifted away from worked examples, and more toward foundational issues. === Subject: Re: New math. Dont read this. > On Aug 1, 7:16 am, David Formosa (aka ? the Platypus) [...] > However, I > think that the overall picture that is being painted is valid, > Could you give me an argument to support your argument that this is a > valid thesis? > [...] > My very liberal usage of the words (1)existence, (2)nonexistence, and > (3) existentially indeterminate, can be thought of or even defined > more rigorously as follows: Definition of existence (as I am using it) > The sum totality of all things which are said to exist. By sum totality you mean something equilverlent to class in the mathmatical sence of the word, if not what is the diffrence? I'm also wondering how you define exist do abstractions exist? You state that sum totalities can exist, what does this mean for one to exist? > Definition of nonexistence (as I am using it) > The sum totality of all things which are said to be nonexistent. This > includes both those things which cannot exist, and those things > which could exist but simply do not. I dont think that it is logical > to attempt to draw a distinction between those two classes (which > are'nt even classes btw). If something is not included in the sum totality of things that do not exist does that mean it exists? Can a sum totality that does not exist include members? Why are they not considered classes? > Definition of the existially indeterminate (as I am using it) > The sum totality of all things which may or may not exist. Given that (a / not(a)) is a tautology everything is existially indeterminate. > I'll post some analysis below to justify this, a worked problem. But to reword things, > [1] The sum totality of all things which are said to exist, exists. Ok this implies that abstractions do exist. > [2] The sum totality of all things which are said to be nonexistent, > clearly do not exist. Its not at all clear to me. Not clearly knowing what you mean by sum totality nor what you mean by a sum totality existing or not existing. It is my view that using the verb exist to discuss abstract objects is an example of fallacy of reifiction and doesn't lead to usefull resoning. > [3] The sum totality of all things which are existially indeterminate, > obviously may or may not exist. Again it is not obvious to mean. [...] [...] > Normally a paradox is an indercator that your axioms have problems. > I think that the paradox here is caused because you cannot perform > acts of logic (like math) upon the nonexistent. Logic and mathics has very strong tools to deal with nonexistent objects, I would argue that mathatics only deals with abstractions and that abstractions don't have any concrete existence[1]. [...] > You cannot make statements about the nonexistent. > In standard mathmatics one can make statements about the properties of > the empty set, however they all end up being trivally true. Empty set is very much like zero. It creates no problems, but I am > pretty sure that there are some things that you are not supposed to do > with it to avoid division by zero type issues. What you cant do is calculate the area of a 4-sided triangle. You can't calculate the area of a 4-sided triangle because the sentence is meaningless. Likewise the sentence The colour of the quadratic equation is also meaningless. [...] > The presence of paradox implies the reflexivity of the property. > Can you please explain this? > Some analysis and a worked problem. First, the central motivation for this approach. > [1] Length can be graduated by Plancklengths. The length of a physical object may be graduated by plancklengths. However length itself is an abstract quality that has no graduations. > [2] The precise position of graduations is indeterminate. The > graduations can slide back and forth freely. > [3] Therefore length is probabilistic, and _any_ length contained in R > can exist in space, despite Plancklength. To consider the idea of > random length, you would have one of two possible cases : That doesn't answer my question, how does a paradox show that a property is reflexive? === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 2, 4:55æam, David Formosa (aka ? the Platypus) > On Aug 1, 7:16 am, David Formosa (aka ? the Platypus) [...] > However, I > think that the overall picture that is being painted is valid, > Could you give me an argument to support your argument that this is a > valid thesis? > [...] > My very liberal usage of the words (1)existence, (2)nonexistence, and > (3) existentially indeterminate, can be thought of or even defined > more rigorously as follows: > Definition of existence (as I am using it) > The sum totality of all things which are said to exist. By sum totality you mean something equilverlent to class in the > mathmatical sence of the word, if not what is the diffrence? I'm also wondering how you define exist do abstractions exist? æ You state that sum totalities can exist, what does this mean for one > to exist? Well, math is a logical model. An abstraction. And so the usage of the concept of existence will always have two very distinct usages. One is abstract, and one is physical. I think that my usage applies to both the physical and the abstract, and so I usually do not bother to draw the distinction. The model is so close to reality that observing this technicality becomes cumbersome, but the distinction will always be there. As far as physical existence of abstractions goes, I have no idea, that happens in the mind somehow and Im not really concerned with whether an abstract mathematical object has some aspect which makes it exist physically... completely different issue and Im not really concerned with it. For your last question, all I can do is explain what I'm seeing. I believe that physical existence is bounded by extreme scales. Everything which exists relative to us is within about +/- 40 orders of magnitude. The boundaries are where you find existential indeterminacy. This is the precise place where math and physics must part and go their separate ways, because mathematics has no Plancklength. To have such a thing as a Plancklength in mathematics, you have to construct it deliberately. So, if the universe were an Oreo cookie, the crispy chocolate outer cracker would be existentially indeterminate, and existence would be defined as the sugary cream filling. Yum. > Definition of nonexistence (as I am using it) > The sum totality of all things which are said to be nonexistent. This > includes both those things which cannot exist, and those things > which could exist but simply do not. I dont think that it is logical > to attempt to draw a distinction between those two classes (which > are'nt even classes btw). If something is not included in the sum totality of things that do not > exist does that mean it exists? Can a sum totality that does not exist include members? Why are they not considered classes? (i) You cannot consider them to be distinct classes because that would constitute performing a logical or mathematical procedure on the nonexistent. If you consider all of the nonexistent cubes, and you have that some are red and some are blue, you cannot divide them into separate classes according to color because that is a mathematical act. That would be the equivalent of dividing by zero. > Definition of the existially indeterminate (as I am using it) > The sum totality of all things which may or may not exist. Given that (a / not(a)) is a tautology everything is existially > indeterminate. > I'll post some analysis below to justify this, a worked problem. > But to reword things, > [1] The sum totality of all things which are said to exist, exists. Ok this implies that abstractions do exist. It might, but I really dont intend for that to be the case. Because the brain is made of molecules, it is very likely that the abstract and the physical do intersect somehow, but I am not really interested in trying to say that abstractions exist physically. > [2] The sum totality of all things which are said to be nonexistent, > clearly do not exist. Its not at all clear to me. æNot clearly knowing what you mean by sum > totality nor what you mean by a sum totality existing or not > existing. æIt is my view that using the verb exist to discuss abstract > objects is an example of fallacy of reifiction and doesn't lead to > usefull resoning. Well, this definition is a bit paradoxical for the same reason as (i) above. There is no sum totality of things nonexistent, for the reason explained in (i). However, it is possible for beings such as ourselves to contemplate the nonexistent, or to determine that something does not exist (real or abstract). This creates a bit of an illusion that there is some collection of objects which are nonexistent, but clearly there is no such collection or set, it is impossible as explained in (i). One cannot attempt to define nonexistence without creating a fallacy of reification. That is because you are trying to define a paradox. Nonexistence is self referential, just like the other existential forms. As an existent being, I have the ability to enumerate many things which do not exist. I could create a very long list which becomes infinitely long. This creates the illusion that there is such a thing as a sum totality of things nonexistent when in fact there is none. You are correct, I did commit the fallacy of reification, but there is no way to avoid it and here is why : You cannot define nonexistence. To define it it to perform a logical act upon it, and it is immune to logic. My fallacy was in attempting to define it in the first place. But this does not alter the thesis that nonexistence is self-referential, and that this property of being self referential is consistent among all of these supposed existential forms. > [3] The sum totality of all things which are existially indeterminate, > obviously may or may not exist. Again it is not obvious to mean. [...] [...] > Normally a paradox is an indercator that your axioms have problems. > I think that the paradox here is caused because you cannot perform > acts of logic (like math) upon the nonexistent. Logic and mathics has very strong tools to deal with nonexistent > objects, I would argue that mathatics only deals with abstractions > and that abstractions don't have any concrete existence[1]. I would agree completely, with the caveat that there may or may not be some kind of intersection of these worlds in the mind (which is not understood). But it seems that what you said would also be true even if there was no such thing as human beings. So, I dont think that the mind is neccesarily requisite to any of this either. > You cannot make statements about the nonexistent. > In standard mathmatics one can make statements about the properties of > the empty set, however they all end up being trivally true. > Empty set is very much like zero. It creates no problems, but I am > pretty sure that there are some things that you are not supposed to do > with it to avoid division by zero type issues. > What you cant do is calculate the area of a 4-sided triangle. You can't calculate the area of a 4-sided triangle because the > sentence is meaningless. æLikewise the sentence The colour > of the quadratic equation is also meaningless. That is true. I just use that as a prop. > The presence of paradox implies the reflexivity of the property. > Can you please explain this? > Some analysis and a worked problem. > First, the central motivation for this approach. > [1] Length can be graduated by Plancklengths. The length of a physical object may be graduated by plancklengths. > However length itself is an abstract quality that has no graduations. Length can be physical or abstract. I would argue that physical length is a tangible thing which is composed of dimension. Mathematical length is merely a model of the physical variety. > [2] The precise position of graduations is indeterminate. The > graduations can slide back and forth freely. > [3] Therefore length is probabilistic, and any length contained in R > can exist in space, despite Plancklength. To consider the idea of > random length, you would have one of two possible cases : That doesn't answer my question, how does a paradox show that a > property is reflexive? Lets assume that all existential forms are self referential. Then you have that nonexistence is nonexistent. We just performed (or attempted to perform) a logical act upon the nonexistent by saying that it does not exist, which is not valid logically. So, the assumption that all existential forms are self referential points to this situation with the particular case of nonexistence where you have paradox. The usage of existential indeterminacy allows us to do quantum mechanics without saying that things pop in and out of existence - which is nonsense. You can also create an analysis based on these ideas but you sacrifice the ability to prove anything because it is quasi-logic. === Subject: Re: New math. Dont read this. > On Aug 2, 4:55æam, David Formosa (aka ? the Platypus) > On Aug 1, 7:16 am, David Formosa (aka ? the Platypus) [...] > My very liberal usage of the words (1)existence, (2)nonexistence, and > (3) existentially indeterminate, can be thought of or even defined > more rigorously as follows: > Definition of existence (as I am using it) > The sum totality of all things which are said to exist. > By sum totality you mean something equilverlent to class in the > mathmatical sence of the word, if not what is the diffrence? > I'm also wondering how you define exist do abstractions exist? æ > You state that sum totalities can exist, what does this mean for one > to exist? Well, math is a logical model. An abstraction. And so the usage of the > concept of existence will always have two very distinct usages. One is > abstract, and one is physical. Now if something physically exists it means that I can (at least in principal) physically interact with it. I can see it, feel it, or detect via verious pysical means. What does it mean for an abstraction to exist? > I think that my usage applies to both the physical and the abstract, > and so I usually do not bother to draw the distinction. In order to communicate clearly and allow us to reason about this without resulting in error we have to be careful about this distinction. [...] > As far as physical existence of abstractions goes, I have no idea, > that happens in the mind somehow and Im not really concerned with > whether an abstract mathematical object has some aspect which makes it > exist physically... completely different issue and Im not really > concerned with it. The problem is that you are making claims that implicitly assumes this. For example you claim that The sum totality of objects that exist exists however this makes a claim about an abstraction (the sum totality of objects that exist) and its existance. Now if you had said something on the lines of Every object that exists, exists this would by tautological. > For your last question, all I can do is explain what I'm seeing. I > believe that physical existence is bounded by extreme scales. > Everything which exists relative to us is within about +/- 40 orders > of magnitude. The boundaries are where you find existential > indeterminacy. This is the precise place where math and physics must > part and go their separate ways, because mathematics has no > Plancklength. To have such a thing as a Plancklength in mathematics, > you have to construct it deliberately. So, if the universe were an Oreo cookie, the crispy chocolate outer > cracker would be existentially indeterminate, and existence would be > defined as the sugary cream filling. Yum. [...] > If something is not included in the sum totality of things that do not > exist does that mean it exists? > Can a sum totality that does not exist include members? > Why are they not considered classes? > (i) You cannot consider them to be distinct classes because that > would constitute performing a logical or mathematical procedure on the > nonexistent. There is nothing in the rules of mathmatics that prevents it from dealing with objects that do not exist. > If you consider all of the nonexistent cubes, and you > have that some are red and some are blue, you cannot divide them into > separate classes according to color because that is a mathematical > act. That would be the equivalent of dividing by zero. One can constract a mathmatical system that permits the division by zero. [...] > But to reword things, > [1] The sum totality of all things which are said to exist, exists. > Ok this implies that abstractions do exist. It might, but I really dont intend for that to be the case. Can you construct your statement in a way that doesn't carry this implication. Indeed can you write your sentence in a way that unambiguously convays your meaning preferably in a way that makes use of commonly shared definitions. > Because > the brain is made of molecules, it is very likely that the abstract > and the physical do intersect somehow, The map is not the landscape. [...] > [2] The sum totality of all things which are said to be nonexistent, > clearly do not exist. > Its not at all clear to me. æNot clearly knowing what you mean by sum > totality nor what you mean by a sum totality existing or not > existing. æIt is my view that using the verb exist to discuss abstract > objects is an example of fallacy of reifiction and doesn't lead to > usefull resoning. Well, this definition is a bit paradoxical for the same reason as (i) > above. There is no sum totality of things nonexistent, for the > reason explained in (i). You still haven't told me what a sum totality is. How can we have meaningful discussions about anything with out knowing what the words we are using mean? [...] > One cannot attempt to define nonexistence without creating a fallacy > of reification. How does defining nonexistence treat an abstraction like its a real thing? [...] > You are correct, I did commit the fallacy of reification, but there is > no way to avoid it and here is why : If your resoning depends on a fallacy then its invalid. If its invalid then there is nothing to support it being truthful. > You cannot define nonexistence. To define it it to perform a logical > act upon it, and it is immune to logic. There is no reson for it to be immune to logic, all standard logics don't have this restriction. [...] > The presence of paradox implies the reflexivity of the property. > Can you please explain this? [...] > That doesn't answer my question, how does a paradox show that a > property is reflexive? Lets assume that all existential forms are self referential. Then you > have that nonexistence is nonexistent. We just performed (or attempted to perform) a logical act upon the > nonexistent by saying that it does not exist, which is not valid > logically. That sequence leads to the conclusion that your inital assumption was not true. So the logical conculsion you get from this is It is not true that all existential forms are self referential or a little less clunky There exist existential forms that are not self referential. > So, the assumption that all existential forms are self referential > points to this situation with the particular case of nonexistence > where you have paradox. However again this doesn't show that the existance property is reflexive. === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 2, 8:42æpm, David Formosa (aka ? the Platypus) > On Aug 2, 4:55æam, David Formosa (aka ? the Platypus) > On Aug 1, 7:16 am, David Formosa (aka ? the Platypus) > [...] > My very liberal usage of the words (1)existence, (2)nonexistence, and > (3) existentially indeterminate, can be thought of or even defined > more rigorously as follows: > Definition of existence (as I am using it) > The sum totality of all things which are said to exist. > By sum totality you mean something equilverlent to class in the > mathmatical sence of the word, if not what is the diffrence? > I'm also wondering how you define exist do abstractions exist? æ > You state that sum totalities can exist, what does this mean for one > to exist? > Well, math is a logical model. An abstraction. And so the usage of the > concept of existence will always have two very distinct usages. One is > abstract, and one is physical. Now if something physically exists it means that I can (at least in > principal) physically interact with it. æI can see it, feel it, or > detect via verious pysical means. æWhat does it mean for an > abstraction to exist? Whather abstractions actually exist or not in the mind, I dont know and am not concerned. I consider math and physics to be distinct. What happens in the mind is beyond the scope of my claims. > I think that my usage applies to both the physical and the abstract, > and so I usually do not bother to draw the distinction. In order to communicate clearly and allow us to reason about this > without resulting in error we have to be careful about this > distinction. [...] Agreed. There is a distinction, but my usage is intended to apply to both physical and abstract cases equally well. If you have points which are existentially indeterminate, then you can assign probabilities to their existence quite easily. You can explain the bending of physical space pretty easy that way. > As far as physical existence of abstractions goes, I have no idea, > that happens in the mind somehow and Im not really concerned with > whether an abstract mathematical object has some aspect which makes it > exist physically... completely different issue and Im not really > concerned with it. The problem is that you are making claims that implicitly assumes > this. æFor example you claim that The sum totality of objects that > exist exists however this makes a claim about an abstraction (the sum > totality of objects that exist) and its existance. æNow if you had > said something on the lines of Every object that exists, exists > this would by tautological. You could consider these objects individually, or collectively. I dont think it makes a difference until you start talking about thinks like Russel's paradox. But that is correct, regarding existent things I have been referring to the totality of all existent things. Technically, it makes no sense to speak of the totality of nonexistent things because there is no such totality, I only do this for convenience. > For your last question, all I can do is explain what I'm seeing. I > believe that physical existence is bounded by extreme scales. > Everything which exists relative to us is within about +/- 40 orders > of magnitude. The boundaries are where you find existential > indeterminacy. This is the precise place where math and physics must > part and go their separate ways, because mathematics has no > Plancklength. To have such a thing as a Plancklength in mathematics, > you have to construct it deliberately. > So, if the universe were an Oreo cookie, the crispy chocolate outer > cracker would be existentially indeterminate, and existence would be > defined as the sugary cream filling. Yum. [...] > If something is not included in the sum totality of things that do not > exist does that mean it exists? > Can a sum totality that does not exist include members? > Why are they not considered classes? > (i) æYou cannot consider them to be distinct classes because that > would constitute performing a logical or mathematical procedure on the > nonexistent. There is nothing in the rules of mathmatics that prevents it from > dealing with objects that do not exist. There are many situations where something can be shown to not exist. This is quite different than performing logical operations on the nonexistent. In fact, the only thing that allows logic to function properly at all is that things are existing nicely and so everything behaves like a perfect Newtonian Clock. Of course math does deal with the nonexistent, but only at a distance. You cannot make rules and expect nonexistent objects to obey them in accordance to any logic, nonexistent objects are nonsensical. That which is existentially indeterminate is somewhere in between. You dont know if it is neccesarily logical, or nonsensical. This is indeterminate in the world of existential indeterminacy. And that it because when you make rules and logical statements, you might be talking about the existent or the nonexistent. You dont know which, and cant. And dont need to. But you do sacrifice the ability to ever prove anything. > If you consider all of the nonexistent cubes, and you > have that some are red and some are blue, you cannot divide them into > separate classes according to color because that is a mathematical > act. That would be the equivalent of dividing by zero. One can constract a mathmatical system that permits the division by > zero. [...] > But to reword things, > [1] The sum totality of all things which are said to exist, exists. > Ok this implies that abstractions do exist. > It might, but I really dont intend for that to be the case. Can you construct your statement in a way that doesn't carry this > implication. æIndeed can you write your sentence in a way that > unambiguously convays your meaning preferably in a way that makes use > of commonly shared definitions. What I really need to do is study some predicate calculus and try to formalize my babble symbolically. Well, I know Meinong was a quack, Russell kicked his ass gloriously and deservedly. Whether abstractions exist or not, I wont attempt to answer that. But I do think that the thesis makes sense physically, and so there must be some way to make sense of it mathematically. While I think that this approach brings math and physics much closer to each other, they will always be distinct unless one is talking about what happens in the conscious mind. I would never make assertions regarding the physical nature of abstractions. > Because > the brain is made of molecules, it is very likely that the abstract > and the physical do intersect somehow, The map is not the landscape. Agreed completely - well stated. > [...] > [2] The sum totality of all things which are said to be nonexistent, > clearly do not exist. > Its not at all clear to me. æNot clearly knowing what you mean by sum > totality nor what you mean by a sum totality existing or not > existing. æIt is my view that using the verb exist to discuss abstract > objects is an example of fallacy of reifiction and doesn't lead to > usefull resoning. > Well, this definition is a bit paradoxical for the same reason as (i) > above. There is no sum totality of things nonexistent, for the > reason explained in (i). You still haven't told me what a sum totality is. æHow can we have > meaningful discussions about anything with out knowing what the words > we are using mean? Ive been applying induction to existent things to derive a totality of all thing existent. It is handy to also apply this to the nonexistent for purposes of illustrating a more important point. But, clearly, one cannot perform induction on the nonexistent to obtain such a sum totality, and so yes I took a shortcut. > One cannot attempt to define nonexistence without creating a fallacy > of reification. How does defining nonexistence treat an abstraction like its a real > thing? [...] > You are correct, I did commit the fallacy of reification, but there is > no way to avoid it and here is why : If your resoning depends on a fallacy then its invalid. æIf its > invalid then there is nothing to support it being truthful. Im not sure that I committed the fallacy, we might be talking about 2 different reifications. I think that you're usage is in the sense of the fallacy that Meinong made, and I would certainly plead innocent to that. > You cannot define nonexistence. To define it it to perform a logical > act upon it, and it is immune to logic. There is no reson for it to be immune to logic, all standard logics > don't have this restriction. So then I should be able to calculate the area of a round square ? > [...] > The presence of paradox implies the reflexivity of the property. > Can you please explain this? [...] > That doesn't answer my question, how does a paradox show that a > property is reflexive? > Lets assume that all existential forms are self referential. Then you > have that nonexistence is nonexistent. > We just performed (or attempted to perform) a logical act upon the > nonexistent by saying that it does not exist, which is not valid > logically. That sequence leads to the conclusion that your inital assumption was > not true. æSo the logical conculsion you get from this is It is not > true that all existential forms are self referential or a little less > clunky There exist existential forms that are not self referential. Disagree. > So, the assumption that all existential forms are self referential > points to this situation with the particular case of nonexistence > where you have paradox. However again this doesn't show that the existance property is > reflexive.- Hide quoted text - It does indeed. === Subject: Re: New math. Dont read this. > On Aug 2, 8:42æpm, David Formosa (aka ? the Platypus) [...] > Now if something physically exists it means that I can (at least in > principal) physically interact with it. æI can see it, feel it, or > detect via verious pysical means. æWhat does it mean for an > abstraction to exist? Whather abstractions actually exist or not in the mind, I dont know > and am not concerned. I consider math and physics to be distinct. What > happens in the mind is beyond the scope of my claims. The problem is that you keep making claims that carry the implication that such abstractions exists. [...] > In order to communicate clearly and allow us to reason about this > without resulting in error we have to be careful about this > distinction. > [...] Agreed. There is a distinction, but my usage is intended to apply to > both physical and abstract cases equally well. Now I'm compleately confused. In the responce just above here you say Whather asbtractions actually exist or not in the mind, I don't know and am not concerned then you claim that your usage of the word exists is intended to apply both to physical and abstract cases. [...] > The problem is that you are making claims that implicitly assumes > this. æFor example you claim that The sum totality of objects that > exist exists however this makes a claim about an abstraction (the sum > totality of objects that exist) and its existance. æNow if you had > said something on the lines of Every object that exists, exists > this would by tautological. You could consider these objects individually, or collectively. I dont > think it makes a difference until you start talking about thinks like > Russel's paradox. Is the sum totality of objects that ... a collection of some sort or a way of saying All objects that have property ... have property .... This is not some technical distinction but a core and fundermental key to understanding what you are writing about. Unless you can make this clear your words can't convay meaning. [...] > There is nothing in the rules of mathmatics that prevents it from > dealing with objects that do not exist. There are many situations where something can be shown to not exist. > This is quite different than performing logical operations on the > nonexistent. Sure you can. I'll given an example. In the theory of computation there exists a device called a halting oracle. However if the strong turing-church therom holds (and I beleave that it does) it isn't possable for a halting oracle to exist. However nothing prevents us from resoning about the behavour of halting oricals and discussing the results of there existance. [...] > Of course math does deal with the nonexistent, but only at a distance. > You cannot make rules and expect nonexistent objects to obey them in > accordance to any logic, nonexistent objects are nonsensical. Yes we can and we do. Even if the objects contain logical inconsitancies we can make use of a paraconsitant logic. [...] > Can you construct your statement in a way that doesn't carry this > implication. æIndeed can you write your sentence in a way that > unambiguously convays your meaning preferably in a way that makes use > of commonly shared definitions. What I really need to do is study some predicate calculus and try to > formalize my babble symbolically. Please do. Formalizing your reasoning is a great way to find loop holes and overlooked fallacies. > Well, I know Meinong was a quack, Russell kicked his ass gloriously > and deservedly. Russell had quite a bit of respect for Meinong. [...] > You still haven't told me what a sum totality is. æHow can we have > meaningful discussions about anything with out knowing what the words > we are using mean? Ive been applying induction to existent things to derive a totality of > all thing existent. I'm sorry you still haven't defined totality or sum totality in a way I can understand. Nor are you using the word induction in a mannor that is standard in mathmatics. Please step back and start defining your terms. [...] > You cannot define nonexistence. To define it it to perform a logical > act upon it, and it is immune to logic. > There is no reson for it to be immune to logic, all standard logics > don't have this restriction. So then I should be able to calculate the area of a round square ? Calculating the area of a round square isn't a logical operation. Logical operations involve words like For every, For all,Not, And, Or and Predicate. [...] > Lets assume that all existential forms are self referential. Then you > have that nonexistence is nonexistent. > We just performed (or attempted to perform) a logical act upon the > nonexistent by saying that it does not exist, which is not valid > logically. > That sequence leads to the conclusion that your inital assumption was > not true. æSo the logical conculsion you get from this is It is not > true that all existential forms are self referential or a little less > clunky There exist existential forms that are not self referential. Disagree. I'm sorry but what you showed me there was a pritty standard reductio ad abserdum (spelling?). This takes the form of Assume X Show that X leads to an absurdity. This proves not(X) is true. > However again this doesn't show that the existance property is > reflexive. It does indeed. What meaning are you using for the word reflexive in this context? === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Aug 3, 5:55æam, David Formosa (aka ? the Platypus) > On Aug 2, 8:42æpm, David Formosa (aka ? the Platypus) [...] > Now if something physically exists it means that I can (at least in > principal) physically interact with it. æI can see it, feel it, or > detect via verious pysical means. æWhat does it mean for an > abstraction to exist? > Whather abstractions actually exist or not in the mind, I dont know > and am not concerned. I consider math and physics to be distinct. What > happens in the mind is beyond the scope of my claims. The problem is that you keep making claims that carry the implication > that such abstractions exists. Existence in mathematics is abstract. Existence in physics is physical reality. Mathematical existence models the physical variety. And my thesis is quite simply that you can refine the model to a great extent by employing existential indeterminacy which is an abstract kind of existence where one assigns probabilities of existence to abstract objects such as points, areas, volumes, whatever. In no way does this imply or even hint at the idea that abstractions exist, or that abstractions are physically real. It is merely an expansion of the concept of existence, which is abstract. And the utility of this method can be demonstrated in a physics lab. It would be pretty funny if physics actually surpassed mathematics due to the limitations of our imagination. Math is usually accused of creating frivolous knowledge, it amazes me that the reverse is true in this case. > [...] > In order to communicate clearly and allow us to reason about this > without resulting in error we have to be careful about this > distinction. > [...] > Agreed. There is a distinction, but my usage is intended to apply to > both physical and abstract cases equally well. Now I'm compleately confused. æIn the responce just above here you say > Whather asbtractions actually exist or not in the mind, I don't know > and am not concerned then you claim that your usage of the word > exists is intended to apply both to physical and abstract cases. I do not confuse abstractions with physically real objects. What I am doing creating an abstract model which models physics more closely. It is a better model, and the model is still abstract. There should be no confusion on that. > [...] > The problem is that you are making claims that implicitly assumes > this. æFor example you claim that The sum totality of objects that > exist exists however this makes a claim about an abstraction (the sum > totality of objects that exist) and its existance. æNow if you had > said something on the lines of Every object that exists, exists > this would by tautological. > You could consider these objects individually, or collectively. I dont > think it makes a difference until you start talking about thinks like > Russel's paradox. Is the sum totality of objects that ... a collection of some sort or > a way of saying All objects that have property ... have property > .... æThis is not some technical distinction but a core and > fundermental key to understanding what you are writing about. æUnless > you can make this clear your words can't convay meaning. Yes. And there are two cases, abstract and physical. To establish the negation of your rebuttal regarding reification and restate properly : All [physical] objects [that exist] that have property [they exist]. All [abstract] objects [that exist] that have property [they exist]. All [physical] objects [that ~exist] that have property [they ~exist]. All [abstract] objects [that ~exist] that have property [they ~exist]. All [physical] objects [that exist probabilistically] that have property [they exist probabilistically]. All [abstract] objects [that exist probabilistically] that have property [they exist probabilistically]. > There is nothing in the rules of mathmatics that prevents it from > dealing with objects that do not exist. > There are many situations where something can be shown to not exist. > This is quite different than performing logical operations on the > nonexistent. Sure you can. æI'll given an example. æIn the theory of computation > there exists a device called a halting oracle. æHowever if the strong > turing-church therom holds (and I beleave that it does) it isn't > possable for a halting oracle to exist. æHowever nothing prevents us > from resoning about the behavour of halting oricals and discussing the > results of there existance. [...] > Of course math does deal with the nonexistent, but only at a distance. > You cannot make rules and expect nonexistent objects to obey them in > accordance to any logic, nonexistent objects are nonsensical. Yes we can and we do. æEven if the objects contain logical > inconsitancies we can make use of a paraconsitant logic. [...] > Can you construct your statement in a way that doesn't carry this > implication. æIndeed can you write your sentence in a way that > unambiguously convays your meaning preferably in a way that makes use > of commonly shared definitions. > What I really need to do is study some predicate calculus and try to > formalize my babble symbolically. Please do. æFormalizing your reasoning is a great way to find loop > holes and overlooked fallacies. > Well, I know Meinong was a quack, Russell kicked his ass gloriously > and deservedly. Russell had quite a bit of respect for Meinong. Im sure he did, but he disagreed completely with his ideas regarding the existence of abstractions. > [...] > You still haven't told me what a sum totality is. æHow can we have > meaningful discussions about anything with out knowing what the words > we are using mean? > Ive been applying induction to existent things to derive a totality of > all thing existent. I'm sorry you still haven't defined totality or sum totality in a > way I can understand. æNor are you using the word induction in a > mannor that is standard in mathmatics. æPlease step back and start > defining your terms. [...] > You cannot define nonexistence. To define it it to perform a logical > act upon it, and it is immune to logic. > There is no reson for it to be immune to logic, all standard logics > don't have this restriction. > So then I should be able to calculate the area of a round square ? Calculating the area of a round square isn't a logical operation. > Logical operations involve words like For every, For all,Not, > And, Or and Predicate. But you just said earlier that I could do math on the nonexistent ? If There is no reson for it to be immune to logic,..., then I should be able to perform the operation. > [...] > Lets assume that all existential forms are self referential. Then you > have that nonexistence is nonexistent. > We just performed (or attempted to perform) a logical act upon the > nonexistent by saying that it does not exist, which is not valid > logically. > That sequence leads to the conclusion that your inital assumption was > not true. æSo the logical conculsion you get from this is It is not > true that all existential forms are self referential or a little less > clunky There exist existential forms that are not self referential. > Disagree. I'm sorry but what you showed me there was a pritty standard reductio > ad abserdum (spelling?). æThis takes the form of Assume X > Show that X leads to an absurdity. > This proves not(X) is true. Reducto ad absurdum is fine in a mathematical or logical system where you have that things either exist or not. If your system is built on just these two choices, then yes, reducto ad absurdum would falsify the premise. But if you have existential indeterminacy, then reducto ad absurdum is actually what you want to achieve. I am not pulling your leg. If I can prove that a premise is both true and false, then in my scheme this does NOT invalidate the premise, but rather proves the presence if indeterminacy. This method is not possible or sensible without existential indeterminacy. === Subject: Re: New math. Dont read this. > On Aug 3, 5:55æam, David Formosa (aka ? the Platypus) [...] > Yes. And there are two cases, abstract and physical. To establish the > negation of your rebuttal regarding reification and restate properly : All [physical] objects [that exist] that have property [they exist]. > All [abstract] objects [that exist] that have property [they exist]. All [physical] objects [that ~exist] that have property [they > ~exist]. > All [abstract] objects [that ~exist] that have property [they > ~exist]. All [physical] objects [that exist probabilistically] that have > property [they exist probabilistically]. > All [abstract] objects [that exist probabilistically] that have > property [they exist probabilistically]. Now we have got to a point where your terms are written in a clear unambiguous way I can say that I aggry that the above are true. [...] > There is no reson for it to be immune to logic, all standard logics > don't have this restriction. > So then I should be able to calculate the area of a round square ? > Calculating the area of a round square isn't a logical operation. > Logical operations involve words like For every, For all,Not, > And, Or and Predicate. But you just said earlier that I could do math on the nonexistent ? Yes but not all forms of maths, logic one can do with nonexistent things, as for the area of a round square its undefined. > If There is no reson for it to be immune to logic,..., then I should > be able to perform the operation. But you didn't ask me to perform an act in the part of maths that is called logic. [...] > I'm sorry but what you showed me there was a pritty standard reductio > ad abserdum (spelling?). æThis takes the form of > Assume X > Show that X leads to an absurdity. > This proves not(X) is true. Reducto ad absurdum is fine in a mathematical or logical system where > you have that things either exist or not. If your system is built on > just these two choices, then yes, reducto ad absurdum would falsify > the premise. So what you are suggesting is some sort of tristate logic? Or some form of constructivism? > But if you have existential indeterminacy, then reducto ad absurdum is > actually what you want to achieve. I am not pulling your leg. If I can > prove that a premise is both true and false, then in my scheme this > does NOT invalidate the premise, but rather proves the presence if > indeterminacy. This method is not possible or sensible without > existential indeterminacy. Have you read up on probility logic and Bayesian reasoning? === Subject: Re: New math. Dont read this. posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > So what you are suggesting is some sort of tristate logic? æOr some > form of constructivism? That is what would be implied, yes. But it's very strange. Not the customary kind of logic, and I dont know much about that aspect oy my approach. I'm definately no logician. But if you start from the point of view of those who invented random variables 80 years ago, just try to put yourself in their shoes. They were trying to avoid certain pitfalls, and instead of sidestepping those pitfalls I have embraced them and attempted to formalize them. That's all it is. > But if you have existential indeterminacy, then reducto ad absurdum is > actually what you want to achieve. I am not pulling your leg. If I can > prove that a premise is both true and false, then in my scheme this > does NOT invalidate the premise, but rather proves the presence if > indeterminacy. This method is not possible or sensible without > existential indeterminacy. Have you read up on probility logic and Bayesian reasoning? I know a little but not enough to brag about. First, I dont know if I am even doing math or not because that is just the nature of the approach. If you cant prove anything, if you are holding logic in one hand and nonsense in the other, you cant really call that math. But, the analysis makes sense. So I spend most of my time in this area trying to create a kind of calculus and explain why it would make any sense at all, and not surprisingly I have been able to convince myself that there are many things that seem sensible, and even usable to physics. And I am probably lucky that I dont have a career in math because I feel that my methods actually violate certain mores of the field and I do feel a bit guilty for pursuing voodoo mathematics. And while I am confident in saying that almost every area of known and accepted mathematics would have some counterpart in an existentially indeterminate system of pseudo-math.....I could not possibly hope to explain all of it in those terms. There's a hell of alot that I dont even know, and so I focus my efforts on existential calculus and foundational issues. If these things make sense, then I'd probably start looking at set theory of something. Maybe differential equations or something. === Subject: Re: Relativity and Lorentz transformations > Ian Parker says... >http://ianparker.g3z.com/Relativity/aviation.htmfor an account of > disc aircraft (Completely classical) tells you how they REALLY work. > File not found >http://ianparker.g3z.com/Relativity/hoax.htmTells you the REAL > motivations for The Einstein Hoax > File not found > You fell into their little free website trap, didn't you? > Soon Alex Trebek and Jessie Ventura will be arriving at your house to > have a little talk about how deceptive Venus can appear to be in a dark > sky...particularly during a solar eclipse, when gravity bends its light > waves. ;-) > Pat >No, the error was in case. The Website is case sensitive. Let us >however start discusing content. Okay, the content seems to be that you are either creating a > parody website or are making fun of an actual website. In either > case, you seem to be pointing out the lunacy of anti-relativity > cranks. To take one line: > Dr. Einstein's Special Relativity (which is easily seen to be a mathematical > copy of the earlier Lorentz Transformations Aether Theory) to propagate the idea > that our physical reality was too subtle to be understood by mortal men and > could only be dealt with using sophisticated mathematics. The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. Another way of looking at it is that the Lorentz transformations do not have to be derived at all. Assume the principle of relativity for inertial frames (what ever they are :). Signaling is with electromangnetic fields. Electormagnetism is governed by experimentally determined laws, such as Faraday's law. The experimentally determined laws can be formulated as a set of partial differential equations. The PDEs imply a wave equation in which the wave disturbance propogates with constant speed, independently of relative, unaccelerated motion. The speed is experimentally derivable from the behavior of magnetic and electric fields. The transformation group for the PDEs is the Lorentz group. Therefore measurements transform by the Lorentz group from frame to frame. A frame is one where the PDEs of electromagnetism govern the behavior of matter. The structure of matter is governed by the PDEs, therefore matter is governed by the Lorentz group. -- Michael Press === Subject: Re: Relativity and Lorentz transformations posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs > Ian Parker says... >http://ianparker.g3z.com/Relativity/aviation.htmfor an account of > disc aircraft (Completely classical) tells you how they REALLY work. > File not found >http://ianparker.g3z.com/Relativity/hoax.htmTells you the REAL > motivations for The Einstein Hoax > File not found > You fell into their little free website trap, didn't you? > Soon Alex Trebek and Jessie Ventura will be arriving at your house to > have a little talk about how deceptive Venus can appear to be in a dark > sky...particularly during a solar eclipse, when gravity bends its light > waves. ;-) > Pat >No, the error was in case. The Website is case sensitive. Let us >however start discusing content. > Okay, the content seems to be that you are either creating a > parody website or are making fun of an actual website. In either > case, you seem to be pointing out the lunacy of anti-relativity > cranks. > To take one line: > Dr. Einstein's Special Relativity (which is easily seen to be a mathematical > copy of the earlier Lorentz Transformations Aether Theory) to propagate the idea > that our physical reality was too subtle to be understood by mortal men and > could only be dealt with using sophisticated mathematics. > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. Another way of looking at it is that the Lorentz > transformations do not have to be derived at all. Assume > the principle of relativity for inertial frames (what > ever they are :). Signaling is with electromangnetic > fields. Electormagnetism is governed by experimentally > determined laws, such as Faraday's law. < http://en.wikipedia.org/wiki/Faraday_paradox See also: Multiple integral Lorentz force Retarded potentials Sue... > Michael Press === Subject: Re: Relativity and Lorentz transformations > Ian Parker says... >http://ianparker.g3z.com/Relativity/aviation.htmfor an account of > disc aircraft (Completely classical) tells you how they REALLY work. > File not found >http://ianparker.g3z.com/Relativity/hoax.htmTells you the REAL > motivations for The Einstein Hoax > File not found > You fell into their little free website trap, didn't you? > Soon Alex Trebek and Jessie Ventura will be arriving at your house to > have a little talk about how deceptive Venus can appear to be in a dark > sky...particularly during a solar eclipse, when gravity bends its light > waves. ;-) > Pat >No, the error was in case. The Website is case sensitive. Let us >however start discusing content. > Okay, the content seems to be that you are either creating a > parody website or are making fun of an actual website. In either > case, you seem to be pointing out the lunacy of anti-relativity > cranks. > To take one line: > Dr. Einstein's Special Relativity (which is easily seen to be a mathematical > copy of the earlier Lorentz Transformations Aether Theory) to propagate the idea > that our physical reality was too subtle to be understood by mortal men and > could only be dealt with using sophisticated mathematics. > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. > Another way of looking at it is that the Lorentz > transformations do not have to be derived at all. Assume > the principle of relativity for inertial frames (what > ever they are :). Signaling is with electromangnetic > fields. Electormagnetism is governed by experimentally > determined laws, such as Faraday's law. < examples, where Faraday's law does not work. A rectangle of photoconductive material slides along > a pair of wires. At a fixed location a strong light > and a strong magnetic field create a narrow > immovable strip of conducting material subject > to a Lorentz force. Figure 4 shows a translating rectangle of material > with a narrow conducting strip subject to a > conducting at a fixed location by, for example, > a strong light beam at this location. The magnetic > field also is confined to the same strip. > The Lorentz force drives a current from the top > rail to the bottom rail through this strip, and the > circuit is completed through leads attached to > the top and bottom conducting rails. In this example, > the circuit does not move, and the magnetic flux > through the circuit is not changing, so Faraday's > law suggests no current flows. However, the > Lorentz force law suggests a current does flow. > This example is based upon one devised by > Richard Feynman to illustrate the inapplicability > of Faraday's law of induction to certain situations > (that is, the version of Faraday's law of induction > which relates EMF to magnetic flux, which he > terms the flux rule). Referring to his example, > Feynman said:[3] > http://en.wikipedia.org/wiki/Faraday_paradox See also: Multiple integral > Lorentz force > Retarded potentials Yes, ok. I quoted Faraday's law as a discovery and formulation in electromagnetism. Early formulations had to be sharpened. The essay gives two examples that purport to falsify FL. One fails to properly apply Leibniz's rule for differentiating a definite integral, and one attempts a shell game with physical components of the circuit. curl E(r, t) = @ /@t B(r, t). -- Michael Press === Subject: Re: Relativity and Lorentz transformations posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs Gecko/20071201 Epiphany/2.20 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > Another way of looking at it is that the Lorentz > transformations do not have to be derived at all. Assume > the principle of relativity for inertial frames (what > ever they are :). Signaling is with electromangnetic > fields. Electormagnetism is governed by experimentally > determined laws, such as Faraday's law. > Inapplicability of Faraday's law Figure 4: An example, based on one of Feynman's examples, where Faraday's law does not work. http://en.wikipedia.org/wiki/Faraday_paradox > See also: > Multiple integral > Lorentz force > Retarded potentials Yes, ok. I quoted Faraday's law as a discovery > and formulation in electromagnetism. Early > formulations had to be sharpened. The essay > gives two examples that purport to falsify FL. > One fails to properly apply Leibniz's rule for > differentiating a definite integral, and one > attempts a shell game with physical components > of the circuit. Charges are not moving points but rather superpositioned volumes of space. When that is considered, it becomes apparent that the integral form is requured for a complete expression. http://en.wikipedia.org/wiki/Multiple_integral#Some_practical_applications This puts the imaginary current (reactive) where it belongs, in the near-field, rather that along the entire path as a Lorentz transformation does. Compare: Retarded potentials http://farside.ph.utexas.edu/teaching/em/lectures/node50.html with Maxwell equations (with moved bodies) http://www.wolfram-stanek.de/maxwell_equations.htm#maxwell_classic_extended The implication are profound for resolving the constancy of light speed and the principle of relativity. < http://www.bartleby.com/173/7.html << The key to understanding special relativity is Einstein's relativity principle, which states that: All inertial frames are totally equivalent for the performance of all physical experiments. In other words, it is impossible to perform a physical experiment which differentiates in any fundamental sense between different inertial frames. By definition, Newton's laws of motion take the same form in all inertial frames. Einstein generalized this result in his special theory of relativity by asserting that all laws of physics take the same form in all inertial frames. > http://farside.ph.utexas.edu/teaching/em/lectures/node108.html See also: Covariant formulation of classical electromagnetism http://en.wikipedia.org/wiki/Formulation_of_Maxwell's_equations_in_special_r elativity Also note that the Purcell derivation of magnetism is circular and widely dicredited. http://en.wikipedia.org/wiki/Relativistic_electromagnetism http://physics.weber.edu/schroeder/mrr/MRRtalk.html Things should be made as simple as possible, but not any simpler. --Albert Einstein Sue... > -- > Michael Press === Subject: Re: Relativity and Lorentz transformations says... > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. Another way of looking at it is that the Lorentz >transformations do not have to be derived at all. I think you are missing the point: no matter how much experimental basis you have -- and we have a lot of it -- at some point, you must do derivation to get the Lorentz transformation. >Assume >the principle of relativity for inertial frames (what >ever they are :). Signaling is with electromangnetic >fields. Electormagnetism is governed by experimentally >determined laws, such as Faraday's law. The experimentally >determined laws can be formulated as a set of partial >differential equations. True, but you are leaving out Maxwell's displacement current, which was NOT experimentally derived. See http://farside.ph.utexas.edu/teaching/em/lectures/node46.html >The PDEs imply a wave equation >in which the wave disturbance propogates with constant >speed, independently of relative, unaccelerated motion. Once you have the term for displacement current, yes. >The speed is experimentally derivable from the behavior >of magnetic and electric fields. The transformation group >for the PDEs is the Lorentz group. Yes, but this is why people were skeptical of Maxwell's equations for so uncomfortably long. They realized this contradicted Newtonian physics, which had been around for much longer than Maxwell and his equations. And then when they finally did accept them, they tried to explain it wiht a physical Lorentz contraction, rather than with Relativity. It was only when Einstein made it clear that ALL the then known laws of physics have to be preserved by the transformation group, and that this could be done without sacrificing Newtonian physics for v << c -- it was then people realized the Relativity route was much better than Lorentc contractions and Galilean relativity. >Therefore measurements >transform by the Lorentz group from frame to frame. But this, of course, is exactly what the Relativity deniers will continue to deny. === Subject: Re: Relativity and Lorentz transformations > says... > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. >Another way of looking at it is that the Lorentz >transformations do not have to be derived at all. I think you are missing the point: no matter how much experimental basis you > have -- and we have a lot of it -- at some point, you must do derivation to get > the Lorentz transformation. Assume >the principle of relativity for inertial frames (what >ever they are :). Signaling is with electromangnetic >fields. Electormagnetism is governed by experimentally >determined laws, such as Faraday's law. The experimentally >determined laws can be formulated as a set of partial >differential equations. True, but you are leaving out Maxwell's displacement current, which was NOT > experimentally derived. No, but it is experimentally verifiable, though not easily, and maybe not with nineteenth century apparatuses. curl B = mu_0 j is an attempt to formulate Faraday's findings. Turns out the equation only holds when div j = 0, which is not always. Yes, I left out much. I cited Faraday's law as an example of an experimental law that was folded into a set of PDEs describing electric and magnetic fields. > See > http://farside.ph.utexas.edu/teaching/em/lectures/node46.html Yes, clear. >The PDEs imply a wave equation >in which the wave disturbance propogates with constant >speed, independently of relative, unaccelerated motion. Once you have the term for displacement current, yes. Yes, of course. I was worried you were telling me something I overlooked. Turns out I had not overlooked anything. Is there something wrong with the view that the Lorentz transformation is implicit in the PDEs of EM? That LT does not have to be derived? Yes the LT are derived from assumptions on ideal bodies and signals. Real bodies and signals are governed by EM, and EM is formulated in a system whose transformation group is the Lorentz group. -- Michael Press === Subject: Re: Relativity and Lorentz transformations | Press | > says... | > | > | > The mathematics used in Einstein's derivation of the Lorentz | > transformations involves nothing more than high school algebra. | > being intentionally dishonest. | > | >Another way of looking at it is that the Lorentz | >transformations do not have to be derived at all. | > | > I think you are missing the point: no matter how much experimental basis you | > have -- and we have a lot of it -- at some point, you must do derivation to get | > the Lorentz transformation. | > | >Assume | >the principle of relativity for inertial frames (what | >ever they are :). Signaling is with electromangnetic | >fields. Electormagnetism is governed by experimentally | >determined laws, such as Faraday's law. The experimentally | >determined laws can be formulated as a set of partial | >differential equations. | > | > True, but you are leaving out Maxwell's displacement current, which was NOT | > experimentally derived. | | No, but it is experimentally verifiable, though not easily, | and maybe not with nineteenth century apparatuses. | | curl B = mu_0 j | is an attempt to formulate Faraday's findings. | Turns out the equation only holds when div j = 0, | which is not always. | | Yes, I left out much. I cited Faraday's law as an example of | an experimental law that was folded into a set of PDEs | describing electric and magnetic fields. | | > See | > http://farside.ph.utexas.edu/teaching/em/lectures/node46.html | | Yes, clear. | | >The PDEs imply a wave equation | >in which the wave disturbance propogates with constant | >speed, independently of relative, unaccelerated motion. | > | > Once you have the term for displacement current, yes. | | Yes, of course. I was worried you were telling me something | I overlooked. Turns out I had not overlooked anything. | Is there something wrong with the view that the Lorentz | transformation is implicit in the PDEs of EM? That LT | does not have to be derived? | | Yes the LT are derived from assumptions on ideal bodies | and signals. Real bodies and signals are governed by EM, | and EM is formulated in a system whose transformation | group is the Lorentz group. | | -- | Michael Press What a load of drooling, babbling, ing NONSENSE, you ranting idiot! Why did Einstein say the speed of light from A to B is c-v, the speed of light from B to A is c+v, the time each way is the same? Your answer goes here: ________________________________________________________ Other answers have been: According to Ian Parker: We are not talking about the speed of light here we are talking classical stability theory. -- Idiot Ian Parker. ______________________________________________________ According to cretin harald.vanlintelButNotThis@epfl.ch Easy: he did NOT say that. ______________________________________________________ According to xxein: It is an artefactual/superficially imposed yin-yang of sorts. ______________________________________________________ According to Lamenting Shubert: Why do you want to know? ______________________________________________________ According to Imbecile Jimmy Black: In neither system (meaning frame of reference in modern-day terminology) is the speed of light c-v or c+v. In both systems the speed of light is c. According to the imbecile Jimmy Black, Einstein did not write the equation ______________________________________________________ According to Dork Bruere ______________________________________________________ According to Spirit of Truth: that math is correct but WRONG ______________________________________________________ 'we establish by definition that the time required by light to travel from A to B equals the time it requires to travel from B to A' because I SAY SO and you have to agree because I'm the great genius, STOOOPID, don't you dare question it. -- Rabbi Albert Einstein === Subject: Re: Relativity and Lorentz transformations : Androcles : Why did Einstein say : the speed of light from A to B is c-v, : the speed of light from B to A is c+v, : the time each way is the same? What he said was moves relatively to the initial point of k, when measured in the stationary system, with velocity c-v (etc). Which, of course, is not the same thing as a velocity in coordinate system k. A distinction the paper pages thoroughly clear in the context you snipped away. He also didn't say the time in the stationary system was the same in those two cases; he substituted the values into two different places on the previous page; the values x'/(c-v) and x'/(c+v) are not equal. The times in the moving system are equal, and the UNequal times in the stationary system are used to derive just how stationary and moving coordinates are related. Stationary and moving being, of course, arbitrary labels, as is also made thoroughly clear. Aaaaaand this is the point where you foam at the mouth and call me names and say various things you think are insulting about Einstein and accuse me of trimming your sacred post and so on and on and on. Good luck with that. Wayne Throop throopw@sheol.org http://sheol.org/throopw === Subject: Re: Relativity and Lorentz transformations > says... > The mathematics used in Einstein's derivation of the Lorentz > transformations involves nothing more than high school algebra. > being intentionally dishonest. >Another way of looking at it is that the Lorentz >transformations do not have to be derived at all. I think you are missing the point: no matter how much experimental basis you > have -- and we have a lot of it -- at some point, you must do derivation to get > the Lorentz transformation. Assume >the principle of relativity for inertial frames (what >ever they are :). Signaling is with electromangnetic >fields. Electormagnetism is governed by experimentally >determined laws, such as Faraday's law. The experimentally >determined laws can be formulated as a set of partial >differential equations. True, but you are leaving out Maxwell's displacement current, which was NOT > experimentally derived. See > http://farside.ph.utexas.edu/teaching/em/lectures/node46.html The PDEs imply a wave equation >in which the wave disturbance propogates with constant >speed, independently of relative, unaccelerated motion. Once you have the term for displacement current, yes. The speed is experimentally derivable from the behavior >of magnetic and electric fields. The transformation group >for the PDEs is the Lorentz group. Yes, but this is why people were skeptical of Maxwell's equations for so > uncomfortably long. They realized this contradicted Newtonian physics, which had > been around for much longer than Maxwell and his equations. And then when they > finally did accept them, they tried to explain it wiht a physical Lorentz > contraction, rather than with Relativity. Do you take the Lorentz transformation to be a convenient bookkeeping tool? I used to take it that way, but eventually adopted the view of Larmor, Lorentz, Einstein, and Poincare'. Insofar as physics deals with the physical, then bodies in paper detailing these matters. How to teach special relativity. Progress in Scientific Culture, Vol 1, No 2, summer 1976. Reprinted in Speakable and unspeakable in quantum mechanics, 1987. > It was only when Einstein made it clear that ALL the then known laws of physics > have to be preserved by the transformation group, and that this could be done > without sacrificing Newtonian physics for v << c -- it was then people realized > the Relativity route was much better than Lorentc contractions and Galilean > relativity. Therefore measurements >transform by the Lorentz group from frame to frame. But this, of course, is exactly what the Relativity deniers will continue to > deny. They have their own frame for reference. Happily, the myriad of laws they labour under do not all apply to me. -- Michael Press === Subject: Re: Relativity and Lorentz transformations says... > True, but you are leaving out Maxwell's displacement current, which was NOT > experimentally derived. See > http://farside.ph.utexas.edu/teaching/em/lectures/node46.html You are welcome. I see the author of those lecture notes also used one of my favorite E&M text, Griffiths, which describes the invention as follows: The problem is in the right side of equation (7.31) [divergence of Ampere'e Law], which should be zero, but isn't. Applying the continuity equation (5.25) and Gauss's law, the offending term can be rewritten: divJ = -dRho/dt = -d(esp[0]divE) =-div(eps[0]dE/dt) [In typing this in, I am using d for the 'round d' that indicates partial differentiation that appears in the text; I also replaced nabla. and nabla-cross with div] It might occur to you that if we were to add the quantity eps[0](dE/dt) to J, in Ampere's law, it would be just right to kill off the extra divergence: curlB = Mu[0]J + mu[0]eps[0]dE/dt. (Maxwell himself had other reasons for wanting to add this quanitty to Ampere's law. To him the rescue of the continuity equation was a happy divident rather than a primary motive. But today we regognize this argument as a far more compellingone than Maxwell's, which was based on a now-discredited model of the ether). p274 Introduction to Electrodynamics, Griffiths, David; Prentice Hall 1981.] >Do you take the Lorentz transformation to be a convenient >bookkeeping tool? Oh, no. I take it as much more than that. It is the group under which the laws of physics are invariants. You see, I take the invention of relativity as a vindication of Hilbert's Erlangen program, the grand idea of founding all the kinds of geometry on the transformation groups, their representations, and especially, their invariants. This is the mathematical idea that lies behind discovering all the symmetries of nature that show up in physics. It does not, however, explain symmetry breaking. > But this, of course, is exactly what the Relativity deniers will continue to > deny. They have their own frame for reference. Happily, the myriad of >laws they labour under do not all apply to me. That is a relief to hear;) === === Subject: A puzzling issue: object with 8 degrees of freedom posting-account=PIdpdAoAAABmOZotWTpUEsX0KIi_Gc24 rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) A friend and I are having a bet. He states that there must be objects or mechanisms with 8 degrees of freedom (not counting translation} which have 3-fold symmetry (at least in some configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners whose angles are not fixed. But such a cubus has - three orientational degrees of freedom - three internal angles which makes a total of only 6 degrees of freedom. A cubus has 3fold symmetry when seen along a diagonal, so that would fit; but 6 are not 8 degrees of freedom. I brought up the idea of a tetrahedral skeleton, (like a methane molecule http://en.wikipedia.org/wiki/Methane ) . It has 8 degrees of freedom, it has 3fold symmetry in some configurations, but we do not see a way to build that in metal or rubber without having more or less than 8 degrees of freedom. On the other hand, I am not able to prove that the puzzle is impossible to solve. Is there another solution? Where can one look for such objects or related theorems? Are there books or sites on these issues? John === Subject: Re: A puzzling issue: object with 8 degrees of freedom > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus .... geometry and robotics. Ken Pledger. === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=PIdpdAoAAABmOZotWTpUEsX0KIi_Gc24 rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John Many answers! Of course a rigid body has only six degrees of freedom. That is why we are thinking about deformable objects or mechanisms. Are there any canonical lists of such mechanisms? We are looking for one with threefold symmetry and 8 degrees of freedom in total. John === Subject: Re: A puzzling issue: object with 8 degrees of freedom >...we are thinking about >deformable objects or mechanisms. >..... >We are looking for one with threefold symmetry and >8 degrees of freedom in total. John Will this one suit your purpose? On the axis of trifold symmetry, a long finger (1) with a central ball joint permitting the one half (1a) to rotate in the axis of symmetry only, At the end of this member, three fingers (2,3,4) attached to it, with pin joints, so they can each rotate in just one plane. At the tip of each of these three members (2,3,4) , a finger joined to each with a pin joint also permitting just one axis of rotation, (labeled 2a,3a,4a). This appears to provide the trifold symmetry you want, in that the mechanism can rotate on the axis of finger (1) and in the colinear axis of finger (1a). Each of three fingers can sweep an angle about the axis of finger (1a) and each of three finger tips (2a,3a,4a) can also sweep an angle with respect to the finger to which they connect. This is only one of numerous way to provide this specification, it seems. Brian W === Subject: Re: A puzzling issue: object with 8 degrees of freedom <117c949m75619f1nl1gnupn9gstmcon4rs@4ax.com> posting-account=PIdpdAoAAABmOZotWTpUEsX0KIi_Gc24 rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) >...we are thinking about >deformable objects or mechanisms. >..... >We are looking for one with threefold symmetry and >8 degrees of freedom in total. >John Will this one suit your purpose? On the axis of trifold symmetry, a long finger (1) with a central > ball joint permitting the one half (1a) to rotate in the axis of > symmetry only, At the end of this member, three fingers (2,3,4) attached to it, with > pin joints, so they can each rotate in just one plane. At the tip of each of these three members (2,3,4) , a finger joined to > each with a pin joint also permitting just one axis of rotation, > (labeled 2a,3a,4a). > This appears to provide the trifold symmetry you want, in that the > mechanism can rotate on the axis of finger (1) and in the > colinear axis of finger (1a). > Each of three fingers can sweep an angle about the axis of finger (1a) > and each of three finger tips (2a,3a,4a) can also sweep an angle > with respect to the finger to which they connect. This is only one of numerous way to provide this specification, it > seems. Brian W Brian, thank you for the proposal. (It almost looks as if it had 4fold symmetry - or am I wrong?) You also mention numerous ways to do this. Can you give a few more? In any case, thank you very much! John === Subject: HANSON! See this one! | | >...we are thinking about | >deformable objects or mechanisms. | >..... | >We are looking for one with threefold symmetry and | >8 degrees of freedom in total. | > | >John | > | > | | Will this one suit your purpose? | | On the axis of trifold symmetry, a long finger (1) with a central | ball joint permitting the one half (1a) to rotate in the axis of | symmetry only, | | At the end of this member, three fingers (2,3,4) attached to it, with | pin joints, so they can each rotate in just one plane. | | At the tip of each of these three members (2,3,4) , a finger joined to | each with a pin joint also permitting just one axis of rotation, | (labeled 2a,3a,4a). | This appears to provide the trifold symmetry you want, in that the | mechanism can rotate on the axis of finger (1) and in the | colinear axis of finger (1a). | Each of three fingers can sweep an angle about the axis of finger (1a) | and each of three finger tips (2a,3a,4a) can also sweep an angle | with respect to the finger to which they connect. | | This is only one of numerous way to provide this specification, it | seems. | | Brian W HAHAHA! http://www.insanesoccer.com/games/files/thefinger.jpg I love it! === Subject: Re: HANSON! See this one! | > | | > | >...we are thinking about deformable objects | > | >or mechanisms. ..... [snipped]..... | > | >We are looking for one with threefold symmetry | > | >and 8 degrees of freedom in total. | > | >John | > | > | > | > | > | Will this one suit your purpose? | > | On the axis of trifold symmetry, a long finger (1) with a central | > | ball joint permitting the one half (1a) to rotate in the axis of | > | symmetry only, | > | At the end of this member, three fingers (2,3,4) attached to it, with | > | pin joints, so they can each rotate in just one plane. | > | | > | At the tip of each of these three members (2,3,4) , a finger joined to | > | each with a pin joint also permitting just one axis of rotation, | > | (labeled 2a,3a,4a). | > | This appears to provide the trifold symmetry you want, in that the | > | mechanism can rotate on the axis of finger (1) and in the | > | colinear axis of finger (1a). | > | Each of three fingers can sweep an angle about the axis of finger (1a) | > | and each of three finger tips (2a,3a,4a) can also sweep an angle | > | with respect to the finger to which they connect. | > | This is only one of numerous way to provide this specification, it | > | seems. | > | Brian W | > | > HAHAHA! | > http://www.insanesoccer.com/games/files/thefinger.jpg | > I love it! | > | ... ahahahaha... Yeah, to you their exchange may sound | funny... ahahahahaha... but John and Brian are simply | having a standard machine shop operator talk. That | is their line and their language. Why they posted that | into sci.physics & sci math that is the funny part.... ahaha... | Both of them are probably laughing louder then you do... | Brian W is definitely having a laugh, John Stanton may have his head up his arse. === Subject: Re: HANSON! See this one! [cut] ahahaha... Is somebody kissing Hanson's ass by any chance? Yep. Once an ass kisser, always an ass kisser. ahahaha... AHAHAHA... ahahaha... Louis Savain Rebel Science News: http://rebelscience.blogspot.com/ === Subject: Re: HANSON! See this one! > Rebel Science News: > http://rebelscience.blogspot.com/ >Shhh... Louis... shhhh ... ahahahaha... >When fancy strikes I will tell you why >your thought after grav .& charge field >gradient velocity is not instantaneous >(for that would lead to infinite regression) >but may turn out to be (on first principles) >a whopping 6*10^71 cm/sec..... which >for practical purposes is instantaneous >and it quantifies Mach's Principle... ahahaha.. >Talk to you later after all the aggrieved Einstein >Dingleberries, in the other thread that you >have started, will have said their prayers... >ahahahaha... ahahahanson Well, for lack of a better word, 'instantaneous' is mostly a manner of speaking since nothing is instantaneous in a universe ruled by cause and effect. Cause must always precede effect by a fundamental interval. What I really want to say is that, as far as gravity and electrostatic fields are concerned, the time between cause and effect does not depend on distance. It's non-local. It's a very short time, though, possibly on the order of Planck time. It's hard to prove experimentally since our instruments cannot measure intervals at that minute scale. However, it should be possible to prove that distance does not affect the measured interval. I'll wait to hear what you have to say about infinite regress. Louis Savain Rebel Science News: http://rebelscience.blogspot.com/ === Subject: Re: A puzzling issue: object with 8 degrees of freedom > Will this one suit your purpose? On the axis of trifold symmetry, a long finger (1) with a central > ball joint permitting the one half (1a) to rotate in the axis of > symmetry only, At the end of this member, three fingers (2,3,4) attached to it, with > pin joints, so they can each rotate in just one plane. At the tip of each of these three members (2,3,4) , a finger joined to > each with a pin joint also permitting just one axis of rotation, > (labeled 2a,3a,4a). > This appears to provide the trifold symmetry you want, in that the > mechanism can rotate on the axis of finger (1) and in the > colinear axis of finger (1a). > Each of three fingers can sweep an angle about the axis of finger (1a) > and each of three finger tips (2a,3a,4a) can also sweep an angle > with respect to the finger to which they connect. This is only one of numerous way to provide this specification, it > seems. So as I stated, re-using the degrees already known is how you would play with the term more than six degrees of freedom. Sadly, You are only using the same 6 degrees of freedom of motions more than once. You have a multiple angles of motion in the same 6 degrees occuring in different places only. Don't ever try and engineer the hair on a shaggy dog. :) -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom > Many answers! Of course a rigid body has only six > degrees of freedom. That is why we are thinking about > deformable objects or mechanisms. Are there any canonical lists of such mechanisms? > We are looking for one with threefold symmetry and > 8 degrees of freedom in total. You are not truly finding 2 extra degrees of freedom You are counting a degree more than once. If you really think that creates multiple degrees of freedom Then a porcupines needles must really blow your mind for degrees of freedom. and boy oh boy Don't even try to think about a forest full of trees and millions (and billions) of branches and leaves etc. :) -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom | > A friend and I are having a bet. He states that there must be | > objects or mechanisms with 8 degrees of freedom | > (not counting translation} which | > have 3-fold symmetry (at least in some | > configurations). But we cannot find any. | > | > He is thinking of objects like a deformable cubus with corners | > whose angles are not fixed. But such a cubus has | > - three orientational degrees of freedom | > - three internal angles | > which makes a total of only 6 degrees of freedom. | > A cubus has 3fold symmetry when seen along | > a diagonal, so that would fit; but 6 are not 8 | > degrees of freedom. | > | > I brought up the idea of a tetrahedral skeleton, | > (like a methane molecule http://en.wikipedia.org/wiki/Methane) . | > It has 8 degrees of freedom, | > it has 3fold symmetry in some configurations, | > but we do not see a way to build that in metal | > or rubber without having more or less than 8 degrees | > of freedom. | > | > On the other hand, I am not able to prove | > that the puzzle is impossible to solve. | > | > Is there another solution? Where can one look for such | > objects or related theorems? Are there books or sites | > on these issues? | > | > | > John | | Many answers! Of course a rigid body has only six | degrees of freedom. That is why we are thinking about | deformable objects or mechanisms. | | Are there any canonical lists of such mechanisms? | We are looking for one with threefold symmetry and | 8 degrees of freedom in total. | | John An object deformed is not symmetric about one of its three axes of symmetry unless the deformation is also symmetric; but that merely returns the 6-DOF of the rigid body. You can't have your cake with a bite out of it. === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=1wPXHwkAAACFV0NiGWX7tZb1o0HYkMjT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; PeoplePal 6.6) w:PACBHO60,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John Just to clarify the meaning of a degree of freedom in my thinking. An independent geometric variable. Angle as a side length appear to be the issue and the basic variable where angle appears dependent was all that need be discussed. | | | here is a right angle and it is clearly independent of side length. A constructed geometric as a whole then allows side to cause angle. IN mechanical interpretation a machine can be described as a geometric function. MY right angle machine above has three apparent degrees of freedom. Two sides and the angle. As soon as a machine is a triangle it has three again, but the angles are side length determined and the meaning as machine itself in abstract appear to be a school idea I am trying to understand. A function of machine applies to any design wheather a robot like device or not. A location of machine parts as function allows a complexity design such as a robot to be discussed, but exact question was apparently unclear to me. A thoughtful man would allow all machine whether a simple constant structure to be a fuunctionally defined structure. A robot type of machine appears the issue. A three fold symmetry mean it has three axises and so all machines with threee axis are functionally equivalent. An inverted triangleoizoid;) was used at the National Insitute of Standards and Technology, NIST as a robot arm/platform. Cables varying the side lengths functionally could cause the tip of the volume to be motionable in an exact functional fashion. It translates left and right and up and down and spins in the center in a circle if required. So all that was required was to allow an actual usage to be stated. I thought it was a crystallike common question and not a mechanical engineeering question. Analogy to crystal was a possible reason for the NIST discovery as a class of robot though. If there is a question concerning the usage of crystal design in robots I would entertain them because there are few machines able to be used in crystal analogy! Here is a small novel machine design based on a crystal geometric. A cubic structure has like maybe ten degree of freedom. And to functionally control side length to cause the function meant the solution would be indeterminate! A rather airfcraft like control function would have to be defined for a cubic to be used as a robot and a test loop in control code would have to prevent nonsolution motion. Making the nIST inverted triangleosoid a truely novel discovery. Maechanical design is very interesting and the basic question here is to either talk of the method of machine analogy or not. === Subject: Re: A puzzling issue: object with 8 degrees of freedom > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. > He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. > I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. > On the other hand, I am not able to prove > that the puzzle is impossible to solve. > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John Just to clarify the meaning of a degree of freedom in my thinking. An independent geometric variable. Angle as a side length appear to be the issue and the basic variable > where angle appears dependent was all that need be discussed. _______ here is a right angle and it is clearly independent of side length. A > constructed geometric as a whole then allows side to cause angle. IN mechanical interpretation a machine can be described as a geometric > function. MY right angle machine above has three apparent degrees of > freedom. Two sides and the angle. As soon as a machine is a triangle > it has three again, but the angles are side length determined and the > meaning as machine itself in abstract appear to be a school idea I am > trying to understand. A function of machine applies to any design wheather a robot like > device or not. A location of machine parts as function allows a > complexity design such as a robot to be discussed, but exact question > was apparently unclear to me. A thoughtful man would allow all machine whether a simple constant > structure to be a fuunctionally defined structure. A robot type of machine appears the issue. A three fold symmetry mean > it has three axises and so all machines with threee axis are > functionally equivalent. An inverted triangleoizoid;) was used at the National Insitute of > Standards and Technology, NIST as a robot arm/platform. Cables > varying the side lengths functionally could cause the tip of the > volume to be motionable in an exact functional fashion. It translates > left and right and up and down and spins in the center in a circle if > required. So all that was required was to allow an actual usage to be stated. I > thought it was a crystallike common question and not a mechanical > engineeering question. Analogy to crystal was a possible reason for > the NIST discovery as a class of robot though. If there is a question concerning the usage of crystal design in > robots I would entertain them because there are few machines able to > be used in crystal analogy! Here is a small novel machine design based on a crystal geometric. A cubic structure has like maybe ten degree of freedom. And to > functionally control side length to cause the function meant the > solution would be indeterminate! A rather airfcraft like control > function would have to be defined for a cubic to be used as a robot > and a test loop in control code would have to prevent nonsolution > motion. Making the nIST inverted triangleosoid a truely novel discovery. Maechanical design is very interesting and the basic question here is > to either talk of the method of machine analogy or not. Hi Doug, The cubic structure still does not have more than 6 degrees of freedom, What you are doing is allowing for the degrees to have a multiple of positions or multiple motions within the normal 6 degrees. That is not extra degrees that is still just an extra motion in the normal 6 degrees of freedom. Any point of the object can have no more than 6 different directions (up, down, left, right, forward, backward) of motion in the 3 planes of 3D space they reside it. Motions that combine lets say up and left, are not an extra degree of freedom. They are a combination of 2 or more of the normal degrees of freedom. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom a truss member has two degree of freedom, namely compress and extension which is the axial force at both ends. a frame member has six degree of freedom, namely translation in the x, y axis plus a rotation at each ends. that means 3 degree of freedom at each end. a cubic would would have three degree of freedom on each faces which is 6 times 3 which is 18 degree of freedom. a material is not measure by it's atomic structure but rather the material property of isotropical or anisotropical which is measure by the modulus of elasticity and the possion ratio.....done > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John > Just to clarify the meaning of a degree of freedom in my thinking. > An independent geometric variable. > Angle as a side length appear to be the issue and the basic variable > where angle appears dependent was all that need be discussed. > _______ > here is a right angle and it is clearly independent of side length. A > constructed geometric as a whole then allows side to cause angle. > IN mechanical interpretation a machine can be described as a geometric > function. MY right angle machine above has three apparent degrees of > freedom. Two sides and the angle. As soon as a machine is a triangle > it has three again, but the angles are side length determined and the > meaning as machine itself in abstract appear to be a school idea I am > trying to understand. > A function of machine applies to any design wheather a robot like > device or not. A location of machine parts as function allows a > complexity design such as a robot to be discussed, but exact question > was apparently unclear to me. > A thoughtful man would allow all machine whether a simple constant > structure to be a fuunctionally defined structure. > A robot type of machine appears the issue. A three fold symmetry mean > it has three axises and so all machines with threee axis are > functionally equivalent. > An inverted triangleoizoid;) was used at the National Insitute of > Standards and Technology, NIST as a robot arm/platform. Cables > varying the side lengths functionally could cause the tip of the > volume to be motionable in an exact functional fashion. It translates > left and right and up and down and spins in the center in a circle if > required. > So all that was required was to allow an actual usage to be stated. I > thought it was a crystallike common question and not a mechanical > engineeering question. Analogy to crystal was a possible reason for > the NIST discovery as a class of robot though. > If there is a question concerning the usage of crystal design in > robots I would entertain them because there are few machines able to > be used in crystal analogy! > Here is a small novel machine design based on a crystal geometric. > A cubic structure has like maybe ten degree of freedom. And to > functionally control side length to cause the function meant the > solution would be indeterminate! A rather airfcraft like control > function would have to be defined for a cubic to be used as a robot > and a test loop in control code would have to prevent nonsolution > motion. > Making the nIST inverted triangleosoid a truely novel discovery. > Maechanical design is very interesting and the basic question here is > to either talk of the method of machine analogy or not. Hi Doug, > The cubic structure still does not have more than 6 degrees of freedom, > What you are doing is allowing for the degrees to have > a multiple of positions or multiple motions within the normal 6 degrees. > That is not extra degrees that is still just an extra motion in the > normal > 6 > degrees of freedom. Any point of the object can have no more than 6 different directions > (up, down, left, right, forward, backward) > of motion in the 3 planes of 3D space they reside it. > Motions that combine lets say up and left, are not an extra > degree of freedom. > They are a combination of 2 or more of the normal degrees of freedom. -- > James M Driscoll Jr > Creator of the Clock Malfunction Theory > Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom >a truss member has two degree of freedom, > namely compress and extension which is > the axial force at both ends. A truss member has at least three degrees of freedom: 1) change in length (+/-) 2) bending off the central axis (up/down, left/right, etc., Euler column) 3) torsion around the central axis David A. Smith === Subject: Re: A puzzling issue: object with 8 degrees of freedom | | >a truss member has two degree of freedom, | > namely compress and extension which is | > the axial force at both ends. | | A truss member has at least three degrees of freedom: | 1) change in length (+/-) A hydraulic ram is free to change in length. It is not a ing truss. A truss has no FREEDOM to change its length, you dork. The six degrees of freedom are x,y,z, pitch, roll, yaw. An aircraft is free to move in any of them. 8-DOF is meaningless drivel. Google 6-DOF. === Subject: Re: A puzzling issue: object with 8 degrees of freedom > a truss member has two degree of freedom, > namely compress and extension which is > the axial force at both ends. A truss member has at least three degrees of freedom: > 1) change in length (+/-) Using only 2 degrees of the 6 known. > 2) bending off the central axis (up/down, left/right, etc., Euler > column) Using 4 degrees of the 6 known. > 3) torsion around the central axis Again Using same 4 degrees of (2) of the 6 known. (up down left right motion of points with a variable of motion for each point) Still only 6 degrees of actual freedom total. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom > a truss member has two degree of freedom, > namely compress and extension which is > the axial force at both ends. > A truss member has at least three degrees of freedom: > 1) change in length (+/-) Using only 2 degrees of the 6 known. No, that is a single degree of freedom. > 2) bending off the central axis (up/down, left/right, > etc., Euler column) Using 4 degrees of the 6 known. Only two. > 3) torsion around the central axis Again Using same 4 degrees of (2) > of the 6 known. (up down left right motion > of points with a variable of motion for each > point) No, this is three degrees of freedom. David A. Smith === Subject: Re: A puzzling issue: object with 8 degrees of freedom /// > A truss member has at least three degrees of freedom: > 1) change in length (+/-) > Using only 2 degrees of the 6 known. No, that is a single degree of freedom. /and so on/ >David A. Smith Dave, I see a problem for you; debating with the folks who have strayed onto an engineering group that actually uses the concept of DoF: it's the one called rassling with pigs.... You WILL get muddy! :-) Better to leave them to campout on sci.physics, sci.maths..... Brian W === Subject: Re: A puzzling issue: object with 8 degrees of freedom /// > A truss member has at least three degrees of freedom: > 1) change in length (+/-) Using only 2 degrees of the 6 known. > No, that is a single degree of freedom. > /and so on/ > David A. Smith Dave, > I see a problem for you; debating with the folks who > have strayed onto an engineering group that actually > uses the concept of DoF: > it's the one called rassling with pigs.... You WILL get muddy! :-) Better to leave them to campout on > sci.physics, sci.maths..... Actually physics does not actually use more than 6 degrees of freedom It is only the math heads that play with such sillyness instead of realizing they are just re-using the same known degrees of freedom already. So it would be best to play with such porcupine needled degrees of freedom that increase with the amount of objects and rubberyness in the math group alone. :) -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom > a truss member has two degree of freedom, > namely compress and extension which is > the axial force at both ends. A truss member has at least three degrees of freedom: > 1) change in length (+/-) > Using only 2 degrees of the 6 known. No, that is a single degree of freedom. No, two directions of motion is 2 degrees of freedom in a single plane of motion. > 2) bending off the central axis (up/down, left/right, > etc., Euler column) > Using 4 degrees of the 6 known. Only two. No, again 4 degrees but now in two planes. I think you are confusing planes with degrees. Each plane has 2 directions of freedom. (2 degrees) > 3) torsion around the central axis > Again Using same 4 degrees of (2) > of the 6 known. (up down left right motion > of points with a variable of motion for each > point) No, this is three degrees of freedom. Nope. It is the same as above. It has 2 directions for for any point in one plane and 2 more directions in the other again, 4 degrees of freedom. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom > a truss member has two degree of freedom, namely compress and > extension which is the axial force at both ends. a frame member has > six degree of freedom, namely translation in the x, y axis plus a > rotation at each ends. that means 3 degree of freedom at each end. a > cubic would would have three degree of freedom on each faces which is > 6 times 3 which is 18 degree of freedom. a material is not measure > by it's atomic structure but rather the material property of > isotropical or anisotropical which is measure by the modulus of > elasticity and the possion ratio.....done Again, The rotation at each end you speak of is still just freedom of motion in the same 6 degrees known. and each end also has a compression factor to allow 6 degrees to still exist at each end. You are merely mixing already known degrees of freedom into extra degrees that are not actually there. You are adding already known degrees of freedom as extra degrees. A cube, has 6 degrees of freedom only, Any single point on the cube or inside the cube also only has 6 degrees of freedom. You can combine any of the degrees for different motion in such free 3D space. But it still only moves with 6 degrees of freedom. Just because it has ends does not give it extra degrees by adding the same degrees. Each end can move in the same 6 degrees the other end can move in. There is no addition of degrees occuring. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom A friend and I are having a bet. He states that there must be >objects or mechanisms with 8 degrees of freedom >(not counting translation} which >have 3-fold symmetry (at least in some >configurations). But we cannot find any. /// >John Better not take the other end of the bet. A rotational degree of freedom occurs in one object capable of rotating with respect to another. An object with several rotatable links can provide several degrees of rotary freedom for each successive link with respect to the base attachment. An object like the Manx three-legged object, can rotate three ways at each ankle with respect to its limb. Brian W === Subject: Re: A puzzling issue: object with 8 degrees of freedom A friend and I are having a bet. He states that there > must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus > with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane molecule > http://en.wikipedia.org/wiki/Methane ) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for > such > objects or related theorems? Are there books or sites > on these issues? If I understood you correctly, the degrees of freedom of which you speak are equal to the cardinal points of the dimensionality. A time-dependent 3 dimensional object (6 degrees of freedom in your context) has two more degrees (6 + 2) where time is a simple parameter of reversible direction. This would, of course, beg treatment in the complex plane, which is inherently 2-dimensional. Tom > John > === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze Filter 1.2.0.72; .NET CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; WWTClient2),gzip(gfe),gzip(gfe) > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? An object with movable appendages, such as the human body, has multiple degrees of freedom. Dave === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=1wPXHwkAAACFV0NiGWX7tZb1o0HYkMjT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; PeoplePal 6.6) w:PACBHO60,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John A pentagram has five sides. A vertex angle makes 5 a symmetry. Making one degree of freedom and one symmetry. A line length or side length makes a five degree of freedom change, each side may be independent of the other. Allowing a legnth as a cause to ratio of side to side then making a ratio symmetry. And a mirror of set of sides allows a ratio of areas. Draw a line between vertexes and mirror. Making the third degree symmetry. And two degrees of freedom for there are only two axis? NO there are three axis, making 9 degree of freedom. SO use a square. A square is a cubic and all cubic exhibit this majic property. Gold as a cubic crystal system allows a functional method of set to be developed. 3 symmetries and 8 degrees of freedom allows a functional set to be designed. D(3) Length(4) Mirror ratio(2) Wait the square has only 7 degree of freedom, sorry! I went through several shapes and found this one. * A triangloid with a certain number of sides. It haa No mirror property because the axis appears a side! SO the angle vertex makes a ratio of side length to side length For all equal sides, two vertexs exist. One degree for each. Allowing the dies to equal the rest of the degrees of freedom. And the third mirror symmetry exists only as a NON-degree of freedom effect. === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=1wPXHwkAAACFV0NiGWX7tZb1o0HYkMjT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; PeoplePal 6.6) w:PACBHO60,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. > He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. > I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. > On the other hand, I am not able to prove > that the puzzle is impossible to solve. > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John > A pentagram has five sides. æ [snip crap] Uncle Al counts 10 external sides. æIdiot. http://www.electricwitch.com/pentagram2.gif -- > Uncle Alhttp://www.mazepath.com/uncleal/ > æ(Toxic URL! Unsafe for children and most mammals)http://www.mazepath.com/uncleal/lajos.htm#a2- Hide quoted text - - Show quoted text - The pentagram was a postulated shape, I then tried a square, then a triangloziod. THe last was OK. * The formal name escapes me now. === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=1wPXHwkAAACFV0NiGWX7tZb1o0HYkMjT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; PeoplePal 6.6) w:PACBHO60,gzip(gfe),gzip(gfe) > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. > He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. > I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. > On the other hand, I am not able to prove > that the puzzle is impossible to solve. > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > John A pentagram has five sides. æA vertex angle makes 5 a symmetry. > Making one degree of freedom and one symmetry. A line length or side length makes a five degree of freedom change, > each side may be independent of the other. Allowing a legnth as a > cause to ratio of side to side then making a ratio symmetry. And a mirror of set of sides allows a ratio of areas. Draw a line > between vertexes and mirror. > Making the third degree symmetry. æAnd two degrees of freedom for > there are only two axis? æNO there are three axis, making 9 degree of > freedom. SO use a square. A square is a cubic and all cubic exhibit this majic property. æGold > as a cubic crystal system allows a functional method of set to be > developed. 3 symmetries and 8 degrees of freedom allows a functional > set to be designed. D(3) > Length(4) > Mirror ratio(2) Wait the square has only 7 degree of freedom, sorry! I went through several shapes and found this one. æ æ æ* A triangloid with a certain number of sides. æIt haa No mirror > property because the axis appears a side! æSO the angle vertex makes a > ratio of side length to side length For all equal sides, two vertexs > exist. æOne degree for each. Allowing the dies to equal the rest of the degrees of freedom. And the third mirror symmetry exists only as a NON-degree of freedom > effect.- Hide quoted text - - Show quoted text - I forgot to mention. A base line or axis drawn through the base to mirror CAN NOT because the Volume of the mirror appears nonexistent. You can not mirror a volume with a line in other words, except as given. A top vertex line only appears to have the property of symmetric formal applied volume, but it appears ZERO. If the base was square | | | | | | | | axis An axis trough the base side can not make a volume mirrior as with the top vertex because the DEGREE of Freedom of the top vertex was a third symmetrical form. It depends as a symmetry on base and side legnth, while the base verticies depende only on square side length. === Subject: Re: A puzzling issue: object with 8 degrees of freedom > A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. I brought up the idea of a tetrahedral skeleton, > (like a methane molecule http://en.wikipedia.org/wiki/Methane ) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. On the other hand, I am not able to prove > that the puzzle is impossible to solve. Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > Hi John, The shape or makeup of an object does not change the freedom of it's motion. Freedom of motion has 6 directions, up- down, forward- backward,left-right. Those are the 6 so called degrees that I would call planes of motion instead. 6 maximum planes of motion only. -- James M Driscoll Jr Creator of the Clock Malfunction Theory Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Aug 2, 6:55 am, Spaceman A friend and I are having a bet. He states that there must be > objects or mechanisms with 8 degrees of freedom > (not counting translation} which > have 3-fold symmetry (at least in some > configurations). But we cannot find any. > He is thinking of objects like a deformable cubus with corners > whose angles are not fixed. But such a cubus has > - three orientational degrees of freedom > - three internal angles > which makes a total of only 6 degrees of freedom. > A cubus has 3fold symmetry when seen along > a diagonal, so that would fit; but 6 are not 8 > degrees of freedom. > I brought up the idea of a tetrahedral skeleton, > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . > It has 8 degrees of freedom, > it has 3fold symmetry in some configurations, > but we do not see a way to build that in metal > or rubber without having more or less than 8 degrees > of freedom. > On the other hand, I am not able to prove > that the puzzle is impossible to solve. > Is there another solution? Where can one look for such > objects or related theorems? Are there books or sites > on these issues? > Hi John, > The shape or makeup of an object does not change the freedom > of it's motion. > Freedom of motion has 6 directions, up- down, forward- backward,left-right. > Those are the 6 so called degrees that I would call planes of motion > instead. No: under the conventions that mathematicians and physicists use (and those *are* the relevant ones, after all, in this math newsgroup) you have described only three degrees of freedom, not six. These three degrees are East-West, up-down and in-out. This just says that there are three lines along which the motion can be projected, or three coordinates needed to describe velocity. For a generally-shaped so- called rigid body there can be two more degrees of freedom, associated with rotational angles and the like (i.e., the object's orientation). For non-rigid bodies there can be additional degrees of freedom, associated with vibrational modes, internal angles, etc. You need to be careful to count these correctly in order to obtain correct figures for specific heats in polyatomic gasses when doing statistical thermodynamics. Anyway, the OP is, presumably, dealing with only orientation and internal-structure degrees of freedom. I'm still not sure about the exact answer to the OP's question. R.G. Vickson > 6 maximum planes of motion only. -- > James M Driscoll Jr > Creator of the Clock Malfunction Theory > Spaceman === Subject: Re: A puzzling issue: object with 8 degrees of freedom | On Aug 2, 6:55 am, Spaceman | > A friend and I are having a bet. He states that there must be | > objects or mechanisms with 8 degrees of freedom | > (not counting translation} which | > have 3-fold symmetry (at least in some | > configurations). But we cannot find any. | > | > He is thinking of objects like a deformable cubus with corners | > whose angles are not fixed. But such a cubus has | > - three orientational degrees of freedom | > - three internal angles | > which makes a total of only 6 degrees of freedom. | > A cubus has 3fold symmetry when seen along | > a diagonal, so that would fit; but 6 are not 8 | > degrees of freedom. | > | > I brought up the idea of a tetrahedral skeleton, | > (like a methane moleculehttp://en.wikipedia.org/wiki/Methane) . | > It has 8 degrees of freedom, | > it has 3fold symmetry in some configurations, | > but we do not see a way to build that in metal | > or rubber without having more or less than 8 degrees | > of freedom. | > | > On the other hand, I am not able to prove | > that the puzzle is impossible to solve. | > | > Is there another solution? Where can one look for such | > objects or related theorems? Are there books or sites | > on these issues? | > | > | > Hi John, | > The shape or makeup of an object does not change the freedom | > of it's motion. | > Freedom of motion has 6 directions, up- down, forward- backward,left-right. | > Those are the 6 so called degrees that I would call planes of motion | > instead. | | No: under the conventions that mathematicians and physicists use (and | those *are* the relevant ones, after all, in this math newsgroup) | you have described only three degrees of freedom, not six. These three | degrees are East-West, up-down and in-out. This just says that there | are three lines along which the motion can be projected, or three | coordinates needed to describe velocity. For a generally-shaped so- | called rigid body there can be two more degrees of freedom, associated | with rotational angles and the like (i.e., the object's orientation). | For non-rigid bodies there can be additional degrees of freedom, | associated with vibrational modes, internal angles, etc. You need to | be careful to count these correctly in order to obtain correct figures | for specific heats in polyatomic gasses when doing statistical | thermodynamics. | | Anyway, the OP is, presumably, dealing with only orientation and | internal-structure degrees of freedom. I'm still not sure about the | exact answer to the OP's question. | | R.G. Vickson 6-DOF x, y, z, pitch, roll, yaw. 6-DOF platforms: http://www.inmotionsimulation.com/images/6-dof-2.jpg http://www.inmotionsimulation.com/images/6-dof-1.jpg http://www.ckas.com.au/CKAS%20V4%206DOF%20Motion%20Platform.jpg