mm-3309 ==== Does anybody know about whether there exists generalizations of Little's Theorem. For example, If the average time a customer spends in a counter is T and the average number of counter a customer needs to go through is N, can we say the average time to spend through the whole process is T*N? Is the above valid without considering the individual probability mass/density funtions? === Subject: Re: Generalization of Little's Theorem > Does anybody know about whether there exists generalizations > of Little's Theorem. For example, I don't know what you mean by Little's Theorem. > If the average time a customer spends in a counter is T and the > average number of counter a customer needs to go through is > N, can we say the average time to spend through the whole > process is T*N? Is the above valid without considering the individual probability > mass/density funtions? I'm not sure precisely what you mean by the average time a customer spends in a counter is T. My guess is: for any given counter, the expected time that each customer that goes through this counter spends there is T. Thus the conditional expectation for the total time spent going through counter #i, given that C_i customers go through it, is T C_i, and the expected total time spent at all the counters is T times the expected total number of visits of customers to counters. If there are m customers, each with an expected number of counter visits of N, the expected total time is m T N, so the expected total time per customer is T N. === Subject: Re: Generalization of Little's Theorem > I don't know what you mean by Little's Theorem. I meant the one in queueing or network theroy: http://en.wikipedia.org/wiki/Little%27s_theorem === Subject: Re: Prolate surface area Mail-To-News-Contact: abuse@dizum.com >> Anyone know the integral and closed form solutions for >> prolate surface area? In the solution I have they don't >> seem to match. >> The oblate surface area seems well documented, but the >> prolate is rather lacking. See (9) at , >for example. David If you compare it with the oblate version something doesn't seem right with Mathworld's. For the oblate I have _(+pi/2) / a2pi / cos(theta)sqrt(a^2sin^2theta+c^2cos^2theta)dtheta _/ (-pi/2) and it matches the closed form. Somewheres else I saw the prolate version just switches a and c _(+pi/2) / c2pi / cos(theta)sqrt(c^2sin^2theta+a^2cos^2theta)dtheta _/ (-pi/2) but it doesn't match the closed form answer. Ugh! === Subject: Re: Prolate surface area >> Anyone know the integral and closed form solutions for >> prolate surface area? In the solution I have they don't >> seem to match. >> The oblate surface area seems well documented, but the >> prolate is rather lacking. See (9) at , >for example. David If you compare it with the oblate version something doesn't > seem right with Mathworld's. If you still think there is something wrong with one of those MathWorld if possible, how you'd propose it should be corrected. > For the oblate I have / > a2pi / cos(theta)sqrt(a^2sin^2theta+c^2cos^2theta)dtheta > _/ > (-pi/2) > and it matches the closed form. > Somewheres else I saw the prolate version just switches > a and c Well, of course, if one just switches a and c, the closed form will be the same as in the other case, just with a and c switched. > / > c2pi / cos(theta)sqrt(c^2sin^2theta+a^2cos^2theta)dtheta > _/ > (-pi/2) > but it doesn't match the closed form answer. Ugh! Are you aware that the same closed form can be used for both the prolate and oblate cases? Curiously, the only case for which the closed form fails is the intermediate case, the sphere itself. But there's a simple remedy for that failure; see David W. Cantrell === Subject: Technology Inside the Black Box on radiosandysprings.com. Meet to doctors from Georgia Tech as they answer some of the most unique questions from tooth paste to well just listen. www.radiosandysprings.com, you'll love it! === Subject: I want to know howto solve ax^2+bx = cy^2+dy (nonlinear diophantine equation) I want to know howto solve ax^2+bx = cy^2+dy (nonlinear diophantine equation) when a +b =c+d and x,y is variable === Subject: Re: I want to know howto solve ax^2+bx = cy^2+dy (nonlinear diophantine equation) > I want to know howto solve ax^2+bx = cy^2+dy (nonlinear diophantine > equation) > when a +b =c+d and x,y is variable The general case is axî + bxy + cyî + dx + ey + f = 0 If a = 0 put everything containing x on the left-hand side and everything else on the right-hand side. On the left-hand side put x outside the parenthesis. Then divide by a polynomial in y so that x = ... remains. If c = 0 do the analogous thing with y. If a, c and at least one of b, d, e is nonzero, throw out the linear terms. Differentiate the whole thing with respect to either x or y and substitute with the result. Your example: axî + bx - cyî - dy = 0 Differentiate w.r.t. x: 2ax + b Substitute: z = 2ax + b zî = 4aîxî + 4abx + bî When you compare this with the 4a-fold of the original equation, you have got rid of one of the linear terms. Do the same thing with the other one as well. -- Helmut Richter === Subject: Re: JSH: Prime counting: code challenge!!! [Tim Peters] > Just FYI, I had some spare time today and coded a prime-counter in > C based on Legendre's phi recurrence, deliberately kept at the > same level as JSH's Java code. That is, it applied standard tricks > for speeding phi evaluation, but did /not/ go as far as Meissel went > in making asymptotic improvements (let alone more recent improvements). ... [and it ran just like JSH's Java P() code, until adding the standard running Meissel backwards phi optimization, at which point it ran substantially faster] ... > Anyway, if anyone wants the source code, ask in email. I only spent > a few hours on it, and have no interest in making it bulletproof or > more portable (e.g., it assumes a 32-bit box, and uses Microsoft- > specific spellings for I/O of 64-bit integers). More importantly, it's > an /obsolete approach/ to computing pi(), slower and much more of a > memory hog than modern approaches -- this was done just to satisfy > idle curiosity. If you really want it anyway, ask before I delete it > <0.5 wink>. Hmm. I don't want to mail this code out anymore. As explained there, it was a throwaway, and I don't want it to spread. Since the /point/ was to illustrate standard phi optimizations, I'm going to post just that part of the code: /* The phi functions assume p_a <= n on entry. This is true at top * level since p_a^2 <= n there. Internally, p_a^2 <= n is forced at * entry, and internal calls either invoke phi(n, i) for i < a, or * phi(n/p_i, i-1) for i <= a. In the latter cases, p_i <= p_a, so * p_i^2 <= n, so p_i <= n/p_i, and of course p_(i-1) < p_i. */ static UI phi_thin(UI n, UI a) { int i; UI result; UI offset = 0; if (a > 1 && primes2[a-1] >= n) { const UI a2 = pi_lookup((UI)sqrt((double)n)); /* pi(n) = phi(n, a2) + a2 - 1, so * phi(n, a2) = pi(n) - a2 + 1, * and we need to subtract a-a2. */ if (n <= largest_prime) return pi_lookup(n) - a + 1; else { offset = a - a2; a = a2; } } if (n <= 1 || a == 0) return n - offset; if (a <= K) { canned* pi = phi_info + a; const UI q = n / pi->primorial; const UI r = n - q * pi->primorial; return q * pi->totient + pi->phitab[r] - offset; } if (a > ACUT && n <= largest_prime && primes[a-1] * primes2[a-1] >= n) { /* Run Meissel backwards; given a >= pi(cbrt(n)), * phi(n, a) = pi(n) - (a - 1) + P2(n, a). * sum(i-1, i, a, b) = (b+a-2)*(b-a+1)/2 */ const UI b = pi_lookup((UI)sqrt((double)n)); UI total = pi_lookup(n) - (b+a-2) * (b-a+1) / 2; for (i = a; i < b; ++i) total += pi_lookup(n / primes[i]); return total - offset; } result = phi_thin(n, K) - offset; for (i = a; --i >= K; ) result -= phi_thin(n / primes[i], i); return result; } `UI` is a typedef for unsigned 32-bit int. primes[i] = p_(i+1), the (i+1)'th prime. primes2[i] = p_(i+1)^2, the square of the (i+1)'th prime. `largest_prime` is the largest prime in primes[]. `pi_lookup(n)` computes pi(n) via binary search in primes[]. Note that for UI `n`, on a box with a floating-point sqrt function conforming to the IEEE-754 standard, it's guaranteed that (int)sqrt(n) is the largest integer <= the true square root of `n`. In the fat phi code (which takes a 64-bit unsigned int for `n`), a fancier method is needed. When p_a > n, phi(n, a) = phi(n, a-1). The comment at the start explains why this case never occurs. When p_a <= n but p_a^2 > n, phi(n, a) = phi(n, a-1) - 1. The first code block checks for that, and zooms down to the largest a for which p_a^2 <= n. In that case, it's usually possible to evaluate the derived phi(n, a) at once as pi(n) + a - 1, but when not the difference between the incoming and outgoing a's is remembered in `offset`. n <= 1 and/or a = 0 are the trivial endcases. The a <= K block uses the standard table-lookup way to evaluate phi(n, a) for small a. The tables are in static array `phi_info`, indexed by `a`. The fat phi code differs only in that `q` (but not `r`) may also be a 64-bit int. The next block is running Meissel backwards when p_a^3 >= n. ACUT is a tuning constant, and happens to be 10: it's faster to keep running the recurrence instead if `a` is small enough, simply due to that binary search is, while very fast, not free. Ideally, ACUT should be boosted very slowly as the original n increases, because binary search takes time logarithmic in the square root of the original n (because O(sqrt(n)) primes are stored in primes[]). If someone wanted to go nuts with micro-optimizing, they could exploit that the argument to pi_lookup() keeps decreasing in the loop, and use a two-argument version of pi_lookup() allowing to specify an upper bound. The last part is derived from taking the basic Legendre phi recurrence: [1] phi(n, a) = phi(n, a-1) - phi(n/p_a, a-1) and iteratively expanding the left addend on the RHS until a=K is reached: [2] phi(n, a) = phi(n, K) - sum for i=K+1 to a of phi(n/p_i, i-1) Note that we can't get to the final part unless a > K, so [2] always makes sense in context. It's /important/ to use [2] instead of [1], because using [1] directly leads to a recursion depth equal to `a`, and that can blow the runtime stack when the input is large. Using [2], the max recursion depth is proportional to log(n) instead (perhaps not obvious -- this has to do with how fast the primorial function grows, and that recursive calls cut the size of the first argument at least that fast). That's it! === Subject: what conditions are required okay, i did not know what subject to put for this post. but i was wondering if anybody could help me with this. say f,g:R^n-->R assume the folowing condition is TRUE. it will not be true for all functions but assume the domain of the functions, and the nature of the functions are such that this is true. f(x)=0 has a solution, which is a (n-1) dimensional surface call it Sf. also assume if g is a very close function to f, then g(x)=0 also has a solution, let this surface be Sg. i want to know what conditions on the functions do i need to put to capture this closeness. one condition that comes to my mind is that to put for any x, sup|f(x)-g(x)|=eps, for some small eps >0. but it does not prevent from being f(x)=g(x) for all x, which is not intended, because we want f,g to be close but different functions. so will a condition like integral of |f(x)-g(x)| > delta for some small positive delta be useful here? the context of the problem is, i'm trying to solve 1) at any point x in Sf, if we take gradient (which is normal to Sf) and cast a ray in the normal direction, that will intersect Sg at some point y, i'm interested in lower bound of the distance |x-y| except at the points where the surfaces intersect. and this lowerbound must not be a trivial zero bound. === Subject: classifying space for SU(N) Does anybody know anything about the explicit construction of the classifying spaces for special unitary groups? I've found a lot about U(N), O(N) and even SO(N), but nothing for SU(N). Pavel. === Subject: The mathematics of poker Hello folks, question 1: Why isn't there yet a strong Poker AI? Of course, the combinatory explosion is even bigger than with chess, but there are more easy and systematic rules to get a grip onto it. What would happen if you combine a good poker play, a good mathematican and fundamental and structured poker knowledge like on http://www.pstrats.net ? Wouldn't it be just a matter of time until there are easy programs that beat human players and thus ruin the poker boom - which feeds many mathematicans like me a lot of money due to our easy conception of statistical data ;) Chris === Subject: Re: The mathematics of poker > question 1: Why isn't there yet a strong Poker AI? > Of course, the combinatory explosion is even bigger than with chess, but > there are more easy and systematic rules to get a grip onto it. The problem is essentially three-fold: knowing card probabilities, best betting strategies, and reading your opponents. Card probabilities are the easiest. Computers can handle those very quickly. Students are given some of the simplest problems in Statistics classrooms, and professional players can usually estimate well enough on the fly to acquit themselves well here. Betting stategies are more involved Game Theory. Some of the very simplest games have been deduced for optimal strategies for two players, but complicated multi-player games with imperfect information, like say Texas Hold 'Em or 7 Card Stud Deuces Wild, are exceptionally complicated. I do not believe all the choices for all eventualities have been computed for these games. Finally, as any Professional Poker player will tell you, the most important aspect of the game is the ability to read your opponents, be able to pick up on his tells, etc. This is more a study of psychology, and possibly physiology, than it is of mathematics. > What would happen if you combine a good poker play, a good mathematican > and fundamental and structured poker knowledge like on > http://www.pstrats.net ? > Wouldn't it be just a matter of time until there are easy programs that > beat human players and thus ruin the poker boom - which feeds many > mathematicans like me a lot of money due to our easy conception of > statistical data ;) What you speak of is theoretically possibly for games of perfect information, such as chess. (And this game being a long, long way from being completely decided.) But games of imperfect information, in which the person with the worst hand can win and the person with the best hand can lose, reading your important will remain a necessary component for anyone wishing to master the game. === Subject: Re: The mathematics of poker > question 1: Why isn't there yet a strong Poker AI? > Of course, the combinatory explosion is even bigger than with chess, but > there are more easy and systematic rules to get a grip onto it. The problem is essentially three-fold: knowing card probabilities, best > betting strategies, and reading your opponents. Card probabilities are the easiest. Computers can handle those very > quickly. Students are given some of the simplest problems in > Statistics classrooms, and professional players can usually estimate > well enough on the fly to acquit themselves well here. Betting stategies are more involved Game Theory. Some of the very > simplest games have been deduced for optimal strategies for two > players, but complicated multi-player games with imperfect information, > like say Texas Hold 'Em or 7 Card Stud Deuces Wild, are exceptionally > complicated. I do not believe all the choices for all eventualities > have been computed for these games. Finally, as any Professional Poker player will tell you, the most > important aspect of the game is the ability to read your opponents, be > able to pick up on his tells, etc. This is more a study of > psychology, and possibly physiology, than it is of mathematics. I gather that a lot of poker is being played over the internet nowadays. I think that takes the physiology out of it. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: The mathematics of poker If there was a good poker AI, do you think you would know about it? Or do you think the author would use it to take all of your money on partypoker.net? -MD On Nov 23, 5:25 pm, Gerry Myerson question 1: Why isn't there yet a strong Poker AI? > Of course, the combinatory explosion is even bigger than with chess, but > there are more easy and systematic rules to get a grip onto it. The problem is essentially three-fold: knowing card probabilities, best > betting strategies, and reading your opponents. Card probabilities are the easiest. Computers can handle those very > quickly. Students are given some of the simplest problems in > Statistics classrooms, and professional players can usually estimate > well enough on the fly to acquit themselves well here. Betting stategies are more involved Game Theory. Some of the very > simplest games have been deduced for optimal strategies for two > players, but complicated multi-player games with imperfect information, > like say Texas Hold 'Em or 7 Card Stud Deuces Wild, are exceptionally > complicated. I do not believe all the choices for all eventualities > have been computed for these games. Finally, as any Professional Poker player will tell you, the most > important aspect of the game is the ability to read your opponents, be > able to pick up on his tells, etc. This is more a study of > psychology, and possibly physiology, than it is of mathematics.I gather that a lot of poker is being played over the internet > nowadays. I think that takes the physiology out of it. -- > Gerry Myerson (g...@maths.mq.edi.ai) (i -> u for email) === Subject: Looking for passionate singles? Find someone now on the largest erotic personals site http://passion.com/go/g822474-pmo FREE signup! Post a FREE ad with 5 photos, flirt in chatrooms, view uncensored live Webcams, hookup for REAL encounters! 30,000 new photos every day! http://passion.com/go/g822474-pmo Go there now! === Subject: integral over bessel function I have the following integral: int_0^{2 pi} J_0[sqrt(a + b cos(beta))] dbeta I would like to get an analytic solution, possibly there exists a solution in the form G(a) G(b) r(a,b) where the function r is relatively simple. Any ideas? Martin === Subject: Re: integral over bessel function I have the following integral: int_0^{2 pi} J_0[sqrt(a + b cos(beta))] dbeta I would like to get an analytic solution, possibly there exists a > solution in the form > G(a) G(b) r(a,b) where the function r is relatively simple. Any ideas? Martin > For a=b=1, I get the answer 2*Pi*J_0(1/sqrt(2))^2 That's because sqrt(1+cos(x)) can be simplified. When |a|<|b|, do you have something in mind for the squareroot when (a + b cos(beta)) is negative? === Subject: Re: integral over bessel function OK, to be more specific, I have a band-pass filter function u(t) and need the following integral int_0^{2pi} U(|vec x + vec y|) d beta (*) where U is the Fourier transform of u and beta the angle between the two (2d)-vectors vec x and vec y. I can write this as 2pi int_0^infinity u(x) x dx int_0^{2pi} J_0(sqrt{x^2 + y^2 + 2 x y cos(beta)}) so a = x^2 + y^2, b = 2 x y. Therefore the expr under the square root is never negative. Alternatively, one can try to solve the first integral (*) directly. I need it for U(x) = J_4(x)/x^2. Using a Gaussian-type filter function U(x) = x^2/2 exp(-x^2/2), (*) has the closed-form solution 1/2 *exp{- (x^2 + y^2)/2} [ (x^2 + y^2) I_0(x y) - 2 x y I_1(x y)]. (This result is also found by Mathematica 4 but not version 5 due to a bug in Integrate. Be wary of StruveL!) function integral. Martin > For a=b=1, I get the answer 2*Pi*J_0(1/sqrt(2))^2 > That's because sqrt(1+cos(x)) can be simplified. When |a|<|b|, do you have something in mind for the squareroot > when (a + b cos(beta)) is negative? === Subject: Re: integral over bessel function > I have the following integral: int_0^{2 pi} J_0[sqrt(a + b cos(beta))] dbeta I would like to get an analytic solution, possibly there exists a > solution in the form > G(a) G(b) r(a,b) where the function r is relatively simple. Any ideas? Sometimes it helps to know where the integral comes from. Han de Bruijn === Subject: Re: integral over bessel function I have the following integral: int_0^{2 pi} J_0[sqrt(a + b cos(beta))] dbeta I would like to get an analytic solution, possibly there exists a > solution in the form > G(a) G(b) r(a,b) where the function r is relatively simple. I doubt that there is a closed-form solution. === Subject: Re: integral over bessel function I have the following integral: int_0^{2 pi} J_0[sqrt(a + b cos(beta))] dbeta I would like to get an analytic solution, possibly there exists a > solution in the form > G(a) G(b) r(a,b) where the function r is relatively simple. I doubt that there is a closed-form solution. Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada int(f) = f.[d(ln(f))]^-1 would exisist if integral closed form exists. analytical closure of integration equation. === Subject: Re: integral over bessel function I have the following integral: int_0^{2 pi} J_0[sqrt(a + b cos(beta))] dbeta I would like to get an analytic solution, possibly there exists a > solution in the form > G(a) G(b) r(a,b) where the function r is relatively simple. I doubt that there is a closed-form solution. Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada int(f) = f.[d(ln(f))]^-1 would exisist if integral closed form exists. analytical closure of integration equation. sorry int(f) = f.SQf Qf = [d(ln(f))]^-1 Sx = sigma [(-x)^n . d^n(x),0,infinity] so you make an infinite sum by parametric substitution of y = e^x and then infinitly use partial integration. === Subject: Jumping through the maze My home town Ruurlo is famous for its hedge maze. My new puzzle at http://home.planet.nl/~p.j.hendriks/ppvdw.htm (click on the Union Jack to get at the english version) is about a maze. Find the shortest road along the largest number of squares. Sounds confusing perhaps, but if you read the puzzle you may like it. I would be delighted to recieve your solutions by email. Have fun with the puzzle !!! Peter === Subject: existence of soln I have to show that there exists x, s.t. Ax=b, x {i}>=0 where A is mxn matrix with m I have to show that there exists x, s.t. Ax=b, x {i}>=0 where A is mxn > matrix with m= 0 for i = 1,...,5. The Phase I LP associated with this problem shows that it is infeasible: every basic solution has at least one negative x i. R.G. Vickson I looked a bit linear programm.88ng theory but now I'm a bit confused > and I do not know how to show existence of basic feasible solution. Any helpful commends? > === Subject: Re: existence of soln > I have to show that there exists x, s.t. Ax=b, x_{i}>=0 where A is mxn > matrix with m= 0 for all i, for some specified i, or for at least one i? In the first two cases, it's not true unless there are other assumptions. Try [ -1 ] [ 1 0 0 ] b = [ -1 ], A = [ 0 1 0 ] === Subject: Re: existence of soln I mean for all i. === Subject: Re: existence of soln ays yazd[CapitalYAcute]: > I have to show that there exists x, s.t. Ax=b, x {i}>=0 where A is mxn > matrix with m and I do not know how to show existence of basic feasible solution. Farkas Lemma is proposed but there also I have to show ceratin inequalities is not satisfied. Any other way? > === Subject: Re: Cantor Confusion > Cantor's proof is one of the most popular topics on this NG. It > seems that people are confused or uncomfortable with it, so > I've tried to summarize it to the simplest terms: 1. Assume there is a list containing all the reals. > 2. Show that a real can be defined/constructed from that list. > 3. Show why the real from step 2 is not on the list. > 4. Conclude that the premise is wrong because of the contradiction. so reflect the integers about the right most decimal point and you have a countable infinity of fractionals, each fractional can be a fractional added to an integer so set size is Z*Z -> countable. this is a direct proof. of the countability of the frationals, or infinite rationals as one may choose to see them === Subject: Re: Cantor Confusion > But in mathematics comparing numbers comes in quite late in > the process of definition. With mathematics you perhaps mean the possibly questionable attempt to formally justify what has proven successful. > It starts when the ordering axioms are > given. In a set a ardering relation may be can be defined. Let us use >= as the basis. Mathematicians like you are hopefully aware of the trifle that the relation >= cannot be applied to the really real numbers. Do not worry, the actual infinity is not taken seriously with ZF axioms. My problem is: Narrow-minded former learner of set theory cannot imagine the basic idea by Dedekind and Cantor wrong, the whole cardinal ordinal stuff reduced to a simple distinction between finite, countably infinite, and uncountable. The particular obstacle is: really real numbers are uncountable fictions and therefore they exhibit properties pretty different from those of genuine, i.e. rational, numbers. === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > But in mathematics comparing numbers comes in quite late in > the process of definition. > > With mathematics you perhaps mean the possibly questionable attempt to > formally justify what has proven successful. What has been shown successful in some cases. The formalisation allows us to find under precisely what conditions it will be successful, and why it is successful under those conditions. In addition it allows to provide (possibly) more general alternatives that work under less strict conditions. > It starts when the ordering axioms are > given. In a set a ardering relation may be can be defined. Let us use >= as the basis. > > Mathematicians like you are hopefully aware of the trifle that the > relation >= cannot be applied to the really real numbers. Oh. What are really real numbers? What you are missing is that mathematics provides an idealisation of arithmetic (amongst others), and from that background provides processes to do real calculations to get results in (amongst others) the physical world. The whole field of numerical mathematics could barely have been build without the idealised mathematical background. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion > Mathematicians like you are hopefully aware of the trifle that the > relation >= cannot be applied to the really real numbers. Oh. What are really real numbers? Those, like the imagined numerical solution to the task pi, which are just fictions. Those which are assumed as basis for DA2, > What you are missing is that mathematics provides an idealisation of > arithmetic (amongst others), and from that background provides > processes to do real calculations to get results in (amongst others) > the physical world. The whole field of numerical mathematics could > barely have been build without the idealised mathematical background. No no. I do not deny that numbers can come as close as you like to the ideal concept of infinity and continuum. Rational numbers already do that job. Eckard Blumschein === Subject: Re: Cantor Confusion > > Mathematicians like you are hopefully aware of the trifle that the > relation >= cannot be applied to the really real numbers. > > Oh. What are really real numbers? > > Those, like the imagined numerical solution to the task pi, which are > just fictions. Those which are assumed as basis for DA2, In that case what are the not-really real numbers? > What you are missing is that mathematics provides an idealisation of > arithmetic (amongst others), and from that background provides > processes to do real calculations to get results in (amongst others) > the physical world. The whole field of numerical mathematics could > barely have been build without the idealised mathematical background. > > No no. I do not deny that numbers can come as close as you like to the > ideal concept of infinity and continuum. Rational numbers already do > that job. positive definite matrices will not work when you are using rationals only. But it is commonly used. The reason is that in ideal mathematics the decomposition will work, and numerical mathematics provides the additional information you need to get it to work in finite precision. === Subject: Re: Cantor Confusion > assume that part of my mathematics belongs to absolute truth which does > not rely on arbitrary axioms. And you do not formulate your absolute truths as axioms, so your absolute truths do not have objective form by which other people (not just you) can verify that a given statement is of one of your absolute truths. Your absolute truths are spoken by you only as verbiage that is at worst vague and at best unformalized. You provide no objective means by which other people (not just you) can verify that your arguments do indeed follow by a specified logic from axioms which assert your absolute truths. In this situation, all discussions can only defer to you personally, as only you personally can say what is or is not absolute truth as you have determined it and as you argue as to its implications by your unspecified logic. On the other hand, even if you disagree that certain mathematical axioms are true, at least they are given in utterly objective form as formulas in a formal language and such that at least in principle one can mechanically check whether a given sequence of formulas is or is not a proof in a given formal system (and for the most basic theorems, even in practice it wouldn't be difficult to fully formalize for mechanical checkability). In other words, at least mathematics can give you - up front, clearly, and precisely - exactly what its axioms are and exactly by which rules theorems may be proven, while your polemics offer no such intellectual courtesy or benefits. MoeBlee === Subject: Re: Cantor Confusion >> I >> assume that part of my mathematics belongs to absolute truth which does >> not rely on arbitrary axioms. And you do not formulate your absolute truths as axioms, so your >absolute truths do not have objective form by which other people (not >just you) can verify that a given statement is of one of your absolute >truths. Your absolute truths are spoken by you only as verbiage that is >at worst vague and at best unformalized. You provide no objective means >by which other people (not just you) can verify that your arguments do >indeed follow by a specified logic from axioms which assert your >absolute truths. In this situation, all discussions can only defer to >you personally, as only you personally can say what is or is not >absolute truth as you have determined it and as you argue as to its >implications by your unspecified logic. On the other hand, even if you disagree that certain mathematical >axioms are true, at least they are given in utterly objective form as >formulas in a formal language and such that at least in principle one >can mechanically check whether a given sequence of formulas is or is >not a proof in a given formal system (and for the most basic theorems, >even in practice it wouldn't be difficult to fully formalize for >mechanical checkability). In other words, at least mathematics can give >you - up front, clearly, and precisely - exactly what its axioms are >and exactly by which rules theorems may be proven, while your polemics >offer no such intellectual courtesy or benefits. Well, Moe, you know I can completely agree with these comments. On the whole they represent a reasonably accurate assessment of the status quo in mathematics And I agree with your observations concerning explicit and objective form. However nothing in explicit and objective form guarantees a correct and accurate assessment of the implicit form and implications of one objective form in relation to others because that assessment is mechanized subjectively and not objectively. In other words you have objective axioms A, B, C, etc. which summarize various properties and predicates believed to be true explicitly. But you still have no way to evaluate the consistency among implications of A, B, C, etc. in relation to one another because those implications are not stated explicitly. ~v~~ === Subject: Re: Cantor Confusion >> Since it _exists_ only as a gedankenexperiment, a sort of mental game, >> such fears are irrelevant. Other fears are more relevant. Such as the fear that infinities in >> mathematics may easily lead to irrelevancies in physics (like e.g. >> with Black Holes or String Theory) As the almost universal consensus of astronomers now is that the center > of many, possibly most, galaxies is a huge black hole, and that many > larger stars too big to eventually collapse into neutron stars collapse > into black holes when they run out of fuel, it appears that HdB is the > unaware of what are irrelevancies in physics. So that HdB is hardly in a position to dictate what should be considered > irrelevancies in areas outside of his supposed area of expertise. Yes. There is widespread consensus about black holes to exist. What about white holes, they are perhaps nor many supporter. When we learn from history we have to realize widespread but unfounded and wrong consensus for more than 100 years declaring Ohm correct but Seebeck wrong. I do not trust in infinity in physics. Infinity and continuum are just ideal concepts. Reality is most likely different from these models. I would like to support Han because his expertise is much closer to physics than the horizont of pure mathematicans. Maybe he even feels equally reluctant to swallow |sign(0)|=0 like me. With aptly defined reals I prefer |sign(0)|=1. Han will possibly not trust me because practicians like him used to operate with numbers, with rational numbers of course. Eckard === Subject: Re: Cantor Confusion Am Tue, 21 Nov 2006 15:49:57 +0100 schrieb Eckard Blumschein: > When we learn from history we know that Ecki is a little stupid girl who is to dumb to learn anything about mathematics. === Subject: Re: Cantor Confusion > I would like to support Han because his expertise is much closer to > physics than the horizont of pure mathematicans. Do you really think so? > Maybe he even feels > equally reluctant to swallow |sign(0)|=0 like me. With aptly defined > reals I prefer |sign(0)|=1. Then, please tell us your definition of reals and of sign. -- David Marcus === Subject: Re: Cantor Confusion > >> If you want to find absolute truth you should not look at mathematics. Perhaps we should replace absolute truth with culturally neutral > truth, or in other words, truth without any cultural, religious, or > philosophical bias. We can reason about this concept by asking the > question: what will the mathematics of advanced alien civilizations > (i.e. from other planets) look like? Thinking about this question > leads most of us to believe that there is a core of mathematics which > every such civilization will accept. I wonder, this is a belief I am sharing wholehartedly. Otherwise I am a notoriously unbelieving person. === Subject: Re: Cantor Confusion Am Tue, 21 Nov 2006 15:22:13 +0100 schrieb Eckard Blumschein: > I wonder, this is a belief I am sharing wholehartedly. > Otherwise I am a notoriously unbelieving person. Otherwise you are a very stupid person. First you dumped the german ng's and now you dump here. We did go through this all the time - but you are snotnosed and dumb. === Subject: Re: Cantor Confusion >> Was Herr Veronese dar?ber in seiner Schrift giebt, halte ich f?r >> Phantastereien und was er gegen mich darin vorbringt, ist unbegr?ndet. >> Ueber seine unendlich gro?en Zahlen sagt er, da? sie auf anderen >> Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber >> auf gar keinen Hypothesen sondern sind unmittelbar aus dem nat?rlichen >> Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von >> Willk?r, wie die endlichen ganzen Zahlen. Briefly: My infinite numbers are founded only on the natural notion of >> sets. They are as necessary and free of arbitriness as the finite whole >> numbers. > > What Cantor may have said in 1895 need not be binding in 2006. It was not binding in 1895 either. However, the Utopia by Dedekind and Cantor was reformed but it never got rid of its basic mistakes. While most views of Veronese were largely correct, his infinitely large numbers are not better than those by Cantor. === Subject: Re: Cantor Confusion > >> Was Herr Veronese dar?ber in seiner Schrift giebt, halte ich f?r >> Phantastereien und was er gegen mich darin vorbringt, ist unbegr?ndet. >> Ueber seine unendlich gro?en Zahlen sagt er, da? sie auf anderen >> Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber >> auf gar keinen Hypothesen sondern sind unmittelbar aus dem nat?rlichen >> Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von >> Willk?r, wie die endlichen ganzen Zahlen. Briefly: My infinite numbers are founded only on the natural notion of >> sets. They are as necessary and free of arbitriness as the finite whole >> numbers. > > What Cantor may have said in 1895 need not be binding in 2006. > As I do not have access to the original documents, and could not read the German well enough anyway, I cannot be certain of it. It was not binding in 1895 either. However, the Utopia by Dedekind and > Cantor was reformed but it never got rid of its basic mistakes. Only in EB's opinion are they mistakes. Others disagree. === Subject: Re: Cantor Confusion > Every set of natural numbers has a superset of natural numbers which is > finite. Every! I am only aware of the unique natural numbers. If one imagines them altogether like a set, then this bag may also be the only lonly one. === Subject: Re: Cantor Confusion Every set of natural numbers has a superset of natural numbers which is > finite. Every! I am only aware of the unique natural numbers. If one imagines them > altogether like a set, then this bag may also be the only lonly one. WM means that every finite set of natural numbers is contained in a larger finite set of natural numbers. Since he believes all sets of natural numbers are finite, he then concludes that every set of natural numbers is contained in a finite set of natural numbers. This seems to be due to an allergy to the word set. -- David Marcus === Subject: Re: Cantor Confusion Every set of natural numbers has a superset of natural numbers which is >> finite. Every! I am only aware of the unique natural numbers. If one imagines them >> altogether like a set, then this bag may also be the only lonly one. WM means that every finite set of natural numbers is contained in a > larger finite set of natural numbers. The sequence of natural numbers has no reason to end, exept if one does not have enough power of imagination or abstraction, respectively, to exclude any physical restriction from the ideal model of natural numbers. > Since he believes all sets of > natural numbers are finite, he then concludes that every set of natural > numbers is contained in a finite set of natural numbers. This seems to > be due to an allergy to the word set. The word set does indeed serve as a deliberately obscuring crutch. It suggests an aprioric point of view, contrasting to Archimedes. WM denies this selfcontradictory abstraction. Following Leibniz, I consider uncountable numbers a useful fiction with a fundamentum in re. === Subject: Re: Cantor Confusion Every set of natural numbers has a superset of natural numbers which is > finite. Every! > I am only aware of the unique natural numbers. If one imagines them > altogether like a set, then this bag may also be the only lonly one. Is that lonely? === Subject: Re: Cantor Confusion > Every set of natural numbers has a superset of natural numbers which is >> finite. Every! >> I am only aware of the unique natural numbers. If one imagines them >> altogether like a set, then this bag may also be the only lonly one. Is that lonely? I meant: There is only one natural sequence of numbers: 1, 2, 3, ... For instance 0, 1, 2, ... or 3, 4, 5, ... or 2, 4, 6, ... are something else. Every implies that there are more than one. === Subject: Re: Cantor Confusion > Every set of natural numbers has a superset of natural numbers which is >> finite. Every! >> I am only aware of the unique natural numbers. If one imagines them >> altogether like a set, then this bag may also be the only lonly one. Is that lonely? I meant: There is only one natural sequence of numbers: 1, 2, 3, ... > For instance 0, 1, 2, ... or 3, 4, 5, ... or 2, 4, 6, ... are something > else. Every implies that there are more than one. There is only one sequence of all natural numbers taken in natural order. But there are uncountably many different sequences of natural numbers if one is not required to include all of them nor to take them in any particular order. === Subject: Re: Cantor Confusion > Every set of natural numbers has a superset of natural numbers which is > finite. Every! > I am only aware of the unique natural numbers. If one imagines them > altogether like a set, then this bag may also be the only lonly one. Is that lonely? I meant: There is only one natural sequence of numbers: 1, 2, 3, ... >> For instance 0, 1, 2, ... or 3, 4, 5, ... or 2, 4, 6, ... are something >> else. Every implies that there are more than one. There is only one sequence of all natural numbers taken in natural > order. > But there are uncountably many different sequences of natural numbers if > one is not required to include all of them nor to take them in any > particular order. Countably many, yes. As many as there are natural numbers. However, none of them is identical with the notion of _the_ natural numbers. === Subject: Re: Cantor Confusion > Every set of natural numbers has a superset of natural numbers which > is > finite. Every! > I am only aware of the unique natural numbers. If one imagines them > altogether like a set, then this bag may also be the only lonly one. Is that lonely? I meant: There is only one natural sequence of numbers: 1, 2, 3, ... >> For instance 0, 1, 2, ... or 3, 4, 5, ... or 2, 4, 6, ... are something >> else. Every implies that there are more than one. There is only one sequence of all natural numbers taken in natural > order. > But there are uncountably many different sequences of natural numbers if > one is not required to include all of them nor to take them in any > particular order. Countably many, yes. As many as there are natural numbers. If one is not required to include all of them nor to take them in any particular order, then least as many of sequences as there are infinite subsets of the infinite set of finite naturals, N, multiplied by the number of bijections from N to N. > However, none of them is identical with the notion of _the_ natural > numbers. Whyever need it be? A sequence is a function whose DOMAIN is the set of naturals in standard order, The values of such a function, even if all naturals, need not cover all naturals nor appear in natural order. === Subject: Re: Cantor Confusion > As you stated Goes on forever is a property of the natural > numbers. The set {1,2,3,...} is just the natural numbers. So > this set must go on forever. Being admittedly not very familiar with set theory, I nonetheless wonder if sets are considered like something going on forever. === Subject: Re: Cantor Confusion As you stated Goes on forever is a property of the natural > numbers. The set {1,2,3,...} is just the natural numbers. So > this set must go on forever. Being admittedly not very familiar with set theory, I nonetheless wonder > if sets are considered like something going on forever. Sequences do, sets do not. When one represents the members of a set as members of a sequence, as {1,2,3,...} does, that little ambiguity should not really mislead anyone. === Subject: Re: Cantor Confusion > As you stated Goes on forever is a property of the natural >> numbers. The set {1,2,3,...} is just the natural numbers. So >> this set must go on forever. Being admittedly not very familiar with set theory, I nonetheless wonder >> if sets are considered like something going on forever. Sequences do, sets do not. When one represents the members of a set as members of a sequence, as > {1,2,3,...} does, that little ambiguity should not really mislead anyone. That little ambiguity? Doesn't it make a categorical difference if a set has been a priori set for good? === Subject: Re: Cantor Confusion > As you stated Goes on forever is a property of the natural >> numbers. The set {1,2,3,...} is just the natural numbers. So >> this set must go on forever. Being admittedly not very familiar with set theory, I nonetheless wonder >> if sets are considered like something going on forever. Sequences do, sets do not. When one represents the members of a set as members of a sequence, as > {1,2,3,...} does, that little ambiguity should not really mislead anyone. That little ambiguity? Doesn't it make a categorical difference if a set > has been a priori set for good? When one speaks of the sequence {1,2,3,...} and another speaks of the set {1,2,3,...}, the notational ambiguity should not override the words sequence and set. === Subject: Re: Cantor Confusion > As you stated Goes on forever is a property of the natural > numbers. The set {1,2,3,...} is just the natural numbers. So > this set must go on forever. > Eckard> Being admittedly not very familiar with set theory, I nonetheless wonder > if sets are considered like something going on forever. >> William>> Sequences do, sets do not. >> Virgil>> When one represents the members of a set as members of a sequence, as >> {1,2,3,...} does, that little ambiguity should not really mislead anyone. >> Eckard>> That little ambiguity? Doesn't it make a categorical difference if a set >> has been a priori set for good? > Virgil> When one speaks of the sequence {1,2,3,...} and another speaks of the > set {1,2,3,...}, the notational ambiguity should not override the words > sequence and set. Perhaps you did not get or at least did not accept my point: Does it not make a categorical difference whether a set is something unchanging beeing already perfect for good or something that just formally includes the vital property of the series of nagtural numbers to have no end at all? The notation {1, 2, 3, ...} is ambiguous with this respect. It depends on the decision whether ... denote potential or actual infinity. Meanwhile I see mounting evidence for my suspicion that set theory is some sort of (self?)-deception. It claims to rule continuity and incorporate the irrational ratios. However axiomatic set theory merely replaces Weierstrass's way to formalize Cauchy's idea. I do not deny that this is clever. However, the whole burden of Dedekind's cuts and Cantor's indeed naive fabrication of cardinality, cardinal and ordinal numbers, transfinite numbers and more than infinitely many numbers does not have any sound basis. Real numbers are defined in a misleading manner. Therefore present mathematical education hinders to understand my reasoning that the genuine continuum IR_g _can_ be symmetrically divided into IR_g+ and IR_g-. Eckard Blumschein, Magdeburg === Subject: Re: Cantor Confusion >> Eckard>> That little ambiguity? Doesn't it make a categorical difference >> if a set > has been a priori set for good? Virgil> When one speaks of the sequence {1,2,3,...} and another speaks >> of the >> set {1,2,3,...}, the notational ambiguity should not override the words >> sequence and set. Perhaps you did not get or at least did not accept my point: >> Does it not make a categorical difference whether a set is something >> unchanging beeing already perfect for good or something that just >> formally includes the vital property of the series of nagtural numbers >> to have no end at all? >> The notation {1, 2, 3, ...} is ambiguous with this respect. It depends >> on the decision whether ... denote potential or actual infinity. Are the labels sequence and set ambiguous to EB? If so perhaps he is > just muddled. The ambiguity resides within the three points ... They may denote either actual or potential infinity. >> Meanwhile I see mounting evidence for my suspicion that set theory is >> some sort of (self?)-deception. EB sees what he tells himself he should see, regardless of whether there > is anything there to see of not. Those who follow the discussion will not immediately change their opinion but should have a chance for doing so. === Subject: Re: Cantor Confusion > >> Eckard>> That little ambiguity? Doesn't it make a categorical difference >> if a set > has been a priori set for good? Virgil> When one speaks of the sequence {1,2,3,...} and another speaks >> of the >> set {1,2,3,...}, the notational ambiguity should not override the words >> sequence and set. Perhaps you did not get or at least did not accept my point: >> Does it not make a categorical difference whether a set is something >> unchanging beeing already perfect for good or something that just >> formally includes the vital property of the series of nagtural numbers >> to have no end at all? >> The notation {1, 2, 3, ...} is ambiguous with this respect. It depends >> on the decision whether ... denote potential or actual infinity. Are the labels sequence and set ambiguous to EB? If so perhaps he is > just muddled. The ambiguity resides within the three points ... > They may denote either actual or potential infinity. >> Meanwhile I see mounting evidence for my suspicion that set theory is >> some sort of (self?)-deception. EB sees what he tells himself he should see, regardless of whether there > is anything there to see of not. Those who follow the discussion will not immediately change their > opinion but should have a chance for doing so. If one has a choice between a red hat and a green hat, merely saying hat does not avoid ambiguity, but prefixing hat with red or green does. Thus sequence {1,2,3...} or set [1,2,3...} are no more ambiguous that red hat or green hat. === Subject: Re: Cantor Confusion >> The ambiguity resides within the three points ... >> They may denote either actual or potential infinity. >> >Meanwhile I see mounting evidence for my suspicion that set theory is > some sort of (self?)-deception. EB sees what he tells himself he should see, regardless of whether there >> is anything there to see of not. Those who follow the discussion will not immediately change their >> opinion but should have a chance for doing so. If one has a choice between a red hat and a green hat, merely saying > hat does not avoid ambiguity, but prefixing hat with red or > green does. Thus sequence {1,2,3...} or set [1,2,3...} are no more ambiguous > that red hat or green hat. As ambiguous as hat in the sense of as many red as you like and hat in the sense the quality green includes something irrational: all of indefinitely much. === Subject: Re: Cantor Confusion > >> The ambiguity resides within the three points ... >> They may denote either actual or potential infinity. >> >Meanwhile I see mounting evidence for my suspicion that set theory is > some sort of (self?)-deception. EB sees what he tells himself he should see, regardless of whether there >> is anything there to see of not. Those who follow the discussion will not immediately change their >> opinion but should have a chance for doing so. If one has a choice between a red hat and a green hat, merely saying > hat does not avoid ambiguity, but prefixing hat with red or > green does. Thus sequence {1,2,3...} or set [1,2,3...} are no more ambiguous > that red hat or green hat. As ambiguous as hat in the sense of as many red as you like and hat > in the sense the quality green includes something irrational: all of > indefinitely much. It would appear that EB cannot distinguish between red hats and green ones. Not only innumerate but colorblind. === Subject: Re: Cantor Confusion >> As ambiguous as hat in the sense of as many red as you like and hat >> in the sense the quality green includes something irrational: all of >> indefinitely much. It would appear that EB cannot distinguish between red hats and green > ones. Not only innumerate but colorblind. Not enough phantasy? === Subject: Re: Cantor Confusion > >> As ambiguous as hat in the sense of as many red as you like and hat >> in the sense the quality green includes something irrational: all of >> indefinitely much. It would appear that EB cannot distinguish between red hats and green > ones. Not only innumerate but colorblind. Not enough phantasy? EB had not enough carrots as a child. === Subject: Re: Cantor Confusion <456304B0.70705@et.uni-magdeburg.de> if sets are considered like something going on forever. Sequences do, sets do not. > Sequences are sets. === Subject: Re: Cantor Confusion > Being admittedly not very familiar with set theory, I nonetheless wonder > if sets are considered like something going on forever. Sequences do, sets do not. Sequences are sets. > Sequences are functions, a very special sort of set, those with domain N. Since not all sets are sequences, not all sets have the properties that sequences must have in order to be sequences. === Subject: Re: Cantor Confusion > Being admittedly not very familiar with set theory, I nonetheless wonder > if sets are considered like something going on forever. Sequences do, sets do not. Sequences are sets. > Sequences are functions, a very special sort of set, those with domain > N. Since not all sets are sequences, not all sets have the properties that > sequences must have in order to be sequences. Have you lost the rest of your mind? I said all sequences are sets, but not all sets are sequences. === Subject: Re: Cantor Confusion > Being admittedly not very familiar with set theory, I nonetheless > wonder > if sets are considered like something going on forever. Sequences do, sets do not. Sequences are sets. > Sequences are functions, a very special sort of set, those with domain > N. Since not all sets are sequences, not all sets have the properties that > sequences must have in order to be sequences. Have you lost the rest of your mind? I said all sequences are sets, > but not all sets are sequences. I have not lost anything, though WM seems to have. I did not contradict what you said, I merely expanded upon it. === Subject: Re: Cantor Confusion >> Being admittedly not very familiar with set theory, I nonetheless wonder >> if sets are considered like something going on forever. Sequences do, sets do not. >Sequences are sets. Perhaps you correctly learned set theory. So I guess, William Hughes was correct and Virgil this time wrong. I conclude, all elements of an infinite sets are a priori set for good while simultaneously the process of setting is going on forever. Hm, pretty strange. Eckard === Subject: Re: Cantor Confusion >> Being admittedly not very familiar with set theory, I nonetheless wonder >> if sets are considered like something going on forever. Sequences do, sets do not. >Sequences are sets. Perhaps you correctly learned set theory. So I guess, William Hughes was > correct and Virgil this time wrong. While all sequences are sets ( as functions from N to some set of values) not all sets are sequences. === Subject: Re: Cantor Confusion > Eckard> Being admittedly not very familiar with set theory, I nonetheless wonder > if sets are considered like something going on forever. > Virgil> Sequences do, sets do not. > WM>> Sequences are sets. >> Eckard>> Perhaps you correctly learned set theory. So I guess, William Hughes was >> correct and Virgil this time wrong. > Virgil> While all sequences are sets ( as functions from N to some set of > values) not all sets are sequences. So you could be correct if referring to sets that are no sequences? Look above: I referred to something going on forever. === Subject: Re: Cantor Confusion >> Virgil> While all sequences are sets ( as functions from N to some set of >> values) not all sets are sequences. So you could be correct if referring to sets that are no sequences? >> Look above: I referred to something going on forever. Such as EB's nonsensical utterances about that of which he displays such > great ignorance? Sorry, the factual question of concern is perhaps meanwhile missing, replaced by nonsense and ignorance. Let's stop now. === Subject: Re: Cantor Confusion >> Virgil> While all sequences are sets ( as functions from N to some set of >> values) not all sets are sequences. So you could be correct if referring to sets that are no sequences? >> Look above: I referred to something going on forever. Which forever? In time, in space, or in something else. As neither time not space, in any physical sense, exists in mathematics, it must be in something, as yet unexplained, else. === Subject: Re: Cantor Confusion > >> >Virgil> While all sequences are sets ( as functions from N to some set of > values) not all sets are sequences. So you could be correct if referring to sets that are no sequences? > Look above: I referred to something going on forever. > Which forever? In time, in space, or in something else. Neither nor. Counting is just a repetitious operation. As neither time not space, in any physical sense, exists in mathematics, > it must be in something, as yet unexplained, else. Not unexplained but best imagined via the application on time or space. === Subject: Re: Cantor Confusion > >> >Virgil> While all sequences are sets ( as functions from N to some set > of > values) not all sets are sequences. So you could be correct if referring to sets that are no sequences? > Look above: I referred to something going on forever. > Which forever? In time, in space, or in something else. Neither nor. Counting is just a repetitious operation. > As neither time not space, in any physical sense, exists in mathematics, > it must be in something, as yet unexplained, else. Not unexplained but best imagined via the application on time or space. Certainly unexplained by EB. And one can easily imagine it as, say, as the succession of points (n-1)/n in the rational open interval (0,1) which is completed in the rational interval [0,1]. === Subject: Re: Cantor Confusion >> Not unexplained but best imagined via the application on time or space. Certainly unexplained by EB. > And one can easily imagine it as, say, as the succession of points > (n-1)/n in the rational open interval (0,1) which is completed in the > rational interval [0,1]. Just replacing ) by ] seems to be quite easy and natural. In my understanding, with really real numbers there is no difference between ] and ] at all. === Subject: Re: Cantor Confusion > >> Not unexplained but best imagined via the application on time or space. Certainly unexplained by EB. > And one can easily imagine it as, say, as the succession of points > (n-1)/n in the rational open interval (0,1) which is completed in the > rational interval [0,1]. Just replacing ) by ] seems to be quite easy and natural. > In my understanding, with really real numbers there is no difference > between ] and ] at all. No one is claiming a difference between ] and ]. But in , say, Dedekind cuts, there is a distinguishable difference between (0,1) and [0,1], based on whether those sets contain or do not contain their LUBs and GLBs as members. === Subject: Re: Cantor Confusion > >> Being admittedly not very familiar with set theory, I nonetheless >> wonder if sets are considered like something going on forever. Sequences do, sets do not. >Sequences are sets. Is the identical sequence n |-> n a set, too? F. N. -- xyz === Subject: Re: Cantor Confusion <456304B0.70705@et.uni-magdeburg.de> <456454a5$0$97224$892e7fe2@authen.yellow.readfreenews.net > Being admittedly not very familiar with set theory, I nonetheless >> wonder if sets are considered like something going on forever. >> Sequences do, sets do not. > Sequences are sets. Is the identical sequence n |-> n a set, too? In ZF and ZFC everything is a set. === Subject: Re: Cantor Confusion > Being admittedly not very familiar with set theory, I > nonetheless wonder if sets are considered like something going > on forever. >> Sequences do, sets do not. > Sequences are sets. >> Is the identical sequence n |-> n a set, too? In ZF and ZFC everything is a set. Can you give us a representation of the identical sequence in terms of sets? F. N. -- xyz === Subject: Re: Cantor Confusion Am Tue, 21 Nov 2006 14:52:48 +0100 schrieb Eckard Blumschein: > Being admittedly not very familiar with set theory, I nonetheless wonder > if sets are considered like something going on forever. You are in special not very familiar with mathematics at all. === Subject: Re: Cantor Confusion > As infinitely many paths pass through each edge no path can own more > than an infinitesimal share of that edge according to that equal > sharing rule. And unless we are in something like Robinson's > non-standard analysis, less than infinitesimal means zero. I wonder. I have to agree with Virgil. Is there something wrong? === Subject: Re: Cantor Confusion As infinitely many paths pass through each edge no path can own more > than an infinitesimal share of that edge according to that equal > sharing rule. And unless we are in something like Robinson's > non-standard analysis, less than infinitesimal means zero. I wonder. I have to agree with Virgil. Is there something wrong? Just because you are wrong in your main points doesn't mean you can't sometimes be right when WM is wrong. It is hard to be always wrong. -- David Marcus === Subject: Re: Cantor Confusion > It is hard to be always wrong. In what, according to your opinion, WM is not wrong? === Subject: Re: Cantor Confusion As infinitely many paths pass through each edge no path can own more > than an infinitesimal share of that edge according to that equal > sharing rule. And unless we are in something like Robinson's > non-standard analysis, less than infinitesimal means zero. I wonder. I have to agree with Virgil. Is there something wrong? We will just have to be more careful in future, I suppose. === Subject: Re: Cantor Confusion WM again uses number where we use set. We have a SET which contains all of the /finite/ naturals or all of the > /finite/ ordinals. Yes, modern set-mathematicians have their Gruenderzeit belief. Gruenderzeit was the time after a won war against France when Germany was reunited for the first time and much money out of French colonies together with growing industry boosted anything including genuine science as well as Cantorian fiction. === Subject: Re: Cantor Confusion > WM again uses number where we use set. We have a SET which contains all of the /finite/ naturals or all of the > /finite/ ordinals. Yes, modern set-mathematicians have their Gruenderzeit belief. Modern anti-set non-mathematicians claim to know what set theory should be and what mathematics should be without knowing damn all about either. === Subject: Re: Cantor Confusion > But note, to define an integer, the definition must be > communicated. We have only a finite number of bits > to use for this communication (this includes comunicating > the details of any compression scheme). So there are > only a limited number of integers that it is possible to define. > Among these is the largest integer that it is possible to > define. If we claim that the only integers that exist are the ones > that have been or will be defined, > then there is a largest possible integer. > > - William Hughes Having looked around and realized a lot of impolite and opiniated persons, I consider you, William Hughes, like a perhaps rare positive exception. Hopefully you will not mistake me if I do no longer hesitate to nonetheless comment on your reasoning. While I do not question your last sentence being a correct conclusion if the premise holds, I cannot not see any possibility to reasonably and concisely quantify a largest possible number. Do we really have to define numbers? I tend to be satisfied when I have an executable rule which may create as many natural numbers as I like, as does Archimede's axiom. Executable means effectively about the same as communicable. It excludes postulated a priori existence. Do not get me wrong. I consider the properties executable and communicable like belonging to the abstract mathematical idea, not to any physical limitation to it. The process of counting indefinitely is never completely executable. Accordingly all created numbers are finite, and there is no principial reason for a largest possible number to be found. Eckard Blumschein === Subject: Re: Cantor Confusion > But note, to define an integer, the definition must be > communicated. We have only a finite number of bits > to use for this communication (this includes comunicating > the details of any compression scheme). So there are > only a limited number of integers that it is possible to define. > Among these is the largest integer that it is possible to > define. If we claim that the only integers that exist are the ones > that have been or will be defined, > then there is a largest possible integer. - William Hughes Having looked around and realized a lot of impolite and opiniated > persons, I consider you, William Hughes, like a perhaps rare positive > exception. Hopefully you will not mistake me if I do no longer hesitate to > nonetheless comment on your reasoning. > While I do not question your last sentence being a correct conclusion if > the premise holds, I cannot not see any possibility to reasonably and > concisely quantify a largest possible number. Do we really have to define numbers? I tend to be satisfied when I have > an executable rule which may create as many natural numbers as I like, > as does Archimede's axiom. Executable means effectively about the same > as communicable. It excludes postulated a priori existence. Do not get me wrong. I consider the properties executable and > communicable like belonging to the abstract mathematical idea, not to > any physical limitation to it. The process of counting indefinitely is > never completely executable. Accordingly all created numbers are finite, > and there is no principial reason for a largest possible number to be > found. This is of course the standard position. It is not WM's position. On the larger question of whether the set of all natural numbers exists. Your executable rule defines what is commonly referred to as potential infinity. The problem is the difference between saying The set of all natural numbers exists and We can generate an arbitrarily large set of natural numbers (i.e. the difference between actual and potential infinity) is mostly one of terminology. There is no element of the set of natural numbers that does not exist as an element of some arbitrarily large set. So there is absolutely no difference between the elements of the infinite set of natural numbers and the elements of the potentially infinite set of natural numbers. About the only thing you can say about the potentially infinite set of natural numbers is that it is not a set. So what? You still have this potential, this thing that allows you to get new natural numbers whenever you want. This potential has all the properties of a set but the name. - William Hughes === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de The problem is the difference between saying The set of all natural numbers exists > and > We can generate an arbitrarily large set > of natural numbers (i.e. the difference between actual and potential infinity) is > mostly one of terminology. No. The first can be disproved, the second cannot be disproved. About the only thing you can say about the > potentially infinite set of natural numbers is that it is > not a set. So what? You still have this potential, > this thing that allows you to get new natural numbers > whenever you want. To produce or construct them, not to take them from the shelf. === Subject: Re: Cantor Confusion The problem is the difference between saying The set of all natural numbers exists > and > We can generate an arbitrarily large set > of natural numbers (i.e. the difference between actual and potential infinity) is > mostly one of terminology. No. The first can be disproved, the second cannot be disproved. In what axiom system can the first be disproved? === Subject: Re: Cantor Confusion For the first time I do not feel unnecessarily lectured and deliberately misunderstood. >> While I do not question your last sentence being a correct conclusion if >> the premise holds, I cannot not see any possibility to reasonably and >> concisely quantify a largest possible number. >> Do we really have to define numbers? I tend to be satisfied when I have >> an executable rule which may create as many natural numbers as I like, >> as does Archimede's axiom. Executable means effectively about the same >> as communicable. It excludes postulated a priori existence. >> Do not get me wrong. I consider the properties executable and >> communicable like belonging to the abstract mathematical idea, not to >> any physical limitation to it. The process of counting indefinitely is >> never completely executable. Accordingly all created numbers are finite, >> and there is no principial reason for a largest possible number to be >> found. This is of course the standard position. It is not WM's position. I just described my position and wonder if it is the standard one. > On the larger question of whether the set of > all natural numbers exists. Your executable rule defines what is commonly referred to > as potential infinity. Exactly. > The problem is the difference between saying The set of all natural numbers exists This is what I call Dedekind/Cantor Utopia. Elsewhere i tried to explain why it deals with sets instead of numbers. > and > We can generate an arbitrarily large set > of natural numbers Arbitrarily large is according to Cantor not really infinite. I share this view of him. > (i.e. the difference between actual and potential infinity) is > mostly one of terminology. Yes, but I got aware that actual and potential infinity are two different views both directed towards the same object of natural numbers from different levels of abstraction. The potential infinite view belongs to counting and is able to separate from each other all single numbers. The actual infinite one provides the fiction of all natural numbers at a higher level of abstraction. You cannot have both views at a time. > There is no element of the set > of natural numbers that does not exist as an element of some > arbitrarily large set. So there is absolutely no difference between > the > elements of the infinite set of natural numbers and the elements > of the potentially infinite set of natural numbers. This is what I meant when writing above the same object. > About the only thing you can say about the > potentially infinite set of natural numbers is that it is > not a set. So what? You still have this potential, > this thing that allows you to get new natural numbers > whenever you want. This potential has > all the properties of a set but the name. And the elusive belief hidden within and conveyed by this name. Aren't I correct? Eckard Blumschein - William Hughes === Subject: Re: Cantor Confusion For the first time I do not feel unnecessarily lectured and deliberately > misunderstood. > While I do not question your last sentence being a correct conclusion if >> the premise holds, I cannot not see any possibility to reasonably and >> concisely quantify a largest possible number. >> Do we really have to define numbers? I tend to be satisfied when I have >> an executable rule which may create as many natural numbers as I like, >> as does Archimede's axiom. Executable means effectively about the same >> as communicable. It excludes postulated a priori existence. >> Do not get me wrong. I consider the properties executable and >> communicable like belonging to the abstract mathematical idea, not to >> any physical limitation to it. The process of counting indefinitely is >> never completely executable. Accordingly all created numbers are finite, >> and there is no principial reason for a largest possible number to be >> found. This is of course the standard position. It is not WM's position. I just described my position and wonder if it is the standard one. > On the larger question of whether the set of > all natural numbers exists. Your executable rule defines what is commonly referred to > as potential infinity. Exactly. The problem is the difference between saying The set of all natural numbers exists This is what I call Dedekind/Cantor Utopia. Elsewhere i tried to explain > why it deals with sets instead of numbers. and > We can generate an arbitrarily large set > of natural numbers Arbitrarily large is according to Cantor not really infinite. I share > this view of him. > (i.e. the difference between actual and potential infinity) is > mostly one of terminology. Yes, but I got aware that actual and potential infinity are two > different views both directed towards the same object of natural numbers > from different levels of abstraction. The potential infinite view > belongs to counting and is able to separate from each other all single > numbers. The actual infinite one provides the fiction of all natural > numbers at a higher level of abstraction. You cannot have both views at > a time. > Why not. There is nothing about assuming the set of all natural numbers exists that precludes counting or the ability to sepatate from each other all single numbers. The elements of the potentially infinite set are exactly the same elements with exactly the same properties as the elements of the actually infinite set. > There is no element of the set > of natural numbers that does not exist as an element of some > arbitrarily large set. So there is absolutely no difference between > the > elements of the infinite set of natural numbers and the elements > of the potentially infinite set of natural numbers. This is what I meant when writing above the same object. About the only thing you can say about the > potentially infinite set of natural numbers is that it is > not a set. So what? You still have this potential, > this thing that allows you to get new natural numbers > whenever you want. This potential has > all the properties of a set but the name. And the elusive belief hidden within and conveyed by this name. > Aren't I correct? > No. You can develop potential set theory. Just define an element of a potential set to be an element of any one of the arbitrary sets that can be produced. Now go through set theory and add the word potential in front of each occurence of the word set. There is no difference between saying a potentially infinite set exists and saying an actually infinite set exists. - William Hughes === Subject: Re: Cantor Confusion >> (i.e. the difference between actual and potential infinity) is >> mostly one of terminology. >> Yes, but I got aware that actual and potential infinity are two >> different views both directed towards the same object of natural numbers >> from different levels of abstraction. The potential infinite view >> belongs to counting and is able to separate from each other all single >> numbers. The actual infinite one provides the fiction of all natural >> numbers at a higher level of abstraction. You cannot have both views at >> a time. Why not. There is nothing about assuming the set of all natural > numbers exists that precludes counting or the ability to sepatate > from each other all single numbers. Actual infinity is not as handsome as it seems to be. > The elements of the > potentially infinite set are exactly the same elements with > exactly the same properties as the elements of the > actually infinite set. I object to exactly this assumption. Actual infinity denotes a diffent quality, not a larger quality, not a quantity at all. >> And the elusive belief hidden within and conveyed by this name. >> Aren't I correct? No. You can develop potential set theory. Quite a while ago, it was a surprize to me: Axiomatic set theory does not really require the actual infinity. > Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. Yes. This is indeed a clever method of obscuration. Nonetheless, if one prefers to distinguish between rational and real numbers, then the reals are only distinguished by being fictitious and therefore uncountable. Genuine reals being really real are only required for theoretical considerations. They do however, have properties quite different from the properties of genuine numbers. This is my original concern. After I already settled the somewhat puzzling matter by means of really real numbers, I took me some time to understand the relationship to the still thaught naive set theory, axiomatic set theory and what a comprehensive understanding of mathematics really needs. Eckard Blumschein - William Hughes > === Subject: Re: Cantor Confusion >> (i.e. the difference between actual and potential infinity) is >> mostly one of terminology. >> Yes, but I got aware that actual and potential infinity are two >> different views both directed towards the same object of natural numbers >> from different levels of abstraction. The potential infinite view >> belongs to counting and is able to separate from each other all single >> numbers. The actual infinite one provides the fiction of all natural >> numbers at a higher level of abstraction. You cannot have both views at >> a time. Why not. There is nothing about assuming the set of all natural > numbers exists that precludes counting or the ability to sepatate > from each other all single numbers. Actual infinity is not as handsome as it seems to be. Non sequitur Do you have any argument to support: assuming that the set of all natural numbers exists precludes counting and/or the ability to separate from each other all single numbers. The elements of the > potentially infinite set are exactly the same elements with > exactly the same properties as the elements of the > actually infinite set. I object to exactly this assumption. Actual infinity denotes a diffent > quality, not a larger quality, not a quantity at all. Again a non sequitur. >> And the elusive belief hidden within and conveyed by this name. >> Aren't I correct? No. You can develop potential set theory. Quite a while ago, it was a surprize to me: Axiomatic set theory does > not really require the actual infinity. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. Yes. This is indeed a clever method of obscuration. Obfuscation of what? Do you agree with the statement: There is no difference between saying a potentially infinite set exists and saying an actually infinite set exists. or not? If not, do you have anything even remotely relevent to say? > Nonetheless, if one > prefers to distinguish between rational and real numbers, then the reals > are only distinguished by being fictitious and therefore uncountable. > Genuine reals being really real are only required for theoretical > considerations. They do however, have properties quite different from > the properties of genuine numbers. This is my original concern. After I > already settled the somewhat puzzling matter by means of really real > numbers, I took me some time to understand the relationship to the still > thaught naive set theory, axiomatic set theory and what a comprehensive > understanding of mathematics really needs. > This semi-coherent ramble has nothing whatsoever to do with the question of whether there are any real (as opposed to cosmetic) differences between potentially infinite and actually infinite sets. If you do not wish to discuss this question fine, but a simple statement to that effect would have been far preferable to the above. (Even simply not replying would have been preferable.) - William Hughes === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de> <4564756D.3070004@et.uni-magdeburg.de There is no difference between > saying a potentially infinite set exists and saying an > actually > infinite set exists. or not? If not, do you have anything even remotely relevent to say? This semi-coherent ramble has nothing whatsoever to do with > the question of whether there are any real (as opposed to cosmetic) > differences between potentially infinite and actually infinite sets. Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for ñactual infinity.î The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets. [Thomas Jech, Set Theory Stanford.htm, Stanford Encyclopedia of Philosophy] [Brouwer] maintains that a veritable continuum which is not denumerable can be obtained as a medium of free development; that is to say, besides the points which exist (are ready) on account of their definition by laws, such as e, pi, etc. other points of the continuum are not ready but develop as so-called choice sequences. [Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: Foundations of Set Theory, North Holland, Amsterdam (1984) p. 255] There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity. [Edward Nelson: Completed versus incomplete infinity in arithmetic, Princeton] A viable and interesting alternative to regarding the numbers as a completed infinity, one that leads to great simplifications in some areas of mathematics and that has strong connections with problems of computational complexity. [Edward Nelson: Completed versus incomplete infinity in arithmetic, Princeton] Wenn man die verschiedenen Ansichten, welche sich in bezug auf unsern Gegenstand, das Aktual-Unendliche (im folgenden K.9frze halber mit A.-U. bezeichnet), im Laufe der Geschichte geltend gemacht haben, .9fbersichtlich gruppieren will, so bieten sich dazu mehrere Gesichtspunkte dar, von denen ich heute nur einen hervorheben m.9achte. Man kann n.8amlich das A.-U. in drei Hauptbeziehungen in Frage stellen: erstens, sofern es in Deo extramundano aeterno omnipotenti sive natura naturante, wo es das Absolute hei¤t, zweitens sofern es in concreto seu in natura naturata vorkommt, wo ich es Transfinitum nenne und drittens kann das A.-U. in abstracto in Frage gezogen werden, d. h. sofern es von der menschlichen Erkenntnis in Form von aktual-unendlichen, oder wie ich sie genannt habe, von transfiniten Zahlen oder in der noch allgemeineren Form der transfiniten Ordnungstypen ??????????????? oder ???????????aufgefa¤t werden k.9anne. [G. Cantor, Gesammelte Anhandlungen, p. 372 ] Da¤ das sogenannte potentiale oder synkategorematische Unendliche (Indefinitum) zu keiner derartigen Einteilung Veranlassung gibt, hat darin seinen Grund, da¤ es ausschlie¤lich als Beziehungsbegriff, als Hilfsvorstellung unseres Denkens Bedeutung hat, f.9fr sich aber keine Idee bezeichnet; in jener Rolle hat es allerdings durch die von Leibniz und Newton erfundene Differential- und Integralrechnung seinen gro¤en Wert als Erkenntnismittel und Instrument unseres Geistes bewiesen; eine weitergehende Bedeutung kann dasselbe nicht f.9fr sich in Anspruch nehmen. [G. Cantor, Gesammelte Anhandlungen, p. 373] Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und aktualen Unendlichen, indem ersteres eine ver.8anderliche endliche, .9fber alle Grenzen hinaus wachsende Gr.9a¤e, letztere ein in sich festes, konstantes, jedoch jenseits aller endlichen Gr.9a¤en liegendes Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, da¤ das eine mit dem andern verwechselt wird. [G. Cantor, Gesammelte Anhandlungen, p. 374] Summary: It is incoherent ramble to believe that the potential infinite yields order types or bijections. (free translation after Cantor) === Subject: Re: Cantor Confusion >> Actual infinity is not as handsome as it seems to be. Non sequitur Do you have any argument to support: assuming > that the set of all natural numbers exists precludes counting > and/or the ability to separate from each other all single numbers. Too many laws correspond to an absolute lack of any obedience. In more formal symbolic description (a=any number): oo * a = oo oo + a = oo oo + 0 = oo oo * 0 = a In other words: Infinity, i.e. the actual indefinitely large, must not be used like a number. In order to have an readable order a, one has to restrict the number of included elements to b if the smallest readable interval is c: b*c=a. For b=oo the only reasonable c equals 0. >> The elements of the >> potentially infinite set are exactly the same elements with >> exactly the same properties as the elements of the >> actually infinite set. >> I object to exactly this assumption. Actual infinity denotes a diffent >> quality, not a larger quality, not a quantity at all. Again a non sequitur. Of course. My objection does not follow from your belief. The definition of infinity like a quality, something that can neither be enlaged nor exhausted is very basic and does not have a reasonable alternative. > And the elusive belief hidden within and conveyed by this name. > Aren't I correct? >> No. You can develop potential set theory. >> Quite a while ago, it was a surprize to me: Axiomatic set theory does >> not really require the actual infinity. >> Just define >> an element of a potential set to be an element of any one >> of the arbitrary sets that can be produced. Now go through >> set theory and add the word potential in front of each >> occurence of the word set. There is no difference between >> saying a potentially infinite set exists and saying an actually >> infinite set exists. >> Yes. This is indeed a clever method of obscuration. Obfuscation of what? Hiding of the categorical difference between discrete and contiuous, also expresses like genuine (i.e. rational) number and really real number, also expressed like countable and uncountable. > Do you agree with the statement: > There is no difference between saying a potentially infinite set exists > and saying an actually infinite set exists. or not? Before I may decide I would like to clarify the meaning of the term set. Cantor's definition of a set has been confessed invalid at least since 1923 with no possiblity of correction in case of infinite sets. Why? Fraenkel confessed paradoxa due to this definition. The reason behind them is ambiguity. Cantor's definition claims to provide each singe element of the set as well as simultaneously the infinite set as an entity. So it is a chimera. If not, do you have anything even remotely relevent to say? Definitely. >> Nonetheless, if one >> prefers to distinguish between rational and real numbers, then the reals >> are only distinguished by being fictitious and therefore uncountable. >> Genuine reals being really real are only required for theoretical >> considerations. They do however, have properties quite different from >> the properties of genuine numbers. This is my original concern. After I >> already settled the somewhat puzzling matter by means of really real >> numbers, I took me some time to understand the relationship to the still >> thaught naive set theory, axiomatic set theory and what a comprehensive >> understanding of mathematics really needs. This semi-coherent ramble Semi-understandable to you. What did you not understand? > has nothing whatsoever to do with > the question of whether there are any real (as opposed to cosmetic) > differences between potentially infinite and actually infinite sets. As long as the mathematical notion of a set lacks a valid definition, I regrett: One cannot utter something reliable concerning infinite sets. While Fraenkel 1923 clearly explained the notions potential and actual infinity, he did neither use potentially infinite set nor actually infinite set. Set theory is de facto based on the illusion that there is no difference between number and real number. > If you do not wish to discuss this question fine, but a simple > statement > to that effect would have been far preferable to the above. > (Even simply not replying would have been preferable.) I cannot warrant that I am able to always reply since the discussion has been grown to an extent exceeding my time recources. Therefore do not interprete a lacking reply of mine a sign of surrender. Eckard Blumschein === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de> <4564756D.3070004@et.uni-magdeburg.de> <45658369.3090702@et.uni-magdeburg.de > Before I may decide I would like to clarify the meaning of the term set. > Cantor's definition of a set has been confessed invalid at least since > 1923 with no possiblity of correction in case of infinite sets. Why? Fraenkel confessed paradoxa due to this definition. The reason behind > them is ambiguity. Cantor's definition claims to provide each singe > element of the set as well as simultaneously the infinite set as an > entity. So it is a chimera. > Why? What is there about providing the infinite set as an entitiy which precludes providing each single element of the set? - William Hughes === Subject: Re: Cantor Confusion >> Before I may decide I would like to clarify the meaning of the term set. >> Cantor's definition of a set has been confessed invalid at least since >> 1923 with no possiblity of correction in case of infinite sets. Why? >> Fraenkel confessed paradoxa due to this definition. The reason behind >> them is ambiguity. Cantor's definition claims to provide each singe >> element of the set as well as simultaneously the infinite set as an >> entity. So it is a chimera. Why? What is there about providing the infinite set as an entitiy > which precludes providing each single element of the set? We may suspect but not dirctly verfy the existence of each element. It can never be reached. === Subject: Re: Cantor Confusion No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in bijection with another infinite set because it does not exist completely. === Subject: Re: Cantor Confusion No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. Not true in any extant axiom system. In every axiom system that has been presented so far, there are finite sets and possibly infinite sets, but nothing in between. Something which is potentially, but not actually, infinite cannot be a set at all in any set theory yet seen here. === Subject: Re: Cantor Confusion > Something which is potentially, but not actually, > infinite cannot be a set at all in any set theory yet seen here. Isn't there consensus that the set IN of natural numbers is countable? How do you imagine bijection with a actually infinite set? Notice: Actual infinity is a Gedankending something fictitious. I do not say it is unconceivable or nonsense. It is just unapproachable. Moreove, potentially and actually infinite are mutually excluding points of view. === Subject: Re: Cantor Confusion Something which is potentially, but not actually, > infinite cannot be a set at all in any set theory yet seen here. Isn't there consensus that the set IN of natural numbers is countable? How do you imagine bijection with a actually infinite set? > Notice: Actual infinity is a Gedankending something fictitious. Al numbers are Gedankendingens (if the is the right plural). > I do not say it is unconceivable or nonsense. It is just unapproachable. I t is conceivable as it has been conceived. Moreove, potentially and actually infinite are mutually excluding points > of view. So one can round file one of them. Mathematicians, by and large, round file the first, anti-mathematicians sometimes round file both. === Subject: Re: Cantor Confusion >> How do you imagine bijection with a actually infinite set? >> Notice: Actual infinity is a Gedankending something fictitious. > Al numbers are Gedankendingens (if the is the right plural). The correct plural would be Gedankendinge. Fraenkel (p. 6) referiert Cantor: das als reines Gedankending offenbar nichts widerspruchsvolles in sich birgt. Indeed, the actually infinite set is selfcontradictory. > >> I do not say it is unconceivable or nonsense. It is just unapproachable. I t is conceivable as it has been conceived. Moreover, potentially and actually infinite are mutually excluding points >> of view. So one can round file one of them. Mathematicians, by and large, round > file the first, anti-mathematicians sometimes round file both. I do not understand your idioms round file and by and large. Notice. Mathematics has to do with potential infinity when dealing with genuine numbers. But strictly speaking it deals with the actual infinity when considering real numbers and the continuum. === Subject: Re: Cantor Confusion > >> How do you imagine bijection with a actually infinite set? >> Notice: Actual infinity is a Gedankending something fictitious. > Al numbers are Gedankendingens (if the is the right plural). The correct plural would be Gedankendinge. > Fraenkel (p. 6) referiert Cantor: das als reines Gedankending offenbar > nichts widerspruchsvolles in sich birgt. Indeed, the actually infinite set is selfcontradictory. Then all numbers are self contradictory. >> I do not say it is unconceivable or nonsense. It is just unapproachable. I t is conceivable as it has been conceived. Moreover, potentially and actually infinite are mutually excluding points >> of view. So one can round file one of them. Mathematicians, by and large, round > file the first, anti-mathematicians sometimes round file both. I do not understand your idioms round file and by and large. Next to my desk I have a round container in which I file things that I do not intend to keep. When it gets full, I transfer its contents to the trash. For by and large see +Large&sa=X&oi=glossary_definition&ct=title Notice. Mathematics has to do with potential infinity when dealing with > genuine numbers. But strictly speaking it deals with the actual infinity > when considering real numbers and the continuum. Notice: All numbers are equally fictitious. None of them are any more genuine than others. They all exist only in the mind. === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de> an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. Not true in any extant axiom system. In every axiom system that has been > presented so far, there are finite sets and possibly infinite sets, but > nothing in between. Something which is potentially, but not actually, > infinite cannot be a set at all in any set theory yet seen here. And just that is the reason why any set theory is waste. === Subject: Re: Cantor Confusion > > No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. Not true in any extant axiom system. In every axiom system that has been > presented so far, there are finite sets and possibly infinite sets, but > nothing in between. Something which is potentially, but not actually, > infinite cannot be a set at all in any set theory yet seen here. And just that is the reason why any set theory is waste. Not to mathematicians. === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de > No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. > Piffle. You need to study your potential-set theory. We have defined what it means to be an element of a potential set. Therefore we can define bijections between potentially infinite sets. Therefore we can define cardinalities of potentially infinite sets. - William Hughes === Subject: Re: Cantor Confusion >> No. You can develop potential set theory. Just define >> an element of a potential set to be an element of any one >> of the arbitrary sets that can be produced. Now go through >> set theory and add the word potential in front of each >> occurence of the word set. There is no difference between >> saying a potentially infinite set exists and saying an actually >> infinite set exists. >> A potentially infinite set has no cardinal number. It cannot be in >> bijection with another infinite set because it does not exist >> completely. Since I am convinced that Dedekind and Cantor were wrong, I avoid the expression cardinal number. Why do you not simply write instead: Something potentially infinite is countable because it is not thought to exhibit the impossible property to include all elements of something actually infinite? > Piffle. I recommend to avoid hurting words. >You need to study your potential-set theory. We have defined what it means to be an element of > a potential set. Therefore we can define bijections between potentially > infinite sets. If one considers something as actually infinite, then this point of view even hinders bijection. One cannot eat the cake and have it. potentially infinite and actually infinite point of view exclude each other as do discrete numbers and continuity. > Therefore we can define cardinalities of potentially infinite > sets. I feel in a position similar to that of an atheist who faces a believing child. Please forgive me my lack of belief. Yes there is Santa Claus. === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de> <45647A52.70908@et.uni-magdeburg.de> No. You can develop potential set theory. Just define >> an element of a potential set to be an element of any one >> of the arbitrary sets that can be produced. Now go through >> set theory and add the word potential in front of each >> occurence of the word set. There is no difference between >> saying a potentially infinite set exists and saying an actually >> infinite set exists. >> A potentially infinite set has no cardinal number. It cannot be in >> bijection with another infinite set because it does not exist >> completely. Since I am convinced that Dedekind and Cantor were wrong, I avoid the > expression cardinal number. Ok call the equivelence classes of the equivalence relation bijection the breadfruits. They still exist. > Why do you not simply write instead: > Something potentially infinite is countable because it is not thought to > exhibit the impossible property to include all elements of something > actually infinite? Because this is nonsense. By defininition a set A is countable if and only if there is a bijection between A and the natural numbers. Because of the way we have defined bijection and potentially infinite set it does not matter at all whether or not A is actually infinite when deciding if such a bijection exists. Piffle. I recommend to avoid hurting words. You need to study your potential-set theory. We have defined what it means to be an element of > a potential set. Therefore we can define bijections between potentially > infinite sets. If one considers something as actually infinite, then this point of view > even hinders bijection. One cannot eat the cake and have it. > potentially infinite and actually infinite point of view exclude each > other as do discrete numbers and continuity. > One cannot eat the cake and have it is not much of an argument. Do you have any other argument why you cannot have bijections between potentially infinite sets. - William Hughes === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de> <45647A52.70908@et.uni-magdeburg.de Why do you not simply write instead: > Something potentially infinite is countable because it is not thought to > exhibit the impossible property to include all elements of something > actually infinite? Because this is nonsense. By defininition a set A is countable > if and only if there is a bijection between A and the natural numbers. > Because of the way we have defined bijection and potentially > infinite set it does not matter at all whether or not A is actually > infinite when deciding if such a bijection exists. All you may have defined is actual infinity = potential infinity. A good example for a false definition. > One cannot eat the cake and have it is not much of an > argument. Do you have any other argument why you cannot > have bijections between potentially infinite sets. Because they are never complete. Try to inform you. Cantor is a very competent source. A potentially infinite variable is a variable which is always finite but can grow without bound. A potentially infinite set does not exist. === Subject: Re: Cantor Confusion >> No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. >> A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. >> Since I am convinced that Dedekind and Cantor were wrong, I avoid the >> expression cardinal number. Ok call the equivelence classes of the equivalence relation > bijection the breadfruits. They still exist. Already Galilei used bijection. Of course, bijection is feasable among genuine numbers without restrictions, no matter whether they constitute a finite set, the infinite series of natural numbers, rational numbers or the like. Therefore any mathematical object which is set together like a plurality of numbers is equivalent to the natural numbers. In other words: It is countable like peas in a cornet. Alternatively, powder in a cornet is equivalent to milk in a cornet in being uncountable. So we have two equivalence classes. I would like to abstain from the very risky preciptious claim that one of them is larger than the other. People who were less trained in abstraction may wonder the peas. My conterargument is: We consider powder like an ideal continuum. When I even reckon Dedekind and Cantor among those who adhere to the more intuitive argument claiming the existence of more powder than peas, I can quote indications for that on request. Let's summarize: We have two drawers, one mutually equivalent in being countable, the other one mutually equivalent in being uncountable. You suggested breadfruits, why not Alaaf? Citizen of Cologne used to shout Alaaf during carnival. Give them peas, and they have to shout Alaaf zero or simply Alaaf. Give them powder, and they have to shout Alaaf one. Have you any idea how to make them shouting Alaaf two? Three times Alaaf is possible: Alaaf! Alaaf! Alaaf! Just give them three times peas. However, what means Alaaf_two? Something can be false or correct. However, what does more than correct mean? >> Why do you not simply write instead: >> Something potentially infinite is countable because it is not thought to >> exhibit the impossible property to include all elements of something >> actually infinite? Because this is nonsense. Be cautious. > By defininition a set A is countable > if and only if there is a bijection between A and the natural numbers. Accepted. > Because of the way we Who? > have defined bijection and potentially > infinite set Cantor's definition of an infinite set has been withdrawn by Fraenkel without substitute because it was ambiguous with respect to the two mutually excluding points of view: actually and potentially. it does not matter at all whether or not A is actually > infinite when deciding if such a bijection exists. Proponents of set theory may hope that. >> If one considers something as actually infinite, then this point of view >> even hinders bijection. One cannot eat the cake and have it. >> potentially infinite and actually infinite point of view exclude each >> other as do discrete numbers and continuity. One cannot eat the cake and have it is not much of an > argument. It just illustrated the preceding sentence. > Do you have any other argument why you cannot > have bijections between potentially infinite sets. I did never deny bijection between natural numbers and any other plurality of genuine numbers. However, I came to the conclusion: There is no bijection between natural numbers, seen one by one, on one hand and the entity of all natural numbers, being understandable only like a fiction, on the other hand. In other words: While the realistic just potentially infinite set of IN is countable, the fictitious actually infinite set of IN is uncountable. Do not try to understand this from the perspective of set theory. Rather understand set theory a rather successful obscuration. Eckard Blumschein === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de > No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. > Piffle. You need to study your potential-set theory. We have defined what it means to be an element of > a potential set. We never have all elements available. Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. Therefore we can define cardinalities of potentially infinite > sets. Yes, we can define that a set is finite or that it is infinite, denoting the latter case conveniently by oo. That's all. === Subject: Re: Cantor Confusion > No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. > Piffle. You need to study your potential-set theory. We have defined what it means to be an element of > a potential set. We never have all elements available. Speak only for yourself, WM. Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. Speak only for yourself, WM. Therefore we can define cardinalities of potentially infinite > sets. Yes, we can define that a set is finite or that it is infinite, > denoting the latter case conveniently by oo. Speak only for yourself, WM. > That's all. And way too much. === Subject: Re: Cantor Confusion No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. > Piffle. You need to study your potential-set theory. We have defined what it means to be an element of > a potential set. We never have all elements available. Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. Piffle. Real numbers are represented by potentially infinite sequences. A list of reals is a function that takes an element of the potentially infinite set of natural numbers and returns a potentially infinite sequence. The diagonal number is a potentially infinite sequence. We show the diagonal number is not a member of the list in exactly the same way as before. Therefore we can define cardinalities of potentially infinite > sets. Yes, we can define that a set is finite or that it is infinite, > denoting the latter case conveniently by oo. That's all. > Defintion. B is a g-subset of a potentially infinite set A if B is a set or a potentially infinite set and any element of B is an element of A. We can now contruct potentially infinite power sets, and the usual theory follows. You are confusing potentially infinite with computable. While it is true that a computable set is either finite or potentially infinite, it is not true that a potentially infinite set is computable. - William Hughes === Subject: Re: Cantor Confusion Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. > Real numbers are represented by potentially infinite sequences. Wrong. Without actual infinity, there are no irrational numbers. Think about it. > A list of reals is a function that takes an element of the potentially > infinite > set of natural numbers and returns a potentially infinite sequence. > The diagonal number is a potentially infinite sequence. Wrong. Ask your set therorist companions. Defintion. B is a g-subset of a potentially > infinite set A if B is a set or a potentially infinite > set and any element of B is an element of A. We can > now contruct potentially infinite power sets, and > the usual theory follows. You are confusing potentially infinite with computable. > While it is true that a computable set is either > finite or potentially infinite, it is not true that a potentially > infinite set is computable. Try to inform you. Cantor is a very competent source. A potentially infinite variable is a variable which is always finite but can grow without bound. A potentially infinite set does not exist. === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. Real numbers are represented by potentially infinite sequences. Wrong. Without actual infinity, there are no irrational numbers. Think > about it. > Piffle. An irrational number is represented by a potentially infinite sequence that does not repeat. > A list of reals is a function that takes an element of the potentially > infinite > set of natural numbers and returns a potentially infinite sequence. > The diagonal number is a potentially infinite sequence. Wrong. Ask your set therorist companions. Piffle. How can a definition be wrong? Defintion. B is a g-subset of a potentially > infinite set A if B is a set or a potentially infinite > set and any element of B is an element of A. We can > now contruct potentially infinite power sets, and > the usual theory follows. You are confusing potentially infinite with computable. > While it is true that a computable set is either > finite or potentially infinite, it is not true that a potentially > infinite set is computable. Try to inform you. Cantor is a very competent source. A potentially > infinite variable is a variable which is always finite but can grow > without bound. A potentially infinite set does not exist. > Piffle. Finite sets exist. A potentially infinite set is defined in terms of finite sets. (Since I am not using Cantor's definitions I fail to see the relevence). - William Hughes === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. > Piffle. You need to study your potential-set theory. We have defined what it means to be an element of > a potential set. We never have all elements available. Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. Piffle. Real numbers are represented by potentially infinite sequences. > A list of reals is a function that takes an element of the potentially > infinite > set of natural numbers and returns a potentially infinite sequence. There is no function that takes an element of the potentially (or actually) infinite sequence of natural numbers and returns a potentially (or actually) infinite sequence in that way that all possible potentially (or actually) infinite sequence are covered. But it is not because there are more infinite sequences than natural numbers. It is just because there exists no sequence of the infinite sequences. Albrecht S. Storz > The diagonal number is a potentially infinite sequence. > We show the diagonal number is not a member of the list > in exactly the same way as before. > Therefore we can define cardinalities of potentially infinite > sets. Yes, we can define that a set is finite or that it is infinite, > denoting the latter case conveniently by oo. That's all. > Defintion. B is a g-subset of a potentially > infinite set A if B is a set or a potentially infinite > set and any element of B is an element of A. We can > now contruct potentially infinite power sets, and > the usual theory follows. You are confusing potentially infinite with computable. > While it is true that a computable set is either > finite or potentially infinite, it is not true that a potentially > infinite set is computable. - William Hughes === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de > No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. > Piffle. You need to study your potential-set theory. We have defined what it means to be an element of > a potential set. We never have all elements available. Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. Piffle. Real numbers are represented by potentially infinite sequences. > A list of reals is a function that takes an element of the potentially > infinite > set of natural numbers and returns a potentially infinite sequence. > There is no function that takes an element of the potentially (or > actually) infinite sequence of natural numbers and returns a > potentially (or actually) infinite sequence in that way that all > possible potentially (or actually) infinite sequence are covered. But > it is not because there are more infinite sequences than natural > numbers. It is just because there exists no sequence of the infinite > sequences. > Since you have no support for that last statement, the entire argument holds no weight. - William Hughes P.S. Any progress on the definition of complete? === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de > No. You can develop potential set theory. Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word potential in front of each > occurence of the word set. There is no difference between > saying a potentially infinite set exists and saying an actually > infinite set exists. A potentially infinite set has no cardinal number. It cannot be in > bijection with another infinite set because it does not exist > completely. Piffle. You need to study your potential-set theory. We have defined what it means to be an element of > a potential set. We never have all elements available. Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. Piffle. Real numbers are represented by potentially infinite sequences. > A list of reals is a function that takes an element of the potentially > infinite > set of natural numbers and returns a potentially infinite sequence. There is no function that takes an element of the potentially (or > actually) infinite sequence of natural numbers and returns a > potentially (or actually) infinite sequence in that way that all > possible potentially (or actually) infinite sequence are covered. But > it is not because there are more infinite sequences than natural > numbers. It is just because there exists no sequence of the infinite > sequences. > Since you have no support for that last statement, the entire > argument holds no weight. But it is already proved. The diagonal argument of G.Cantor shows exact this fact. Just use the only meaningful axiom of infinity: all infinite collectivities have one aspect in common: they are infinite, endless, unfinishable. - William Hughes P.S. Any progress on the definition of complete I work on it. In the meantime I use yours. Albrecht S. Storz === Subject: Re: Cantor Confusion <4562FE28.8010908@et.uni-magdeburg.de> <456319F5.8090505@et.uni-magdeburg.de > P.S. Any progress on the definition of complete I work on it. In the meantime I use yours. Unfortunately you need a definition of complete that does not need a context. Your repeated claim is that infinite sets cannot be complete. However, you never give a context. The definitions of complete I use all require a context. A set is said to be complete with respect to something. None of my definitions will work for you. - William Hughes === Subject: Re: Cantor Confusion >> Yes. But, sorry to see, it is the fundament of modern mathematics. >> Finished infinities is your wording. Precisely describing the fundament of modern mathematics. It may describe WM's fundament, but need not describe anyone elses'. The style of discussion seems to correspond to the subject: Cantor Confusion. Confusion due to Cantor. Confusion how to interpret Cantor. Confusion of Cantor himself .... Cantor and Confusion are almost synonymous: Infinite confusion on infinity. To what extent criticism by WM might be justified? Let me tell you my position: WM is most likely wrong if he sees the finished infinities the fundament of modern mathematics. At first, modern mathematics is not at all really based on set theory but on valuable precantorian tradition and independent continuation of it, combined with a rather pragmatic ignorance of principles. Secondly, nothing would be really wrong with the apparent logical contradiction between the infinite divisibility of continuum and the finite amount of attributed numbers available. carefully hidden categorical gap between the primary notion of continuum and the so called Hausdorff continuum of real numbers. Mathematics learned from Cantor logically splitted thinking. On one hand, geometry requires continuity. As made obvious by DA2, this would demand a definition of the reals like fictitious limits ore like perfectly endless continued decimals or the like, i.e. fictions. On the other hand, algebra requires genuine numbers. So all set theory may claim to include infinity. It only nurtures a pertaining illusion. Real numbers as defined e.g. by nested intervals are strictly speaking just very fine grained rationals based on the potential infinity. In so far, WM is absolutely correct. Isn't he? Ich .9fbersetze frei: In welchem Umfang hat WM Recht? Ich meine: WM irrt sich h.9achstwahrscheinlich, wenn er das beendete Nieendende ironisch als das Fundament moderner Mathematik benzeichnet. Erstens basiert die moderne Mathematik .9fberhaupt nicht wirklich auf der Mengenlehre sondern auf wertvoller Vorcantor-Tradition und deren unabh.8angigem Ausbau, kombiniert mit einer ziemlich pragmatischen Mi¤achtung von Grunds.8atzen. Zweitens w.8are nichts einzuwenden gegen dem scheinbaren logischen Widerspruch zwischen unendlicher Teilbarkeit des Kontinuums und der begrenzten Menge von zurechenbaren verf.9fgbaren Zahlen. Ich schrieb w.8are weil ich einen entweder .9fbersehenen oder, was wahrscheinlicher ist, sorgf.8altig versteckten kategorischen Unterschied zwischen dem originalen Kontinuumsbegriff und dem sogenannten Hausdorff-Kontinuum der reellen Zahlen aufgedeckt habe. Die Mathematik hat von Cantor gelernt logisch zweigleisig zu denken. Einerseits braucht die Geometrie die Stetigkeit. Wie das DA2 verdeutlicht, w.9frde dies eine Definition der reellen Zahlen als fiktive Grenzwerte oder mit perfekt endlosen Nachkommastellen erfordern, also Fiktionen. Andererseits erfordert die Algebra echte Zahlen. Somit mag alle Mengenlehre behaupten das Unendliche einzuschlie¤en. Dies n.8ahrt nur entsprechende Illusionen. Reelle Zahlen, so wie man sie beispielsweise mittels Intervallschachtelung definiert, sind genau genommen nur sehr hochaufl.9asende Rationalzahlen, beruhend auf potentieller Unendlichkeit. Insoweit hat WM v.9allig Recht. Oder etwa nicht? Eckard Blumschein === Subject: Re: Cantor Confusion > To what extent criticism by WM might be justified? > Let me tell you my position: WM is most likely wrong if he sees the > finished infinities the fundament of modern mathematics. Quite on the contrary. Eckard, please learn more about Wolfgang Mueckenheim in the first place: http://www.fh-augsburg.de/~mueckenh/ Wild guess: his ideas will turn out to be not so uncomfortable with you. Han de Bruijn === Subject: Re: Cantor Confusion > >> To what extent criticism by WM might be justified? >> Let me tell you my position: WM is most likely wrong if he sees the >> finished infinities the fundament of modern mathematics. Quite on the contrary. Eckard, please learn more about Wolfgang Mueckenheim in the first place: http://www.fh-augsburg.de/~mueckenh/ Wild guess: his ideas will turn out to be not so uncomfortable with you. Hallo Han, First of all I would like to urge you for restricting the discussion to sci.mat by removal of de.sci.mathematik. Nice to meet you again. I am also glad that you are trying to defend WM. I benefitted a lot from his booklet Die Geschichte des Unendlichen which was sent to me by him personally. Presumably you got me wrong concerning what I consider the fundament of modern mathematics. I consider set theory a toxic coat of modern mathematics rather than its genuine fundament. Modern mathematics is inconceivable without the contributions e.g. by Thales, Eudoxos, Pythagoras, Archimedes, Euclid, Fermat, Vieta, Descartes, Galilei, Newton, Leibniz, Euler, Bernoulli, Laplace, Lagrange, Fourier, Hermite, Lindemann, Gauss, Cauchy, Kronecker, Poincar.8e, Shannon, Kolgomoroff, and even Bill Gates. Dedekind and others contributed an illusion necessarily leading to endless quarrels. WM has to adapt too much to present mathematical terminology. When he confused Paul with Emile this was an excusable mistake to me. However I cannot follow him completely. Eckard === Subject: Re: Cantor Confusion > Presumably you got me wrong concerning what I consider the fundament of > modern mathematics. I consider set theory a toxic coat of modern > mathematics rather than its genuine fundament. [ ... snip ... ] It could be that we agree completely on this: Han de Bruijn === Subject: Re: Cantor Confusion > >> Presumably you got me wrong concerning what I consider the fundament of >> modern mathematics. I consider set theory a toxic coat of modern >> mathematics rather than its genuine fundament. [ ... snip ... ] It could be that we agree completely on this: > Han de Bruijn Having experienced non-capitalist systems and having close connections to Muslem scientists, I shoud be able understand your suspicion. No. I do not share it. I consider set theory a toxic coat of modern mathematics. This implies, set theory did not have a positive stimulating effect on mathematics at all. It is not really the fundament of modern mathematics. Free mathematics is merely an excuse for those who are unable to envision usefully applicable and well matching together mathematical structures. Economy is not merely an outdated coat. Free market economy has and will have very drastic consequences. Let's either try to realistically elucidate some relationship between politics and mathematics or strictly avoid this topic. It is definitely true that there is no parity between the comparatively small Jewish population and the many black people among mathematicians. Things may possibly change towards more important mathematicians of Chinese descent or from India. I see education at school very dangerous if it leads to hypocrisy and blind obedience due to enforced believe in set theory. I wonder why apparently nobody seems to be interested in the question Why at all do we consider sets instead of numbers?. My feeling says apriorism is a special kind of stupidity and a very stubborn one. Eckard === Subject: Re: Cantor Confusion > >> Presumably you got me wrong concerning what I consider the fundament of >> modern mathematics. I consider set theory a toxic coat of modern >> mathematics rather than its genuine fundament. [ ... snip ... ] It could be that we agree completely on this: > Han de Bruijn Having experienced non-capitalist systems and having close connections > to Muslem scientists, I shoud be able understand your suspicion. No. I do not share it. I consider set theory a toxic coat of modern > mathematics. This implies, set theory did not have a positive > stimulating effect on mathematics at all. It is not really the fundament > of modern mathematics. Free mathematics is merely an excuse for those > who are unable to envision usefully applicable and well matching > together mathematical structures. Economy is not merely an outdated > coat. Free market economy has and will have very drastic consequences. Let's either try to realistically elucidate some relationship between > politics and mathematics or strictly avoid this topic. It is definitely > true that there is no parity between the comparatively small Jewish > population and the many black people among mathematicians. Things may > possibly change towards more important mathematicians of Chinese descent > or from India. I see education at school very dangerous if it leads to > hypocrisy and blind obedience due to enforced believe in set theory. > I wonder why apparently nobody seems to be interested in the question > Why at all do we consider sets instead of numbers?. My feeling says > apriorism is a special kind of stupidity and a very stubborn one. Those whose feelings tell them that they are Napolean Buonaparte have feelings as relevant to the structure of mathematics as EB's. === Subject: Re: Cantor Confusion >> Why at all do we consider sets instead of numbers?. My feeling says >> apriorism is a special kind of stupidity and a very stubborn one. > Those whose feelings tell them that they are Napolean Buonaparte have > feelings as relevant to the structure of mathematics as EB's. Admittedly, the word apriorism does not always mean dogmatism. Look into httr://wiki.cotch.net/index.php/Apriorism as to find: Apriorism: Explanation: You commit this fallacy if you reason from abstract principles to facts not vice versa. Demarcation: You do not commit this fallacy if you draw conclusions from abstract principles, unless you call these conclusions facts. Attributing concreteness to the abstract belongs to the reification fallacy. For further reading look into http://www.philosophyprofessor.com/philosophies/apriorism.php So I have to describe more precisely what I meant with apriorism. === Subject: Re: Cantor Confusion > >To what extent criticism by WM might be justified? >Let me tell you my position: WM is most likely wrong if he sees the >finished infinities the fundament of modern mathematics. >>Quite on the contrary. >>Eckard, please learn more about Wolfgang Mueckenheim in the first place: >>http://www.fh-augsburg.de/~mueckenh/ >>Wild guess: his ideas will turn out to be not so uncomfortable with you. Hallo Han, First of all I would like to urge you for restricting the discussion to > sci.mat by removal of de.sci.mathematik. Sorry. I didn't notice this crosspost. > Nice to meet you again. I am also glad that you are trying to defend WM. > I benefitted a lot from his booklet Die Geschichte des Unendlichen > which was sent to me by him personally. Okay. That's better. > Presumably you got me wrong concerning what I consider the fundament of > modern mathematics. I consider set theory a toxic coat of modern > mathematics rather than its genuine fundament. Modern mathematics is > inconceivable without the contributions e.g. by Thales, Eudoxos, > Pythagoras, Archimedes, Euclid, Fermat, Vieta, Descartes, Galilei, > Newton, Leibniz, Euler, Bernoulli, Laplace, Lagrange, Fourier, Hermite, > Lindemann, Gauss, Cauchy, Kronecker, Poincar.8e, Shannon, Kolgomoroff, > and even Bill Gates. Dedekind and others contributed an illusion > necessarily leading to endless quarrels. Don't forget Chebyshev, an excellent mathematician I've discovered quite recently. His polynomials turn out to be essential for some recent work of mine. Maybe read the last paragraph in: http://hdebruijn.soo.dto.tudelft.nl/jaar2006/drievoud.pdf > WM has to adapt too much to present mathematical terminology. When he > confused Paul with Emile this was an excusable mistake to me. > However I cannot follow him completely. Han de Bruijn === Subject: Re: Cantor Confusion <45548d8f$0$97235$892e7fe2@authen.yellow.readfreenews.net> <4555ddd5$0$97239$892e7fe2@authen.yellow.readfreenews.net> <45582033$0$97251$892e7fe2@authen.yellow.readfreenews.net> <4562B7D6.6000806@et.uni-magdeburg.de> <94442$456408cc$82a1e228$3787@news1.tudelft.nl> Hi Han, Some time ago, we were in a discussion about sampling the natural integers at random. I wonder about this: biject the unit interval of real numbers, of which it is said a uniform distribution exists, to the elements of NxNxNx..., N^N, the sequences of natural numbers. Sample, discarding samples that don't have only and exactly one of each (finite) element in the naturals. Then, of one of those, select the first element and it's a natural integer at uniform random. What do you think about that? Basically it says that if the reals and set of choice functions of the naturals is equivalent, then the natural integers can be uniformly sampled. Basically talk about well-ordering the reals, cardinals between N and P(N), the continuum hypothesis, functions on the naturals that are not real functions, and staccato pointillism introduce some complexities to the consideration. Ross === Subject: Re: Cantor Confusion > >> Yes. But, sorry to see, it is the fundament of modern mathematics. >> Finished infinities is your wording. Precisely describing the fundament of modern mathematics. It may describe WM's fundament, but need not describe anyone elses'. The style of discussion seems to correspond to the subject: > Cantor Confusion. Cantor seems to confuse anti-mathematicians like EB and WM. But he does not confuse mathematicians. > WM is absolutely correct. Isn't he? Not hardly. === Subject: Re: Cantor Confusion > >> WM is absolutely correct. Isn't he? Not hardly. Nobody is always absolutely correct as he is also not always absolutely wrong. It depends. Serious people ask on what it depends. False Virgil's don't. === Subject: Re: Cantor Confusion >> WM is absolutely correct. Isn't he? Not hardly. Nobody is always absolutely correct as he is also not always absolutely > wrong. It depends. Serious people ask on what it depends. False Virgil's > don't. Since EB agrees with me that nobody, presumably including WM, is absolutely correct, what is his problem? === Subject: Re: Cantor Confusion <4562B7D6.6000806@et.uni-magdeburg.de> Am Tue, 21 Nov 2006 09:24:54 +0100 schrieb Eckard Blumschein: > Ich schrieb w.8are weil ich einen entweder .9fbersehenen oder, was > wahrscheinlicher ist, sorgf.8altig versteckten kategorischen Unterschied > zwischen dem originalen Kontinuumsbegriff und dem sogenannten > Hausdorff-Kontinuum der reellen Zahlen aufgedeckt habe. You are only a stupid little girl from germany. Du hast nur gezeigt, wie dumm Du bist. Sach mal, an der Uni Sommerpause oder wieso kannst du auf einmal Deine ganze Zeit wieder hier verplempern? === Subject: Re: Cantor Confusion > Am Tue, 21 Nov 2006 09:24:54 +0100 schrieb Eckard Blumschein: > >> Ich schrieb w.8are weil ich einen entweder .9fbersehenen oder, was >> wahrscheinlicher ist, sorgf.8altig versteckten kategorischen Unterschied >> zwischen dem originalen Kontinuumsbegriff und dem sogenannten >> Hausdorff-Kontinuum der reellen Zahlen aufgedeckt habe. You are only a stupid little girl from germany. Du hast nur gezeigt, wie dumm Du bist. Sach mal, an der Uni Sommerpause oder wieso kannst du auf einmal Deine > ganze Zeit wieder hier verplempern? === Subject: Re: Cantor Confusion >> Am Tue, 21 Nov 2006 09:24:54 +0100 schrieb Eckard Blumschein: > Ich schrieb w.8are weil ich einen entweder .9fbersehenen oder, was > wahrscheinlicher ist, sorgf.8altig versteckten kategorischen Unterschied > zwischen dem originalen Kontinuumsbegriff und dem sogenannten > Hausdorff-Kontinuum der reellen Zahlen aufgedeckt habe. >> You are only a stupid little girl from germany. >> Du hast nur gezeigt, wie dumm Du bist. >> Sach mal, an der Uni Sommerpause oder wieso kannst du auf einmal Deine >> ganze Zeit wieder hier verplempern? Sorry, irgendwie schiefgelaufen, diese posting ... :-( Gru¤, Rainer === Subject: Re: Cantor Confusion > > No *finite* representation. But Cantor's list does allow *infinite* > representations. More specific, his diagonal proof was about lists > where each element consisted of an *infinite* sequence of symbols. > > Why do you think that my tree does not allow infinite sequences of > symbols (nodes)? It has. But each node is a finite distance away from the root. > i.e., Cantor's original list contains only the set of finite sequences > of m and w. I agree. > > No i.e. and wrong. The list was a list of *infinite* sequences. > > Why do you think that my tree does not allow to represent such > sequences with 0 and 1 instead of m a nd w? But there is *no* node that represents an infinite sequence. Each and every node represents a finite sequence. > Above you said The tree contains only the set of finite binary > representations. > > Yup. There is *no* node 1/3. > > There is no symbol 1/3 in Cantors list. Again confusion. But, 0.333... is a proper symbol in Cantors list. > But you had said that it were the nodes that represented the numbers > in the tree, so the paths are irrelevant. > > I never said so. The nodes represent the bits 0 or 1. So a node is either 0 or 1. > There are infinite paths > in your tree, but they do not contain a node that represents (for > instance) 1/3. So, if the nodes represent numbers (as you have said), > > Do you have a reference? Not needed. Just above you state that the nodes represent the bits 0 or 1. I have shown how you could concatenate the representation of a node with the representations of its parent nodes to get a number. > 1/3 is not in your tree. You are not clear about what the numbers in > your tree are. Are they the nodes? Are they the paths? Sometimes > you say one thing other times you say something different. So to get > proper understanding. What are the things that represent numbers? > > Infinite paths. I do not understand. You stated the nodes represent bits. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion in your tree, but they do not contain a node that represents (for > instance) 1/3. So, if the nodes represent numbers (as you have said), Do you have a reference? Not needed. Just above you state that the nodes represent the bits 0 or > 1. I have shown how you could concatenate the representation of a node > with the representations of its parent nodes to get a number. A number consists of bits. Some numbers consist even of one bit. But you must not mix up these terms. A number like 1/2 consist of the bit sequence 0.1000.... That is a path in my tree: 0. | 1 | 0 | 0 ... 1/3 is not in your tree. Of course it is, like 0.333... is in Cantor's list of decimals. > You are not clear about what the numbers in > your tree are. Are they the nodes? Are they the paths? Sometimes > you say one thing other times you say something different. So to get > proper understanding. What are the things that represent numbers? Infinite paths. I do not understand. You stated the nodes represent bits. Yes. That's the stuff numbers are built from. === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > There are infinite paths > in your tree, but they do not contain a node that represents (for > instance) 1/3. So, if the nodes represent numbers (as you have said), > > Do you have a reference? > > Not needed. Just above you state that the nodes represent the bits 0 or > 1. I have shown how you could concatenate the representation of a node > with the representations of its parent nodes to get a number. > > A number consists of bits. Some numbers consist even of one bit. But > you must not mix up these terms. > A number like 1/2 consist of the bit sequence 0.1000.... That is a path > in my tree: As an infinite bit sequence, yes, but there is *no* node in your tree that represents that bit sequence. > 1/3 is not in your tree. > > Of course it is, like 0.333... is in Cantor's list of decimals. No, there is *no* node in your tree that represents 1/3. Because there is *no* node in your tree that represents an infinite sequence. On the other hand, Cantor's diagonal proof is about infinite sequences. > You are not clear about what the numbers in > your tree are. Are they the nodes? Are they the paths? Sometimes > you say one thing other times you say something different. So to get > proper understanding. What are the things that represent numbers? > > Infinite paths. > > I do not understand. You stated the nodes represent bits. > > Yes. That's the stuff numbers are built from. This makes it still less clear. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion > You are not clear about what the numbers in > your tree are. Are they the nodes? Are they the paths? Sometimes > you say one thing other times you say something different. So to get > proper understanding. What are the things that represent numbers? Infinite paths. I do not understand. You stated the nodes represent bits. Yes. That's the stuff numbers are built from. That is the stuff /numerals/ may be built from, but the numbers themselves need not be dependent on any particular representations. Is the number represented by VIII and different from the number represented by 8, or by 2^3 ? === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > But you do not allow elliptic or hyperbolic > geometry? If not, why not? > > I do not forbid it. It is clear that already the simple geometry on a > sphere does not yield two parallels. Euclid simply did not consider > such kind of plane. In another system we have other axioms (or better > fundamental truths). > > Well, the same in set theory. In one form it has AC as a fundamental > truth, in another version it is not a fundamental truth. What is the > essential difference between the cases? > > That there is not an underlying plane which it is related to. So there should be an underlying plane, whatever that may mean? What is the underlying plane of hyperbolic geometry? What is it in elliptic geometry? > I do not ask what you think. Reread my question. What is true > about a set of numbers in nature or reality? > > There are so many truths. Take order, 1 < 11, commutativity of addition > and multiplication, n + m = m + n. These things do not become invalid > or to be proved only because matrix multiplication or quaternions were > invented. > > Again, no answer. > > If you cannot understand this answer, then we should stop here. No I do not understand it. It does not explain at all what is true about a set of numbers in nature or reality. So you can stop. > I think there is a great difference. It is not necessary to call > negative solutions false solutions as even Descartes did, (because it > was customary at his time. Although this custom was justified as long > as only positive numbers were called numbers.) But it is necessary to > distinguish between negative and positive numbers or real and complex > numbers or Euclidean and non Euclidean spaces. > > And you were vehemently objecting at calling the irrational numbers > numbers. > > The reason is that these numbers have no decimal representation, > hence cannot be used to prove that they are uncountably many. So what was your vehemence directed at? Not at calling them numbers, because you do so yourself. That they have no decimal representation? But that was not in discussion at that time. So I wonder. > What is the essential difference between all those cases? I once asked > you for a definition of number, and you did never supply a proper > definition. Now you say yourself that the definition has been changed in > the course of time. I may note that there is no trichonomy between the > complex numbers, it is not possible to consistently order them. > > You may call these entities numbers or not. My opposition stems from > their use in Cantor's list and the lacking trichotomy. So, now we may call them numbers (as you do). And, we are back to basics. Your misunderstanding of Cantor's argument and whatever. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion truth, in another version it is not a fundamental truth. What is the > essential difference between the cases? That there is not an underlying plane which it is related to. So there should be an underlying plane, whatever that may mean? For geometry, it is required, yes, for arithmetic it is not. > What > is the underlying plane of hyperbolic geometry? What is it in elliptic > geometry? The corresponding spaces and surfaces. > The reason is that these numbers have no decimal representation, > hence cannot be used to prove that they are uncountably many. So what was your vehemence directed at? Not at calling them numbers, > because you do so yourself. That they have no decimal representation? > But that was not in discussion at that time. So I wonder. What is the essential difference between all those cases? I once asked > you for a definition of number, and you did never supply a proper > definition. Now you say yourself that the definition has been changed in > the course of time. I may note that there is no trichonomy between the > complex numbers, it is not possible to consistently order them. You may call these entities numbers or not. My opposition stems from > their use in Cantor's list and the lacking trichotomy. So, now we may call them numbers (as you do). And, we are back to basics. > Your misunderstanding of Cantor's argument and whatever. They have no representation which could be used for any form of Cantor's list. === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > Well, the same in set theory. In one form it has AC as a fundamental > truth, in another version it is not a fundamental truth. What is the > essential difference between the cases? > > That there is not an underlying plane which it is related to. > > So there should be an underlying plane, whatever that may mean? > > For geometry, it is required, yes, for arithmetic it is not. And you do not think there is an underlying something to which set theory with AC and without AC are related? I still do not understand the essential difference. Moreover, AC has *nothing* to do with standard arithmetic. > You may call these entities numbers or not. My opposition stems from > their use in Cantor's list and the lacking trichotomy. > > So, now we may call them numbers (as you do). And, we are back to basics. > Your misunderstanding of Cantor's argument and whatever. > > They have no representation which could be used for any form of > Cantor's list. Well, according to set theory they have such representations. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion Well, the same in set theory. In one form it has AC as a fundamental > truth, in another version it is not a fundamental truth. What is the > essential difference between the cases? That there is not an underlying plane which it is related to. So there should be an underlying plane, whatever that may mean? For geometry, it is required, yes, for arithmetic it is not. There are geometries for which there is no underlying plane. One dimensional geometry, for example. And various finite geometries and projective geometries have no underlying plane. === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > This matrix has length omega and width omega. And its diagonal has > length omega. No line has length omega. Therefore the width is larger > than any line. And the diagonal is longer than any line. This is > impossible. > > No, that is very possible. > > If you assert that there is no line longer than the diagonal, you have > good reasons, which can be proved. Right. > If you assert that the diagonal can be longer than any line, then you > have no reasons, because the diagonal consists of line elements and > cannot be where no line is. So your second assertion is outside of > logic and outside of any mathematics. Therefore I am not willing to > discuss this topic further. You do not want to discuss it because you are not able to prove it. But indeed, let's stop this non-discussion. Unless you come with a proof. You are not even able to comprehend potential infinity. > The correct result: There must be at least one line which is exactly as > long as the diagonal. There must be an infinite natural number. Never ever do you give a proof. Only assertions. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion [. . .] >Never ever do you give a proof. Only assertions. Weeell one might make the same observation about mathematikers. ~v~~ === Subject: Re: Cantor Confusion have no reasons, because the diagonal consists of line elements and > cannot be where no line is. So your second assertion is outside of > logic and outside of any mathematics. Therefore I am not willing to > discuss this topic further. You do not want to discuss it because you are not able to prove it. The set D of indexes of the diagonal is a subset of the sets L_n of indexes of the lines.: D - UL_n = empty set. > But > indeed, let's stop this non-discussion. Unless you come with a proof. > You are not even able to comprehend potential infinity. Tell me which point is not accepted: 1) Every line which contains an index of the diagonal, contains all preceding indexes of the diagonal too. 2) Every index of the diagonal is in a line. 3) In order to show that there is no line containing all indexes of the diagonal, there must be found at least one index, which is in the diagonal but not in any line. This is impossible. 4) There is no line with infinitely many indexes. 5) Conclusion: There is no diagonal with infinitely many indexes. === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl ... > But > indeed, let's stop this non-discussion. Unless you come with a proof. > You are not even able to comprehend potential infinity. > > Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This is impossible. > 4) There is no line with infinitely many indexes. > 5) Conclusion: There is no diagonal with infinitely many indexes. I do not accept (3). It effectively states: forall{n in N}thereis{m in N} (index m is not in line n) -> thereis{m in N}forall{n in N} (index m is not in line n) which is false. Quantifier dislexia again. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion > Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line and a last member of the diagonal. It is certainly false in ZF or NBG, where such an assumption is also false. It is quite enough to show that for every index in the diagonal, except 1, there is some line not containing that index. Then no line can contain every index. > 4) There is no line with infinitely many indexes. > 5) Conclusion: There is no diagonal with infinitely many indexes. This is false too, as it depends on the truth of a false claim. > === Subject: Re: Cantor Confusion Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line > and a last member of the diagonal. We assume not a last line, but we assume that eery line has finitely many indexes. And this is true. It is certainly false in ZF or NBG, where such an assumption is also > false. For finite indexes it is correct. It is quite enough to show that for every index in the diagonal, > except 1, there is some line not containing that index. Then no line > can contain every index. Name two finite indexes which cannot be in one line. === Subject: Re: Cantor Confusion > Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line > and a last member of the diagonal. We assume not a last line, but we assume that eery line has finitely > many indexes. And this is true. What you allege in (3) does not follow from this. It is true that given any line there will be a diagonal element not in that line. It is false that given a diagonal element there is no ine which contains it. WM again dyslexes his quantifiers. It is certainly false in ZF or NBG, where such an assumption is also > false. For finite indexes it is correct. Infinite things are not constrained to behave in all respects like finite ones. That is because infinite means not finite. It is quite enough to show that for every index in the diagonal, > except 1, there is some line not containing that index. Then no line > can contain every index. Name two finite indexes which cannot be in one line. That is not at all relevant to what I said. If WM cannot read English, he should get someone to translate it for him. I know I cannot read German well enough to be sure of meanings so I do not pretend otherwise. === Subject: Re: Cantor Confusion Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line > and a last member of the diagonal. We assume not a last line, but we assume that eery line has finitely > many indexes. And this is true. What you allege in (3) does not follow from this. > It is true that given any line there will be a diagonal element not in > that line. > It is false that given a diagonal element there is no ine which contains > it. WM again dyslexes his quantifiers. It is certainly false in ZF or NBG, where such an assumption is also > false. For finite indexes it is correct. Infinite things are not constrained to behave in all respects like > finite ones. That is because infinite means not finite. But *the lines are all finite*. That's just why I chose the EIT as example. It is quite enough to show that for every index in the diagonal, > except 1, there is some line not containing that index. Then no line > can contain every index. Name two finite indexes which cannot be in one line. That is not at all relevant to what I said. It is relevant since every line is finite. So we have to deal with finite indexes only. And every line has finitely many indexes. So we have to deal with finitely many indexes only, as far as the lines are concernded. (That's why you raised that silly argument with the diagonal loger than every line.) So your magic belief is easily destroyed. === Subject: Re: Cantor Confusion Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line > and a last member of the diagonal. We assume not a last line, but we assume that eery line has finitely > many indexes. And this is true. What you allege in (3) does not follow from this. > It is true that given any line there will be a diagonal element not in > that line. > It is false that given a diagonal element there is no ine which contains > it. WM again dyslexes his quantifiers. It is certainly false in ZF or NBG, where such an assumption is also > false. For finite indexes it is correct. Infinite things are not constrained to behave in all respects like > finite ones. That is because infinite means not finite. But *the lines are all finite*. That's just why I chose the EIT as example. It is quite enough to show that for every index in the diagonal, > except 1, there is some line not containing that index. Then no line > can contain every index. Name two finite indexes which cannot be in one line. That is not at all relevant to what I said. It is relevant since every line is finite. So we have to deal with finite indexes only. And every line has finitely many indexes. So we have to deal with finitely many indexes only, as far as the lines are concernded. (That's why you raised that silly argument with the diagonal loger than every line.) So your magic belief is easily destroyed. === Subject: Re: Cantor Confusion Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line > and a last member of the diagonal. We assume not a last line, but we assume that eery line has finitely > many indexes. And this is true. What you allege in (3) does not follow from this. > It is true that given any line there will be a diagonal element not in > that line. > It is false that given a diagonal element there is no ine which contains > it. WM again dyslexes his quantifiers. It is certainly false in ZF or NBG, where such an assumption is also > false. For finite indexes it is correct. Infinite things are not constrained to behave in all respects like > finite ones. That is because infinite means not finite. But *the lines are all finite*. That's just why I chose the EIT as example. It is quite enough to show that for every index in the diagonal, > except 1, there is some line not containing that index. Then no line > can contain every index. Name two finite indexes which cannot be in one line. That is not at all relevant to what I said. It is relevant since every line is finite. So we have to deal with finite indexes only. And every line has finitely many indexes. So we have to deal with finitely many indexes only, as far as the lines are concernded. (That's why you raised that silly argument with the diagonal loger than every line.) So your magic belief is easily destroyed. === Subject: Re: Cantor Confusion As I have already answered it once, this is merely to note his repetition. === Subject: Re: Cantor Confusion > Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line > and a last member of the diagonal. We assume not a last line, but we assume that eery line has finitely > many indexes. And this is true. What you allege in (3) does not follow from this. > It is true that given any line there will be a diagonal element not in > that line. > It is false that given a diagonal element there is no ine which contains > it. WM again dyslexes his quantifiers. It is certainly false in ZF or NBG, where such an assumption is also > false. For finite indexes it is correct. Infinite things are not constrained to behave in all respects like > finite ones. That is because infinite means not finite. But *the lines are all finite*. That's just why I chose the EIT as example. It is quite enough to show that for every index in the diagonal, > except 1, there is some line not containing that index. Then no line > can contain every index. Name two finite indexes which cannot be in one line. That is not at all relevant to what I said. It is relevant since every line is finite. So we have to deal with finite indexes only. And every line has finitely many indexes. So we have to deal with finitely many indexes only, as far as the lines are concernded. (That's why you raised that silly argument with the diagonal loger than every line.) So your magic belief is easily destroyed. === Subject: Re: Cantor Confusion > Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of > the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line > and a last member of the diagonal. We assume not a last line, but we assume that eery line has finitely > many indexes. And this is true. What you allege in (3) does not follow from this. > It is true that given any line there will be a diagonal element not in > that line. > It is false that given a diagonal element there is no ine which contains > it. WM again dyslexes his quantifiers. It is certainly false in ZF or NBG, where such an assumption is also > false. For finite indexes it is correct. Infinite things are not constrained to behave in all respects like > finite ones. That is because infinite means not finite. But *the lines are all finite*. The *set* of lines in WM's so-called EIT is not finite. > That's just why I chose the EIT as > example. It is quite enough to show that for every index in the diagonal, > except 1, there is some line not containing that index. Then no line > can contain every index. Name two finite indexes which cannot be in one line. That is not at all relevant to what I said. It is relevant since every line is finite. WM again has cart-before-horse-itis. What is relevant is not whether one can always find a line holding a given element but whether one can always find a particular element in a given line. Quantifier dyslexia hits WM again. > So your magic belief is easily > destroyed. Not by those who cannot keep their quantifiers straight. === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > If you want to find absolute truth you should not look at mathematics. > > Not at that what today is called mathematics, I agree. > > I + I = II is very fine and reliable mathematics. Absolutely true. And > this approach can be put forward --- very far. What do you mean with those symbols? How do you define + and =? What is the meaning of I and II? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion this approach can be put forward --- very far. What do you mean with those symbols? How do you define + and =? > What is the meaning of I and II? What do you mean with There or exists or a? How do you define set and which and has? What is the meaning of no and elements? I think these symbols and expressions deserve closer examination than + and II. === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl ... > I + I = II is very fine and reliable mathematics. Absolutely true. And > this approach can be put forward --- very far. > > What do you mean with those symbols? How do you define + and =? > What is the meaning of I and II? > > What do you mean with There or exists or a? > How do you define set and which and has? > What is the meaning of no and elements? > > I think these symbols and expressions deserve closer examination than > + and II. Oh, perhaps. But now you are entering the field of philosphy end exiting the field of mathematics. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion ... > That's the question. By means of axioms you can produce conditional > truth at most. I am interested in absolute truth. Axioms will not help > us to find it. I don't think we need any axioms. > > If you want to find absolute truth you should not look at mathematics. > > Really? There are two groups of order 4; could any truth be more > absolute than that? What is the absolute truth of the axioms and definitions you are using? Within some set of axioms and definitions it is indeed an absolute true. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion > So lets count the set of all natural numbers {1,2,3,...} > There are no natural number left. So we stop > using natural numbers and use ordinals > (and to nobody's surprise a few things change). > > This is wrong. There is no ordinal needed to count the elements of the > set of all natural numbers. You can count until you weigh an ounce (;-)) > but you will never finish. Neither the elements you wish to count will > be exhausted nor the numbers with which you count. I think this is > potential infinity. On the other hand, when you ask how many elements > there are in N, you need an infinity (and this is, I think, actual > infinity). But all this hinges quite closely on the semantics of the > word count. If seen as a process, you do not need an infinity; when > seen as the result of a process, you do need an infinity. In many > languages (German and Dutch amongst others) there are different words > for the two meanings, but the meanings are conflated in English. > > Are you discussing languages or math? What would a mathematical meaning > for count the set of all natural numbers be? I am discussing both language and maths. In German (as in Dutch) there is a clear distinction between Abz.8ahlen and Z.8ahlen. And as you are in discussion with somebody from German origin, it is important to keep this in mind. The first word means the process of counting, the second means counting to get a result. In English for both the verb to count is used (and is given as translation in dictionaries). I think to enumerate is a better translation of Abz.8ahlen. So let met rephrase the position of the opponents: To enumerate the natural numbers you do not need ordinal numbers, this is potential infinity. To count the natural numbers you do need ordinal numbers. This is actual infinity. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion I am discussing both language and maths. In German (as in Dutch) > there is a clear distinction between Abz.8ahlen and Z.8ahlen. And as > you are in discussion with somebody from German origin, it is important > to keep this in mind. The first word means the process of counting, the > second means counting to get a result. In English for both the verb > to count is used (and is given as translation in dictionaries). I > think to enumerate is a better translation of Abz.8ahlen. So let met rephrase the position of the opponents: > To enumerate the natural numbers you do not need [transfinite] ordinal numbers, > this is potential infinity. To count the natural numbers you do > need [transfinite] ordinal numbers. This is actual infinity. The first sentence is correct, but not exciting and not what Cantor meant. The second sentence is Cantor's position: Abz.8ahlen ins Unendliche. === Subject: Re: Cantor Confusion > So lets count the set of all natural numbers {1,2,3,...} > There are no natural number left. So we stop > using natural numbers and use ordinals > (and to nobody's surprise a few things change). This is wrong. There is no ordinal needed to count the elements of > the > set of all natural numbers. You can count until you weigh an ounce > (;-)) > but you will never finish. Neither the elements you wish to count > will > be exhausted nor the numbers with which you count. I think this is > potential infinity. On the other hand, when you ask how many > elements there are in N, you need an infinity (and this is, I think, > actual > infinity). But all this hinges quite closely on the semantics of the > word count. If seen as a process, you do not need an infinity; > when > seen as the result of a process, you do need an infinity. In many > languages (German and Dutch amongst others) there are different words > for the two meanings, but the meanings are conflated in English. Are you discussing languages or math? What would a mathematical meaning > for count the set of all natural numbers be? I am discussing both language and maths. In German (as in Dutch) > there is a clear distinction between Abz.8ahlen and Z.8ahlen. And as > you are in discussion with somebody from German origin, it is important > to keep this in mind. The first word means the process of counting, the > second means counting to get a result. In English for both the verb > to count is used (and is given as translation in dictionaries). I > think to enumerate is a better translation of Abz.8ahlen. As a native speaker of German, I can't confirm this. There is a difference between z.8ahlen and abz.8ahlen, because one can't use these words interchangeably, and this difference seems to have something to do with However this may be, in discussions with M.9fckenheim it doesn't matter anyway, because M.9fckenheim plainly doesn't understand German, or, more precisely, Cantor's original papers. M.9fckenheim gives ample proof of this fact in his publications on the arxiv. In his paper On a severe inconsistency... M.9fckenheims ascribes some assertions to Cantor, and confirms this by quoting from Cantor's papers, which are obviously wrong. If one checks this quotation, one sees that M.9fckenheim ignored assumptions which Cantor had made previously in his text. In another paper whose title I don't remember M.9fckenheim asserts to give a simpler proof of a theorem of Cantor's, namely that the real plane stays connected if all points whose coordinates are rational are left out. Comparing with Cantor's original, one notices that Cantor had proved a much more general result, namely that the real plane stays connected if an arbitrary countable set of points is left out. The set of points with rational coordinates he later mentions as an example, and M.9fckenheim took that example for the main issue, ascribing some other weird opinions to Cantor on the way. Although, surprisingly, M.9fckenheims simplified proof of a restricted assertion is in principle correct, it is also telling that he seems to consider it worth to be published what actually is a mildly interesting exercise in basic point set topology. It simply doesn't make any sense, and doesn't lead anywhere, to discuss mathematics with M.9fckenheim, neither in English nor in German. Ralf === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl ... > I am discussing both language and maths. In German (as in Dutch) > there is a clear distinction between Abz.8ahlen and Z.8ahlen. And as > you are in discussion with somebody from German origin, it is important > to keep this in mind. The first word means the process of counting, the > second means counting to get a result. In English for both the verb > to count is used (and is given as translation in dictionaries). I > think to enumerate is a better translation of Abzíóhlen. > > As a native speaker of German, I can't confirm this. There is a difference > between z.8ahlen and abz.8ahlen, because one can't use these words > interchangeably, and this difference seems to have something to do with I know it is not as clear cut. In Dutch we have the same distinction as in German. At least in Dutch, and also in German I think, tellen when you z.8ahl the elements of a set you want to know the cardinality of the set. On the other hand, when you z.8ahl the elements of a set ab (aftellen in Dutch) you are assigning labels to the elements of the set, and I think this is also the case in German. So when it is stated (in German) that a set is abz.8ahlbar it means that you can assign labels (from the set of integral numbers) to each element of the set. On the other hand, when you can assign a number to the size of the set you might state that it is z.8ahlbar. And I think that it is this very distinction that makes the difference between potential and actual infinity strong for the German speakers (and the Dutch), but not for the English speakers (because they do not have that distinction with the word count). > However this may be, in discussions with M.9fckenheim it doesn't matter > anyway, because M.9fckenheim plainly doesn't understand German, or, more > precisely, Cantor's original papers. I know. Whenever I state that the paper that contains the diagonal proof is not about reals, M.9fckenheim always comes back with the second part of the first paragraph of that paper to show that it is. > Comparing with Cantor's original, > one notices that Cantor had proved a much more general result, namely that > the real plane stays connected if an arbitrary countable set of points is > left out. The set of points with rational coordinates he later mentions as > an example, and M.9fckenheim took that example for the main issue, Pretty similar to my experience. > It simply doesn't make any sense, and doesn't lead anywhere, to discuss > mathematics with M.9fckenheim, neither in English nor in German. But it has entertainment value. I still wonder what he will do now that It also did lead me to read what Cantor actually did write, and I found that he was not without errors in his writing. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion fact in his publications on the arxiv. In his paper On a severe > inconsistency... M.9fckenheims ascribes some assertions to Cantor, and > confirms this by quoting from Cantor's papers, which are obviously wrong. Please give an example. > If one checks this quotation, one sees that M.9fckenheim ignored assumptions > which Cantor had made previously in his text. Please give an example. > In another paper whose title > I don't remember It is the appendix to On Cantor's important theorems: http://arxiv.org/ftp/math/papers/0306/0306200 > M.9fckenheim asserts to give a simpler proof of a theorem of > Cantor's, namely that the real plane stays connected if all points whose > coordinates are rational are left out. Comparing with Cantor's original, > one notices that Cantor had proved a much more general result, namely that > the real plane stays connected if an arbitrary countable set of points is > left out. The set of points with rational coordinates he later mentions as > an example, Later, that is after the 5th line: Was die abz.8ahlbaren Punktmengen betrifft, so bieten sie eine merkw.9frdige Erscheinung dar, welche ich im folgenden zum Ausdruck bringen m.9achte. Betrachten wir irgendeine Punktmenge (M), welche innerhalb eines n-dimensionalen stetig zusammenh.8angenden Gebietes A .9fberalldicht verbreitet ist und die Eigenschaft der Abz.8ahlbarkeit besitzt, so da¤ die zu (M) geh.9arigen Punkte sich in der Reihenform ... vorstellen lassen; als Beispiel diene die Menge aller derjenigen Punkte unseres dreidimensionalen Raumes, deren Koordinaten in bezug auf ein orthogonales Koordinatensystem x, y, z alle drei algebraische Zahlenwerte haben. > and M.9fckenheim took that example for the main issue, ascribing > some other weird opinions to Cantor on the way. Although, surprisingly, > M.9fckenheims simplified proof of a restricted assertion is in principle > correct, it is also telling that he seems to consider it worth to be > published what actually is a mildly interesting exercise in basic point set > topology. Unfortunately you missed the clue. The simplified version has only been developed, because it can easily be extended to uncountable sets. I showed that the real plane even stays connected if an *uncountable* set of points is left out. So Cantor's theorem is useless. It doesn't show a difference between countable and uncountable, as Cantor might have tried to suggest. So, by your stupidity or your carelessness, you have again proven papers, even in the appendices, and for advertising them. === Subject: Re: Cantor Confusion and Mueckenheim took that example for the main issue, ascribing > some other weird opinions to Cantor on the way. Although, surprisingly, > Mueckenheim's simplified proof of a restricted assertion is in principle > correct, it is also telling that he seems to consider it worth to be > published what actually is a mildly interesting exercise in basic point set > topology. Unfortunately you missed the clue. The simplified version has only been > developed, because it can easily be extended to uncountable sets. I showed that the real plane even stays connected if an *uncountable* > set of points is left out. So Cantor's theorem is useless. It doesn't > show a difference between countable and uncountable, as Cantor might > have tried to suggest. So, it turns out that you were proving a different theorem than Ralph thought. That's reassuring. For a moment there, we thought you might have given a correct proof of something. Care to show us your proof that the plane is connected even if an uncountable number of points is deleted? -- David Marcus === Subject: Re: Cantor Confusion and Mueckenheim took that example for the main issue, ascribing > some other weird opinions to Cantor on the way. Although, surprisingly, > Mueckenheim's simplified proof of a restricted assertion is in principle > correct, it is also telling that he seems to consider it worth to be > published what actually is a mildly interesting exercise in basic point > set > topology. Unfortunately you missed the clue. The simplified version has only been > developed, because it can easily be extended to uncountable sets. I showed that the real plane even stays connected if an *uncountable* > set of points is left out. So Cantor's theorem is useless. It doesn't > show a difference between countable and uncountable, as Cantor might > have tried to suggest. So, it turns out that you were proving a different theorem than Ralph > thought. That's reassuring. For a moment there, we thought you might > have given a correct proof of something. Care to show us your proof that > the plane is connected even if an uncountable number of points is > deleted? I will accept that it can be connected if the right uncountable set is removed, but I can think of uncountable sets of points whose removal at least appears to totally disconnect the set of those remaining. For example if one removes all the points in the Cartesian plane which have either coordinate non-integral, what is left is about as disconnected as one can imagine. === Subject: Re: Cantor Confusion some other weird opinions to Cantor on the way. Although, surprisingly, > Mueckenheim's simplified proof of a restricted assertion is in principle > correct, it is also telling that he seems to consider it worth to be > published what actually is a mildly interesting exercise in basic point > set > topology. Unfortunately you missed the clue. The simplified version has only been > developed, because it can easily be extended to uncountable sets. I showed that the real plane even stays connected if an *uncountable* > set of points is left out. So Cantor's theorem is useless. It doesn't > show a difference between countable and uncountable, as Cantor might > have tried to suggest. So, it turns out that you were proving a different theorem than Ralph > thought. That's reassuring. For a moment there, we thought you might > have given a correct proof of something. Care to show us your proof that > the plane is connected even if an uncountable number of points is > deleted? I will accept that it can be connected if the right uncountable set is > removed, but I can think of uncountable sets of points whose removal at > least appears to totally disconnect the set of those remaining. Of course, for example if you remove all points. For example if one removes all the points in the Cartesian plane which > have either coordinate non-integral, what is left is about as > disconnected as one can imagine. Cantor wanted to suggest that the removal of a countable set leaves the plane connected while the removal of an uncountable set does not. As an example he chose the algebraic numbers (obviously in order to distinguish them from the transcendental numbers). I proved that there is by far a simpler proof for the algebraic numbers. I proved that this proof can also be applied to the transcendental numbers. See the appendix of http://arxiv.org/pdf/math.GM/0306200 === Subject: Re: Cantor Confusion > >> and Mueckenheim took that example for the main issue, ascribing >> some other weird opinions to Cantor on the way. Although, >> surprisingly, Mueckenheim's simplified proof of a restricted >> assertion is in principle correct, it is also telling that he seems >> to consider it worth to be published what actually is a mildly >> interesting exercise in basic point set >> topology. >> Unfortunately you missed the clue. The simplified version has only >> been developed, because it can easily be extended to uncountable >> sets. >> I showed that the real plane even stays connected if an *uncountable* >> set of points is left out. So Cantor's theorem is useless. It doesn't >> show a difference between countable and uncountable, as Cantor might >> have tried to suggest. >> So, it turns out that you were proving a different theorem than Ralph >> thought. That's reassuring. For a moment there, we thought you might >> have given a correct proof of something. Care to show us your proof >> that the plane is connected even if an uncountable number of points is >> deleted? >> I will accept that it can be connected if the right uncountable set is >> removed, but I can think of uncountable sets of points whose removal at >> least appears to totally disconnect the set of those remaining. Of course, for example if you remove all points. >> For example if one removes all the points in the Cartesian plane which >> have either coordinate non-integral, what is left is about as >> disconnected as one can imagine. Cantor wanted to suggest that the removal of a countable set leaves the > plane connected while the removal of an uncountable set does not. As an > example he chose the algebraic numbers (obviously in order to > distinguish them from the transcendental numbers). for the main issue, ascribing some other weird opinions to Cantor on the way.. That Cantor wanted to suggest that the removal of an uncountable set does not leave the plane connected is nothing but your phantasy. It is trivial to give examples of uncountable sets whose removal does not disconnect the plane. E.g. the points inside a circle or rectangle, and I'm pretty sure that Cantor was aware of this. Because uncountability has nothing to do with connectedness in this way, it is remarkable that countability has - it seems to be much more plausible to me that this is how Cantor thought. > I proved that there is by far a simpler proof for the algebraic > numbers. > I proved that this proof can also be applied to the transcendental > numbers. Yes, and this proof even seems to be correct, but it is trivial, that is: on the level of an easy exercise in a beginner's topology course. > See the appendix of http://arxiv.org/pdf/math.GM/0306200 It is telling that you consider such a crumb of dust worth to be published. You have no idea about the current state of the art. BTW, I am not obliged to explain your errors to your satisfaction. Ralf === Subject: Re: Cantor Confusion For example if one removes all the points in the Cartesian plane which >> have either coordinate non-integral, what is left is about as >> disconnected as one can imagine. Cantor wanted to suggest that the removal of a countable set leaves the > plane connected while the removal of an uncountable set does not. As an > example he chose the algebraic numbers (obviously in order to > distinguish them from the transcendental numbers). for the main issue, ascribing > some other weird opinions to Cantor on the way. advance? You look into the future? What a wretched excuse! > That Cantor wanted to > suggest that the removal of an uncountable set does not leave the plane > connected is nothing but your phantasy. Cantor emphasized: Was die abz.8ahlbaren Punktmengen betrifft, so bieten sie eine merkw.9frdige Erscheinung dar ... Concerning the countable sets of points we observe a strange phenomenon ... This means, that he did *not* consider uncountable sets. > It is trivial to give examples of > uncountable sets whose removal does not disconnect the plane. In fact, after I showed you such sets. But with Cantor's original proof this is impossible to see, because just the countability is the tool transfinite set theory veils even most simple structures. > E.g. the > points inside a circle or rectangle, and I'm pretty sure that Cantor was > aware of this. Because uncountability has nothing to do with connectedness > in this way, it is remarkable that countability has - it seems to be much > more plausible to me that this is how Cantor thought. the plaine *being connected*. But it is not remarkable that removing an uncountable set leaves the plaine being connected? What a foolish assertion. I proved that there is by far a simpler proof for the algebraic > numbers. > I proved that this proof can also be applied to the transcendental > numbers. Yes, and this proof even seems to be correct, but it is trivial, that is: on > the level of an easy exercise in a beginner's topology course. Why then do you say it seems correct? It seems you are not quite sure, because you do not even understand this admittedly simple piece of mathematics. And that you missed the clue of my paper completely is obvious from your last posting. See the appendix of http://arxiv.org/pdf/math.GM/0306200 It is telling that you consider such a crumb of dust worth to be published. > You have no idea about the current state of the art. Which art are you dreaming of? The art to slander and slaver? BTW, I am not obliged to explain your errors to your satisfaction. and not even able to explain simple mathematics your own satisfaction. You should have recognized at least that the coordinates considered by rational coordinates. Yes, there is a difference! === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl ... > the plaine *being connected*. But it is not remarkable that removing > an uncountable set leaves the plaine being connected? What a foolish > assertion. But removing an uncountable set can leave the plane either connected or disconnected. What is remarkable about that? Remove the line x=0 from the plane. You remove an uncountable set and the result is clearly disconnected. What is remarkable is that when you remove a countable set the result is *always* connected. So to get an disconnected set you *must* remove an uncountable set, but not any uncountable set will do. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion > > and Mueckenheim took that example for the main issue, ascribing > some other weird opinions to Cantor on the way. Although, > surprisingly, > Mueckenheim's simplified proof of a restricted assertion is in > principle > correct, it is also telling that he seems to consider it worth to be > published what actually is a mildly interesting exercise in basic > point > set > topology. Unfortunately you missed the clue. The simplified version has only been > developed, because it can easily be extended to uncountable sets. I showed that the real plane even stays connected if an *uncountable* > set of points is left out. So Cantor's theorem is useless. It doesn't > show a difference between countable and uncountable, as Cantor might > have tried to suggest. So, it turns out that you were proving a different theorem than Ralph > thought. That's reassuring. For a moment there, we thought you might > have given a correct proof of something. Care to show us your proof that > the plane is connected even if an uncountable number of points is > deleted? I will accept that it can be connected if the right uncountable set is > removed, but I can think of uncountable sets of points whose removal at > least appears to totally disconnect the set of those remaining. Of course, for example if you remove all points. The remaining empty set is trivially connected. For example if one removes all the points in the Cartesian plane which > have either coordinate non-integral, what is left is about as > disconnected as one can imagine. Cantor wanted to suggest that the removal of a countable set leaves the > plane connected while the removal of an uncountable set does not. Those who try to read Cantors mind often deceive themselves. And WM is foremost among those self-deceivers. > As an > example he chose the algebraic numbers (obviously in order to > distinguish them from the transcendental numbers). I proved that there is by far a simpler proof for the algebraic > numbers. > I proved that this proof can also be applied to the transcendental > numbers. > See the appendix of http://arxiv.org/pdf/math.GM/0306200 I have seen it. No serious mathematician would dare present such a sloppy paper for publication. Nor would any well-referreed mathematical journal publish it. The alleged proofs, at least as far as I bothered to read, are fatally flawed, and do not establish their claimed results. === Subject: Re: Cantor Confusion I will accept that it can be connected if the right uncountable set is > removed, but I can think of uncountable sets of points whose removal at > least appears to totally disconnect the set of those remaining. Of course, for example if you remove all points. The remaining empty set is trivially connected. That is nonsense. If there is nothing to be connected then there is nothing connected! For example if one removes all the points in the Cartesian plane which > have either coordinate non-integral, what is left is about as > disconnected as one can imagine. Cantor wanted to suggest that the removal of a countable set leaves the > plane connected while the removal of an uncountable set does not. Those who try to read Cantors mind often deceive themselves. I read his words. > As an > example he chose the algebraic numbers (obviously in order to > distinguish them from the transcendental numbers). I proved that there is by far a simpler proof for the algebraic > numbers. > I proved that this proof can also be applied to the transcendental > numbers. > See the appendix of http://arxiv.org/pdf/math.GM/0306200 I have seen it. No serious mathematician would dare present such a > sloppy paper for publication. Nor would any well-referreed mathematical > journal publish it. > The alleged proofs, at least as far as I bothered > to read, are fatally flawed, In fact, this was the opinion of the mathematicians at Cantor's time already. Therefore he had problems to get his papers published. > and do not establish their claimed results. With other words, you did not understand them. But our question concerned the appendix only. The proof given there is very simple, so simple that even Mr. Bader asserted to have understood it (it seemed so to him, at least). You cannot confirm that my proof is correct? PS: You really did not see that Cantor's arguing in his first proof is valid for rational numbers as well as for transfinite numbers? It is the same situation as in the appendix. === Subject: Re: Cantor Confusion I will accept that it can be connected if the right uncountable set is > removed, but I can think of uncountable sets of points whose removal at > least appears to totally disconnect the set of those remaining. Of course, for example if you remove all points. The remaining empty set is trivially connected. That is nonsense. If there is nothing to be connected then there is > nothing connected! Topologically, a set is connected unless it is a union of two or more non-empty disjoint open sets. See http://en.wikipedia.org/wiki/Connected_space According to this definition, an empty set is connected. What definition of connectedness does WM use which disconnects the empty set? > Cantor wanted to suggest that the removal of a countable set leaves the > plane connected while the removal of an uncountable set does not. Those who try to read Cantors mind often deceive themselves. I read his words. Not very well. As an > example he chose the algebraic numbers (obviously in order to > distinguish them from the transcendental numbers). I proved that there is by far a simpler proof for the algebraic > numbers. > I proved that this proof can also be applied to the transcendental > numbers. > See the appendix of http://arxiv.org/pdf/math.GM/0306200 I have seen it. No serious mathematician would dare present such a > sloppy paper for publication. Nor would any well-referreed mathematical > journal publish it. > The alleged proofs, at least as far as I bothered > to read, are fatally flawed, In fact, this was the opinion of the mathematicians at Cantor's time > already. Therefore he had problems to get his papers published. and do not establish their claimed results. With other words, you did not understand them. But our question > concerned the appendix only. The proof given there is very simple, so > simple that even Mr. Bader asserted to have understood it (it seemed so > to him, at least). You cannot confirm that my proof is correct? In reading the beginning of that paper, I found enough errors to dissuade me from bothering. > PS: You really did not see that Cantor's arguing in his first proof is > valid for rational numbers as well as for transfinite numbers? If WM thinks this, he has not understood the first proof. There is an increasing sequence of rationals whose LUB is sqrt(2), e.g., f(n) = floor( 2^n * sqrt(2) ) / 2^n. And a decreasing sequence of rationals whose GLB is sqrt(2), e.g., g(n) = Ceiling( 2^n * sqrt(2) ) / 2^n If Cantor's first proof held for the rationals, as WM claims, then WM would have proved that sqrt(2) is rational. === Subject: Re: Cantor Confusion > So lets count the set of all natural numbers {1,2,3,...} > There are no natural number left. So we stop > using natural numbers and use ordinals > (and to nobody's surprise a few things change). This is wrong. There is no ordinal needed to count the elements of > the > set of all natural numbers. You can count until you weigh an ounce > (;-)) > but you will never finish. Neither the elements you wish to count > will > be exhausted nor the numbers with which you count. I think this is > potential infinity. On the other hand, when you ask how many > elements there are in N, you need an infinity (and this is, I think, > actual > infinity). But all this hinges quite closely on the semantics of the > word count. If seen as a process, you do not need an infinity; > when > seen as the result of a process, you do need an infinity. In many > languages (German and Dutch amongst others) there are different words > for the two meanings, but the meanings are conflated in English. Are you discussing languages or math? What would a mathematical meaning > for count the set of all natural numbers be? I am discussing both language and maths. In German (as in Dutch) > there is a clear distinction between Abz.8ahlen and Z.8ahlen. And as > you are in discussion with somebody from German origin, it is important > to keep this in mind. The first word means the process of counting, the > second means counting to get a result. In English for both the verb > to count is used (and is given as translation in dictionaries). I > think to enumerate is a better translation of Abz.8ahlen. As a native speaker of German, I can't confirm this. There is a difference > between z.8ahlen and abz.8ahlen, because one can't use these words > interchangeably, and this difference seems to have something to do with However this may be, in discussions with M.9fckenheim it doesn't matter > anyway, because M.9fckenheim plainly doesn't understand German, or, more > precisely, Cantor's original papers. M.9fckenheim gives ample proof of this > fact in his publications on the arxiv. In his paper On a severe > inconsistency... M.9fckenheim ascribes some assertions to Cantor, and > confirms this by quoting from Cantor's papers, which are obviously wrong. > If one checks this quotation, one sees that M.9fckenheim ignored assumptions > which Cantor had made previously in his text. In another paper whose title > I don't remember M.9fckenheim asserts to give a simpler proof of a theorem of > Cantor's, namely that the real plane stays connected if all points whose > coordinates are rational are left out. Comparing with Cantor's original, > one notices that Cantor had proved a much more general result, namely that > the real plane stays connected if an arbitrary countable set of points is > left out. The set of points with rational coordinates he later mentions as > an example, and M.9fckenheim took that example for the main issue, ascribing > some other weird opinions to Cantor on the way. Although, surprisingly, > M.9fckenheim's simplified proof of a restricted assertion is in principle > correct, Quite surprising. > it is also telling that he seems to consider it worth to be > published what actually is a mildly interesting exercise in basic point set > topology. > It simply doesn't make any sense, and doesn't lead anywhere, to discuss > mathematics with M.9fckenheim, neither in English nor in German. I'm pretty sure we all are well aware of this. For that matter, I don't think we've been discussing mathematics with WM. -- David Marcus === Subject: Re: Cantor Confusion > So lets count the set of all natural numbers {1,2,3,...} > There are no natural number left. So we stop > using natural numbers and use ordinals > (and to nobody's surprise a few things change). This is wrong. There is no ordinal needed to count the elements of the > set of all natural numbers. You can count until you weigh an ounce (;-)) > but you will never finish. Neither the elements you wish to count will > be exhausted nor the numbers with which you count. I think this is > potential infinity. On the other hand, when you ask how many elements > there are in N, you need an infinity (and this is, I think, actual > infinity). But all this hinges quite closely on the semantics of the > word count. If seen as a process, you do not need an infinity; when > seen as the result of a process, you do need an infinity. In many > languages (German and Dutch amongst others) there are different words > for the two meanings, but the meanings are conflated in English. Are you discussing languages or math? What would a mathematical meaning > for count the set of all natural numbers be? I am discussing both language and maths. In German (as in Dutch) > there is a clear distinction between Abz.8ahlen and Z.8ahlen. And as > you are in discussion with somebody from German origin, it is important > to keep this in mind. The first word means the process of counting, the > second means counting to get a result. In English for both the verb > to count is used (and is given as translation in dictionaries). I > think to enumerate is a better translation of Abz.8ahlen. So let met rephrase the position of the opponents: > To enumerate the natural numbers you do not need ordinal numbers, > this is potential infinity. To count the natural numbers you do > need ordinal numbers. This is actual infinity. I think you are saying that enumeration is the process that we use to determine a count. So, a potential infinity is an infinite process while an actual infinity is an infinite count. I'm not sure whether there is such a distinction in mathematics. Maybe a potential infinity could be defined as a function with an infinite domain while an actual infinity could be defined as an infinite ordinal. With this definition, we don't need infinite ordinal numbers to define a potential infinity, e.g., we could define f:N -> N, f(x) := x + 1. Although, the domain of such a function would still be an infinite set. Usually, when mathematics drops distinctions that were important in the past, the decision is correct. If someone tries to reinvent all of mathematics for themselves, it isn't too surprising if they don't make it up to the present, but instead get stuck in the past. -- David Marcus === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > What are the things that represent numbers? > > My humble trial to answer this question is: > > Infinite trees represent real numbers. > All nodes represent rational numbers. This clearly is a non-answer. I ask about the things that represent numbers. I do not ask about real numbers or rational numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion ... > Why are square circles unimaginable? They are not. With the Manhattan measure of the plane, each circle is a square. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion >... > Why are square circles unimaginable? They are not. With the Manhattan measure of the plane, each circle is >a square. Then what is a square? ~v~~ === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl >... > Why are square circles unimaginable? > >They are not. With the Manhattan measure of the plane, each circle is >a square. > > Then what is a square? Pray, tell me. I would say that a straight line in the Euclidean plane is a line of the form ax + by = c. With the standard formulas for angles it is easy enough to get rectangles. And I would say that a rectangle is a square when the sides have equal length (this is the point where the measure creeps in). So we have a rectangle enclosed by the lines: x + y = 1 x - y = 1 - x + y = 1 - x - y = 1 Now define the Manhattan measure: d((x1,y1), (x2,y2)) = ||x1 - x2| + |y1 - y2|| and we see easily enough that the figure enclosed by the lines above is a square with sides with length 2. A circle is a figure where each point has the same distance to a common centre, and it is also easy to show that the points on the boundary of that square have the same distance to the origin: 0. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion >... > Why are square circles unimaginable? They are not. With the Manhattan measure of the plane, each circle is >a square. Then what is a square? Pray, tell me. I would say that a straight line in the Euclidean plane >is a line of the form ax + by = c. With the standard formulas for angles >it is easy enough to get rectangles. And I would say that a rectangle is >a square when the sides have equal length (this is the point where the >measure creeps in). So we have a rectangle enclosed by the lines: > x + y = 1 > x - y = 1 > - x + y = 1 > - x - y = 1 >Now define the Manhattan measure: > d((x1,y1), (x2,y2)) = ||x1 - x2| + |y1 - y2|| >and we see easily enough that the figure enclosed by the lines above is >a square with sides with length 2. A circle is a figure where each point has the same distance to a common >centre, and it is also easy to show that the points on the boundary of >that square have the same distance to the origin: 0. Well for one thing points equidistant from any point define a sphere not a circle unless one assumes on a plane when the Euclidean plane isn't defined to begin with. But my question was directed not at the definition of a circle or square on a Euclidean plane but at the definition of a square with the Manhattan measure. It looks to me that you've just defined a square with the Manhattan metric with the properties of a circle in Euclidean plane metric. What's the point of that if you don't define a square with different properties in the Manhattan metric? In other words we have two different figures defined in the Euclidean metric, one as a curve and one with straight lines. Now I'm not trying to quibble over the modern math definition of either figure at the moment, just trying to point out that you have certain characteristics and properties defined in the Euclidean metric and then apparently claim that if you use some other metric and don't use the Euclidean metric the two figures are the same. I suppose I should ask instead are there any Euclidean curves in the Manhattan metric? If not the issue is moot. But my primary concern is that I see people all the time defining lines, figures, etc. with the Euclidean metric then going on to do non Euclidean mathematics with them. I mean Euclidean definitions require the Euclidean metric. And if you want to use some other metric you need some other definition using that metric. In which case my original question should be rephrased why are square circles unimaginable in the plane Euclidean metric? For example is it possible to define plane squares with non Euclidean metrics? And can we define right angles without the parallel postulate people simply ellide when operating with non Euclidean geometries? No. Then people complain that what they do gives the same answers. But that's only because they're using the same words yet asking different questions, with concepts defined in the Euclidean metric but operated on in some non Euclidean metric but not defined in that metric. When I ask a question such as why are square circles unimaginable the defining metric for squares and circles is Euclidean and not Manhattan and I don't expect answers that if we look at Euclidean figures through some other metric we'll find they are imaginable. ~v~~ === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl ... >Pray, tell me. I would say that a straight line in the Euclidean plane >is a line of the form ax + by = c. With the standard formulas for angles >it is easy enough to get rectangles. And I would say that a rectangle is >a square when the sides have equal length (this is the point where the >measure creeps in). So we have a rectangle enclosed by the lines: > x + y = 1 > x - y = 1 > - x + y = 1 > - x - y = 1 >Now define the Manhattan measure: > d((x1,y1), (x2,y2)) = ||x1 - x2| + |y1 - y2|| >and we see easily enough that the figure enclosed by the lines above is >a square with sides with length 2. > >A circle is a figure where each point has the same distance to a common >centre, and it is also easy to show that the points on the boundary of >that square have the same distance to the origin: 0. > > Well for one thing points equidistant from any point define a sphere > not a circle unless one assumes on a plane when the Euclidean plane > isn't defined to begin with. But my question was directed not at the > definition of a circle or square on a Euclidean plane but at the > definition of a square with the Manhattan measure. plane without measure (i.e. distance function). With that we can at most define a rectangle. Neither a square, nor a circle. > In other words we have two different figures defined in the Euclidean > metric, one as a curve and one with straight lines. Now I'm not trying > to quibble over the modern math definition of either figure at the > moment, just trying to point out that you have certain characteristics > and properties defined in the Euclidean metric and then apparently > claim that if you use some other metric and don't use the Euclidean > metric the two figures are the same. I do not use Euclidean metric at all. Where, above, do I use Euclidean metric? > For example is it possible to define plane squares with non Euclidean > metrics? Of course. > And can we define right angles without the parallel postulate > people simply ellide when operating with non Euclidean geometries? No. I think you can. But I used Euclidean geometry above. > When I > ask a question such as why are square circles unimaginable the > defining metric for squares and circles is Euclidean and not > Manhattan and I don't expect answers that if we look at Euclidean > figures through some other metric we'll find they are imaginable. In that case you should use better formulations in your questions. And when I follow-up to point out that in the Manhattan measure all circles are squares (but not the other way around) you should state that your formulation was insufficient. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion Perhaps we should replace absolute truth with culturally neutral >> truth, or in other words, truth without any cultural, religious, or >> philosophical bias. [...] Thinking about this question >> leads most of us to believe that there is a core of mathematics which >> every such civilization will accept. > without axioms, yes. For instance: I + I = II (after translating + >> and =). Therefore I call this an absolute truth. > Which axioms are you using to describe the + and = operators? Axioms? For which purpose? For the purpose of making sense of the phrase I + I = II. > Do you think the symbols constituting the > words constituting the axioms are easier or clearer to understand than > the symbols + and =? Without words that define those symbols, the symbols are meaningless. I can say that x =& y, but without any definition of x, =&, and y it is just a meaningless string of symbols. > Take an apple and then another apple. Show > the apples first apart and then together. Repeat with oranges or > fingers or mixed objects, possibly. That defines all that is needed. So then we can take yours words there as axiomatic definitions of + and =? Or are you still claiming that I + I = II means something without axiomatic definitions? === Subject: Re: Cantor Confusion Perhaps we should replace absolute truth with culturally neutral >> truth, or in other words, truth without any cultural, religious, or >> philosophical bias. [...] Thinking about this question >> leads most of us to believe that there is a core of mathematics which >> every such civilization will accept. >> without axioms, yes. For instance: I + I = II (after translating + >> and =). Therefore I call this an absolute truth. >> Which axioms are you using to describe the + and = operators? > Axioms? For which purpose? For the purpose of making sense of the phrase I + I = II. > Do you think the symbols constituting the > words constituting the axioms are easier or clearer to understand than > the symbols + and =? Without words that define those symbols, the symbols are > meaningless. I can say that x =& y, but without any definition > of x, =&, and y it is just a meaningless string of symbols. Without symbols and actions that define those words, the words are meaninless. > Take an apple and then another apple. Show > the apples first apart and then together. Repeat with oranges or > fingers or mixed objects, possibly. That defines all that is needed. So then we can take yours words there as axiomatic definitions of > + and =? Or are you still claiming that I + I = II means > something without axiomatic definitions? What do you mean with There or exists or a? How do you define set and which and has? What is the meaning of no and elements? I think these symbols and expressions (like exists and in particula, there) deserve closer examination than + and II. === Subject: Re: Cantor Confusion >> Perhaps we should replace absolute truth with culturally neutral >> truth, or in other words, truth without any cultural, religious, or >> philosophical bias. [...] Thinking about this question >> leads most of us to believe that there is a core of mathematics which >> every such civilization will accept. > >> without axioms, yes. For instance: I + I = II (after translating + >> and =). Therefore I call this an absolute truth. > >> Which axioms are you using to describe the + and = operators? Axioms? For which purpose? For the purpose of making sense of the phrase I + I = II. Do you think the symbols constituting the > words constituting the axioms are easier or clearer to understand than > the symbols + and =? Without words that define those symbols, the symbols are > meaningless. I can say that x =& y, but without any definition > of x, =&, and y it is just a meaningless string of symbols. Take an apple and then another apple. Show > the apples first apart and then together. Repeat with oranges or > fingers or mixed objects, possibly. That defines all that is needed. So then we can take yours words there as axiomatic definitions of > + and =? Or are you still claiming that I + I = II means > something without axiomatic definitions? Since WM has his own meaning for the word definition, he may have his own meaning for the word axiom (or axiomatic). -- David Marcus === Subject: Re: Cantor Confusion <451b5130$0$28952$afc38c87@news.optusnet.com.au But the set of all lists is countable (as is any quantized or > discontinuous set), so is the set of all list entries. That's absurd. First of all, I'm not really clear what discontinous means when applied to a set, but if it means connected (or, path connected) then it's certainly false that any disconnected set is countable. The irrationals are no-where-connected (ie, they have no connected subset with more than one element) and yet they're not countable. Sara === Subject: Re: Cantor Confusion <451b5130$0$28952$afc38c87@news.optusnet.com.au But the set of all lists is countable (as is any quantized or > discontinuous set), so is the set of all list entries. That's absurd. First of all, I'm not really clear what discontinous > means when applied to a set, Why then do you call it absurd? Here are some explanations: A list is a (injective) sequence. Discontinuous means: The difference between two elements (i.e. natural numbers enumerating a list) has a finite minimum value. Every list occupies a non vanishing part of the space of he universe. The volume of the (accessible) universe is finite. === Subject: Re: Cantor Confusion But the set of all lists is countable (as is any quantized or > discontinuous set), so is the set of all list entries. That's absurd. First of all, I'm not really clear what discontinous > means when applied to a set, Why then do you call it absurd? Here are some explanations: > A list is a (injective) sequence. A list is a function from N to some set in which the listed objects are to be found. While one often wishes to have such functions injective, there is no inherent reason why an object may not be listed more than once is a list, and there are lists allowing this. > Discontinuous means: The difference between two elements (i.e. natural > numbers enumerating a list) has a finite minimum value. That is not a standard mathematical meaning for continuous. Continuity/discontinuity is more standardly only defined for functions between topological spaces. There are only two commonly occurring topologies for the set of naturals, the discrete topology and the co-finite topology, and under neither of them does WM's definition make any sense. There is a property of orderings similar to what WM wants but the name of that property is discreteness not discontinuity. An ordered set is discretely ordered when there are at most finitely many members of the set between any two given members of the set. A topological space is discrete when every subset is open. > Every list occupies a non vanishing part of the space of he universe. > The volume of the (accessible) universe is finite. Those are restrictions in physics, not in math. A mathematical list can be like a point in an uncountable universe of points. === Subject: Re: Cantor Confusion Here are some explanations: > A list is a (injective) sequence. A list is a function from N to some set in which the listed objects are > to be found. While one often wishes to have such functions injective, > there is no inherent reason why an object may not be listed more than > once is a list, and there are lists allowing this. Perhaps we should be grateful that WM actually gave a definition of something (and one we can understand) rather than take him to task for using an unusual meaning for a word. It only took us a few months to convince him that not everyone knows that he means injective when he says list. -- David Marcus === Subject: Re: Cantor Confusion , I claim case iia: There is a potentially infinite sequence N = > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > treat all of its elements. In particular we can never complete this > set. We can never put it into a list OK. Knock yourself out. I read this several times from you. What does it mean? It means a fool who persists in his folly will be made wise, with the further implication that the speaker will no longer attempt to dissuade his interlocutor from a declared course of action. -- Michael Press === Subject: Re: Cantor Confusion On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical definition is >> but nonetheless want to pretend you do. If a mathematical definition >> were just an abbreviation as you claim you wouldn't have any way to >> tell one mathematical definition from another. Why not? Suppose I make the following definitions. Let N denote the set of natural numbers. > Let R denote the set of real numbers. Then I can tell N and R are different because their defintions are >different. If I write 0.5 is not in N, then this means the same as 0.5 is not in the set of natural numbers. And, it means something different from 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are only abbreviations. Granted I suppose even mathematikers can tell the difference between N and R in typographical terms. I mean they may be too lazy and stupid to demonstrate the truth of what they say but even they can see differences in typography. But in terms of abbreviations alone we can't really say what the difference is between N and R because you insist their definitions are only abbreviations and not their conceptual content. ~v~~ === Subject: Re: Cantor Confusion > On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical definition is >> but nonetheless want to pretend you do. If a mathematical definition >> were just an abbreviation as you claim you wouldn't have any way to >> tell one mathematical definition from another. Why not? Suppose I make the following definitions. Let N denote the set of natural numbers. > Let R denote the set of real numbers. Then I can tell N and R are different because their defintions are >different. If I write 0.5 is not in N, then this means the same as 0.5 is not in the set of natural numbers. And, it means something different from 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are > only abbreviations. Granted I suppose even mathematikers can tell > the difference between N and R in typographical terms. I mean they may > be too lazy and stupid to demonstrate the truth of what they say but > even they can see differences in typography. But in terms of > abbreviations alone we can't really say what the difference is between > N and R because you insist their definitions are only abbreviations > and not their conceptual content. You said there was no way to tell two definitions apart. The typographical difference suffices to tell the definitions apart (as you just admitted). -- David Marcus === Subject: Re: Cantor Confusion On Sun, 19 Nov 2006 17:53:12 -0500, David Marcus >> On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical definition is > but nonetheless want to pretend you do. If a mathematical definition > were just an abbreviation as you claim you wouldn't have any way to > tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. >> Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >>different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are >> only abbreviations. Granted I suppose even mathematikers can tell >> the difference between N and R in typographical terms. I mean they may >> be too lazy and stupid to demonstrate the truth of what they say but >> even they can see differences in typography. But in terms of >> abbreviations alone we can't really say what the difference is between >> N and R because you insist their definitions are only abbreviations >> and not their conceptual content. You said there was no way to tell two definitions apart. The >typographical difference suffices to tell the definitions apart (as you >just admitted). So what exactly is the difference between definitions N and R in conceptual terms if definitions are only abbreviations? I mean you say certain things about definitions which are mutually inconsistent. If definitions were only abbreviations as you indicate then your definitions for N and R would be restricted to those abbreviations N and R. Instead you append certain properties to each and pretend that they're part of the definitions for N and R which we'll all can see are not part of their abbreviations such that your definition for definitions is only abbreviations which are not only abbreviations. Obviously this kind of logic extends way beyond your doctoral thesis in philosophy but is nonetheless true. ~v~~ === Subject: Re: Cantor Confusion > On Sun, 19 Nov 2006 17:53:12 -0500, David Marcus >> On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical definition is > but nonetheless want to pretend you do. If a mathematical definition > were just an abbreviation as you claim you wouldn't have any way to > tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. >> Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >>different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are >> only abbreviations. Granted I suppose even mathematikers can tell >> the difference between N and R in typographical terms. I mean they may >> be too lazy and stupid to demonstrate the truth of what they say but >> even they can see differences in typography. But in terms of >> abbreviations alone we can't really say what the difference is between >> N and R because you insist their definitions are only abbreviations >> and not their conceptual content. You said there was no way to tell two definitions apart. The >typographical difference suffices to tell the definitions apart (as you >just admitted). So what exactly is the difference between definitions N and R in > conceptual terms if definitions are only abbreviations? I mean you > say certain things about definitions which are mutually inconsistent. > If definitions were only abbreviations as you indicate then your > definitions for N and R would be restricted to those abbreviations N > and R. Instead you append certain properties to each and pretend that > they're part of the definitions for N and R which we'll all can see > are not part of their abbreviations such that your definition for > definitions is only abbreviations which are not only abbreviations. > Obviously this kind of logic extends way beyond your doctoral thesis > in philosophy but is nonetheless true. That's impressive. Either you are trolling or you have completely misunderstood what people mean when they say definitions are abbreviations. -- David Marcus === Subject: Re: Cantor Confusion On Mon, 20 Nov 2006 14:35:20 -0500, David Marcus >> On Sun, 19 Nov 2006 17:53:12 -0500, David Marcus > On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical definition is >> but nonetheless want to pretend you do. If a mathematical definition >> were just an abbreviation as you claim you wouldn't have any way to >> tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. > Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are > only abbreviations. Granted I suppose even mathematikers can tell > the difference between N and R in typographical terms. I mean they may > be too lazy and stupid to demonstrate the truth of what they say but > even they can see differences in typography. But in terms of > abbreviations alone we can't really say what the difference is between > N and R because you insist their definitions are only abbreviations > and not their conceptual content. >>You said there was no way to tell two definitions apart. The >>typographical difference suffices to tell the definitions apart (as you >>just admitted). So what exactly is the difference between definitions N and R in >> conceptual terms if definitions are only abbreviations? I mean you >> say certain things about definitions which are mutually inconsistent. >> If definitions were only abbreviations as you indicate then your >> definitions for N and R would be restricted to those abbreviations N >> and R. Instead you append certain properties to each and pretend that >> they're part of the definitions for N and R which we'll all can see >> are not part of their abbreviations such that your definition for >> definitions is only abbreviations which are not only abbreviations. >> Obviously this kind of logic extends way beyond your doctoral thesis >> in philosophy but is nonetheless true. That's impressive. Either you are trolling or you have completely >misunderstood what people mean when they say definitions are >abbreviations. Or quite possibly you misunderstand what people mean by abbreviations. Not quite the same as the sloth and professional turpitude of mathematical definitions you're used to I daresay. N and R are abbreviations. Whatever you imagine by what you attach to abbreviations are not abbreviations. But please do go on and don't allow me to distract you from your modern mathematical definition of abbreviations which I'm quite confindent will afford us many a pleasant evening of muffled laughter. ~v~~ === Subject: Re: Cantor Confusion > On Mon, 20 Nov 2006 14:35:20 -0500, David Marcus >> On Sun, 19 Nov 2006 17:53:12 -0500, David Marcus > On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical definition is >> but nonetheless want to pretend you do. If a mathematical definition >> were just an abbreviation as you claim you wouldn't have any way to >> tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. > Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are > only abbreviations. Granted I suppose even mathematikers can tell > the difference between N and R in typographical terms. I mean they may > be too lazy and stupid to demonstrate the truth of what they say but > even they can see differences in typography. But in terms of > abbreviations alone we can't really say what the difference is between > N and R because you insist their definitions are only abbreviations > and not their conceptual content. >>You said there was no way to tell two definitions apart. The >>typographical difference suffices to tell the definitions apart (as you >>just admitted). So what exactly is the difference between definitions N and R in >> conceptual terms if definitions are only abbreviations? I mean you >> say certain things about definitions which are mutually inconsistent. >> If definitions were only abbreviations as you indicate then your >> definitions for N and R would be restricted to those abbreviations N >> and R. Instead you append certain properties to each and pretend that >> they're part of the definitions for N and R which we'll all can see >> are not part of their abbreviations such that your definition for >> definitions is only abbreviations which are not only abbreviations. >> Obviously this kind of logic extends way beyond your doctoral thesis >> in philosophy but is nonetheless true. That's impressive. Either you are trolling or you have completely >misunderstood what people mean when they say definitions are >abbreviations. Or quite possibly you misunderstand what people mean by > abbreviations. Not quite the same as the sloth and professional > turpitude of mathematical definitions you're used to I daresay. N and > R are abbreviations. Whatever you imagine by what you attach to > abbreviations are not abbreviations. But please do go on and don't > allow me to distract you from your modern mathematical definition of > abbreviations which I'm quite confindent will afford us many a > pleasant evening of muffled laughter. And, we can let U.S. stand for the United States. Is this abbreviation/definition/whatever true or false, in your opinion? -- David Marcus === Subject: Re: Cantor Confusion >> I'm saying that you don't understand what a mathematical definition is but nonetheless want to pretend you do. If a mathematical definition were just an abbreviation as you claim you wouldn't have any way to tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. >> Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >>different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are >> only abbreviations. Granted I suppose even mathematikers can tell >> the difference between N and R in typographical terms. I mean they may >> be too lazy and stupid to demonstrate the truth of what they say but >> even they can see differences in typography. But in terms of >> abbreviations alone we can't really say what the difference is between >> N and R because you insist their definitions are only abbreviations >> and not their conceptual content. >>You said there was no way to tell two definitions apart. The >typographical difference suffices to tell the definitions apart (as you >just admitted). So what exactly is the difference between definitions N and R in > conceptual terms if definitions are only abbreviations? I mean you > say certain things about definitions which are mutually inconsistent. > If definitions were only abbreviations as you indicate then your > definitions for N and R would be restricted to those abbreviations N > and R. Instead you append certain properties to each and pretend that > they're part of the definitions for N and R which we'll all can see > are not part of their abbreviations such that your definition for > definitions is only abbreviations which are not only abbreviations. > Obviously this kind of logic extends way beyond your doctoral thesis > in philosophy but is nonetheless true. >>That's impressive. Either you are trolling or you have completely >>misunderstood what people mean when they say definitions are >>abbreviations. Or quite possibly you misunderstand what people mean by >> abbreviations. Not quite the same as the sloth and professional >> turpitude of mathematical definitions you're used to I daresay. N and >> R are abbreviations. Whatever you imagine by what you attach to >> abbreviations are not abbreviations. But please do go on and don't >> allow me to distract you from your modern mathematical definition of >> abbreviations which I'm quite confindent will afford us many a >> pleasant evening of muffled laughter. And, we can let U.S. stand for the United States. Is this >abbreviation/definition/whatever true or false, in your opinion? That is an abbreviation not a definition. I never suggested there couldn't be abbreviations in definitions only that definitions were not only abbreviations as you claim mathematics requires. If an abbreviation denotes a false definition it is equally false to the extent it denotes definitional content and not just typography. But the bottom line seems to be that you're only an abbreviation for a philosophy whore playing grabass with modern mathematics, and what might in polite society otherwise be termed an abbreviation for a lying sack of . ~v~~ === Subject: Re: Cantor Confusion <9bm1m21rp3hsofrhhio2r5c7l7f8he0fjh@4ax.com> <61a4m255p98vr2301tbvp4g0dnnriv0t9m@4ax.com> <6qi6m2d2kenetrr8ma42r98pahna4o5n50@4ax.com On Mon, 20 Nov 2006 17:40:59 -0500, David Marcus > On Mon, 20 Nov 2006 14:35:20 -0500, David Marcus > On Sun, 19 Nov 2006 17:53:12 -0500, David Marcus >> On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical definition is but nonetheless want to pretend you do. If a mathematical definition were just an abbreviation as you claim you wouldn't have any way to tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. >> Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >>different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. >> But the problem, sport, is you claim mathematical definitions are >> only abbreviations. Granted I suppose even mathematikers can tell >> the difference between N and R in typographical terms. I mean they may >> be too lazy and stupid to demonstrate the truth of what they say but >> even they can see differences in typography. But in terms of >> abbreviations alone we can't really say what the difference is between >> N and R because you insist their definitions are only abbreviations >> and not their conceptual content. >>You said there was no way to tell two definitions apart. The >typographical difference suffices to tell the definitions apart (as you >just admitted). >> So what exactly is the difference between definitions N and R in > conceptual terms if definitions are only abbreviations? I mean you > say certain things about definitions which are mutually inconsistent. > If definitions were only abbreviations as you indicate then your > definitions for N and R would be restricted to those abbreviations N > and R. Instead you append certain properties to each and pretend that > they're part of the definitions for N and R which we'll all can see > are not part of their abbreviations such that your definition for > definitions is only abbreviations which are not only abbreviations. > Obviously this kind of logic extends way beyond your doctoral thesis > in philosophy but is nonetheless true. >>That's impressive. Either you are trolling or you have completely >>misunderstood what people mean when they say definitions are >>abbreviations. >> Or quite possibly you misunderstand what people mean by >> abbreviations. Not quite the same as the sloth and professional >> turpitude of mathematical definitions you're used to I daresay. N and >> R are abbreviations. Whatever you imagine by what you attach to >> abbreviations are not abbreviations. But please do go on and don't >> allow me to distract you from your modern mathematical definition of >> abbreviations which I'm quite confindent will afford us many a >> pleasant evening of muffled laughter. And, we can let U.S. stand for the United States. Is this >abbreviation/definition/whatever true or false, in your opinion? That is an abbreviation not a definition. Definitions are abbreviations. N is an abbreviation for the set of natural numbers. The set of natural numbers is an abbreviation for the minimal set which obeys... [Peano axioms]. The conceptual content is contained in those axioms, which are not definitions and are not abbreviations. - Randy === Subject: Re: Cantor Confusion On 21 Nov 2006 13:30:37 -0800, Randy Poe > On Mon, 20 Nov 2006 17:40:59 -0500, David Marcus [. . .] >>And, we can let U.S. stand for the United States. Is this >>abbreviation/definition/whatever true or false, in your opinion? >> That is an abbreviation not a definition. Definitions are abbreviations. This is truly stupifying, Randy. >N is an abbreviation for the set of natural numbers. However it's not a definition for the set of natural numbers. >The set of natural numbers is an abbreviation for the >minimal set which obeys... [Peano axioms]. And the set of natural numbers is an abbreviation for your asshole. >The conceptual content is contained in those axioms, which >are not definitions and are not abbreviations. And the conceptual content of those axioms is an abbreviation for your stupidity. ~v~~ === Subject: Re: Cantor Confusion >And, we can let U.S. stand for the United States. Is this >>abbreviation/definition/whatever true or false, in your opinion? >> That is an abbreviation not a definition. Definitions are abbreviations. This is truly stupifying, Randy. You'd think so, but it's a trivial concept that you seem to have enormous trouble grasping. No problem, I'll step you through it. >N is an abbreviation for the set of natural numbers. However it's not a definition for the set of natural numbers. No, you've got that backwards. It's a definition of the symbol N. When we say X is an abbreviation of Y, that defines X. Y must be defined elsewhere. >The set of natural numbers is an abbreviation for the >minimal set which obeys... [Peano axioms]. And the set of natural numbers is an abbreviation for your asshole. No, that's not correct. Natural numbers is a meaningless phrase until we provide a definition for what concept that phrase is a shorthand. And the concept in question is the set which satisfies the Peano axioms. >The conceptual content is contained in those axioms, which >are not definitions and are not abbreviations. And the conceptual content of those axioms is an abbreviation for your > stupidity. No, that's not correct. - Randy === Subject: Re: Cantor Confusion On 21 Nov 2006 14:49:52 -0800, Randy Poe > On 21 Nov 2006 13:30:37 -0800, Randy Poe > On Mon, 20 Nov 2006 17:40:59 -0500, David Marcus >> [. . .] >And, we can let U.S. stand for the United States. Is this >abbreviation/definition/whatever true or false, in your opinion? >> That is an abbreviation not a definition. >>Definitions are abbreviations. >> This is truly stupifying, Randy. You'd think so, but it's a trivial concept that you seem to have >enormous trouble grasping. No problem, I'll step you through it. DM's contention that mathematical definitions are only abbreviations? >>N is an abbreviation for the set of natural numbers. >> However it's not a definition for the set of natural numbers. No, you've got that backwards. It's a definition of the symbol N. >When we say X is an abbreviation of Y, that defines X. Y must >be defined elsewhere. So we've got a definition of X that must be defined elsewhere? Real bright. I've got an idea: for the definition of X why not just define X instead of claiming it's defined elsewhere? >>The set of natural numbers is an abbreviation for the >>minimal set which obeys... [Peano axioms]. >> And the set of natural numbers is an abbreviation for your asshole. No, that's not correct. Natural numbers is a meaningless >phrase until we provide a definition for what concept that >phrase is a shorthand. And the concept in question is the >set which satisfies the Peano axioms. All you're saying is the predicates you use to define one thing have their own definitions. Big ing deal. No one said they don't. But the combination and recombination of predicates used to define one thing are more than the definition of those predicates in isolation. Consider the definition of X=Y Z. Y and Z each have their own definitions considered individually. And X is an abbreviation for those predicates in combination. But the definition of X combines Y and Z in a certain order which denotes the recombination of their definitions. This isn't rocket science and nothing in this observation justifies DM's conclusion that definitions are only abbreviations. If all you're going to do to define natural numbers is denote the Peano axioms then the definition for natural numbers are the Peano axioms and natural numbers is an abbreviation for those axioms. But just putting together another name for the Peano axioms doesn't define anything. They're still the Peano axioms and nothing more and you're equation of the Peano axioms to the natural numbers is just a facile assumption of truth having no necessary connection to what people actually mean when they use the phrase natural numbers. Let me spell it out for you. DM maintains definitions are only abbreviations which is nonsense. If you or he maintained definitions are sequences of predicates and abbreviations for predicates in combination I would agree. He doesn't maintain that. However definitions are more than simple denotations. In point of fact there would be no reason to denote the Peano axioms as natural numbers because it would be just as easy to say the Peano axioms as it would natural numbers. In fact N is not the set of natural numbers; it is a subset of the natural numbers and the point of defining N is to show the combination of predicates which produces the subset N of the natural numbers. The same is also true for natural numbers themselves. And just saying the definition of natural numbers is the Peano axioms doesn't show how that happens. In the first place when you define natural numbers as the Peano axioms you're not defining the natural numbers at all. At best all you're doing is recapitulating your conviction that natural numbers are generated in accordance with the assumptions represented by the Peano axioms. I could do exactly the same by saying 1 and 1+1 or unity and unity plus unity. But the Peano axioms are not the definition for anything but the Peano axioms and what is generated in accordance with the Peano axioms is nothing more than what is generated by 1 and 1+1. The difficulty with such inane definitions is that they're designed to conceal the defining and definitional element, which has nothing to do with unity at all but with the +. That's the problem and no amount of pretentious temporizing can conceal the fact that the predicate + represents an undemonstrated and undemonstrable assumption of truth. >>The conceptual content is contained in those axioms, which >>are not definitions and are not abbreviations. >> And the conceptual content of those axioms is an abbreviation for your >> stupidity. No, that's not correct. Yadayada whatever. ~v~~ === Subject: Re: Cantor Confusion > On Mon, 20 Nov 2006 14:35:20 -0500, David Marcus >> On Sun, 19 Nov 2006 17:53:12 -0500, David Marcus > On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical >> definition is >> but nonetheless want to pretend you do. If a mathematical >> definition >> were just an abbreviation as you claim you wouldn't have any way >> to >> tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. > Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are > only abbreviations. Granted I suppose even mathematikers can tell > the difference between N and R in typographical terms. I mean they > may > be too lazy and stupid to demonstrate the truth of what they say but > even they can see differences in typography. But in terms of > abbreviations alone we can't really say what the difference is > between > N and R because you insist their definitions are only abbreviations > and not their conceptual content. >>You said there was no way to tell two definitions apart. The >>typographical difference suffices to tell the definitions apart (as you >>just admitted). So what exactly is the difference between definitions N and R in >> conceptual terms if definitions are only abbreviations? I mean you >> say certain things about definitions which are mutually inconsistent. >> If definitions were only abbreviations as you indicate then your >> definitions for N and R would be restricted to those abbreviations N >> and R. Instead you append certain properties to each and pretend that >> they're part of the definitions for N and R which we'll all can see >> are not part of their abbreviations such that your definition for >> definitions is only abbreviations which are not only abbreviations. >> Obviously this kind of logic extends way beyond your doctoral thesis >> in philosophy but is nonetheless true. That's impressive. Either you are trolling or you have completely >misunderstood what people mean when they say definitions are >abbreviations. Or quite possibly you misunderstand what people mean by > abbreviations. Not quite the same as the sloth and professional > turpitude of mathematical definitions you're used to I daresay. N and > R are abbreviations. Whatever you imagine by what you attach to > abbreviations are not abbreviations. But please do go on and don't > allow me to distract you from your modern mathematical definition of > abbreviations which I'm quite confindent will afford us many a > pleasant evening of muffled laughter. And, we can let U.S. stand for the United States. Is this > abbreviation/definition/whatever true or false, in your opinion? There is no point in trying to communicate with Zick, as he is not the least bit interested in communicating. Your best response to his blather is kill filing him. === Subject: Re: Cantor Confusion > On Mon, 20 Nov 2006 14:35:20 -0500, David Marcus > On Sun, 19 Nov 2006 17:53:12 -0500, David Marcus >> On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> I'm saying that you don't understand what a mathematical definition is but nonetheless want to pretend you do. If a mathematical definition were just an abbreviation as you claim you wouldn't have any way to tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. >> Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >>different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are >> only abbreviations. Granted I suppose even mathematikers can tell >> the difference between N and R in typographical terms. I mean they >> may >> be too lazy and stupid to demonstrate the truth of what they say but >> even they can see differences in typography. But in terms of >> abbreviations alone we can't really say what the difference is >> between >> N and R because you insist their definitions are only abbreviations >> and not their conceptual content. >>You said there was no way to tell two definitions apart. The >typographical difference suffices to tell the definitions apart (as you >just admitted). So what exactly is the difference between definitions N and R in > conceptual terms if definitions are only abbreviations? I mean you > say certain things about definitions which are mutually inconsistent. > If definitions were only abbreviations as you indicate then your > definitions for N and R would be restricted to those abbreviations N > and R. Instead you append certain properties to each and pretend that > they're part of the definitions for N and R which we'll all can see > are not part of their abbreviations such that your definition for > definitions is only abbreviations which are not only abbreviations. > Obviously this kind of logic extends way beyond your doctoral thesis > in philosophy but is nonetheless true. >>That's impressive. Either you are trolling or you have completely >>misunderstood what people mean when they say definitions are >>abbreviations. Or quite possibly you misunderstand what people mean by >> abbreviations. Not quite the same as the sloth and professional >> turpitude of mathematical definitions you're used to I daresay. N and >> R are abbreviations. Whatever you imagine by what you attach to >> abbreviations are not abbreviations. But please do go on and don't >> allow me to distract you from your modern mathematical definition of >> abbreviations which I'm quite confindent will afford us many a >> pleasant evening of muffled laughter. And, we can let U.S. stand for the United States. Is this >> abbreviation/definition/whatever true or false, in your opinion? There is no point in trying to communicate with Zick, as he is not the >least bit interested in communicating. Yes, but I must note that you're not the least bit interested in communicating what Zick is not interested in communicating. >Your best response to his blather is kill filing him. Yes well since Aatu called you pathetic have you also killfiled him? ~v~~ === Subject: Re: Cantor Confusion And, we can let U.S. stand for the United States. Is this > abbreviation/definition/whatever true or false, in your opinion? There is no point in trying to communicate with Zick, as he is not the > least bit interested in communicating. You are right, but I felt compelled to make one last post. Sometimes these urges are hard to resist. > Your best response to his blather is kill filing him. If I continue to have these unhealthy urges, I'll do that. -- David Marcus === Subject: Re: Cantor Confusion misunderstood what people mean when they say definitions are > abbreviations. Lester misunderstands pretty much anything anyone says. I suppose that is what happens when someone perversely uses their own personal vocabulary for too long. Stephen === Subject: Re: Cantor Confusion >> That's impressive. Either you are trolling or you have completely >> misunderstood what people mean when they say definitions are >> abbreviations. Lester misunderstands pretty much anything anyone says. Lester certainly misunderstands pretty much everything which is not demonstrably true. Occupational hazard Lester expects when one deals with truth. Which Lester also expects doesn't include pretty much everything Stephen says. > I suppose >that is what happens when someone perversely uses their own personal >vocabulary for too long. Probably the very difficulty David is experiencing with his private definitions for truth and abbreviations. ~v~~ === Subject: Re: Cantor Confusion > On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus > >> I'm saying that you don't understand what a mathematical definition is >> but nonetheless want to pretend you do. If a mathematical definition >> were just an abbreviation as you claim you wouldn't have any way to >> tell one mathematical definition from another. Why not? Suppose I make the following definitions. Let N denote the set of natural numbers. > Let R denote the set of real numbers. Then I can tell N and R are different because their defintions are >different. If I write 0.5 is not in N, then this means the same as 0.5 is not in the set of natural numbers. And, it means something different from 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are > only abbreviations. Granted I suppose even mathematikers can tell > the difference between N and R in typographical terms. I mean they may > be too lazy and stupid to demonstrate the truth of what they say but > even they can see differences in typography. But in terms of > abbreviations alone we can't really say what the difference is between > N and R because you insist their definitions are only abbreviations > and not their conceptual content. You said there was no way to tell two definitions apart. The > typographical difference suffices to tell the definitions apart (as you > just admitted). Responding to Zick is a waste. Even reading him is a waste. I found killfiling him to be much more profitable. === Subject: Re: Cantor Confusion > On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >I'm saying that you don't understand what a mathematical definition is > but nonetheless want to pretend you do. If a mathematical definition > were just an abbreviation as you claim you wouldn't have any way to > tell one mathematical definition from another. >>Why not? Suppose I make the following definitions. >> Let N denote the set of natural numbers. >> Let R denote the set of real numbers. >>Then I can tell N and R are different because their defintions are >>different. If I write >> 0.5 is not in N, >>then this means the same as >> 0.5 is not in the set of natural numbers. >>And, it means something different from >> 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are >> only abbreviations. Granted I suppose even mathematikers can tell >> the difference between N and R in typographical terms. I mean they may >> be too lazy and stupid to demonstrate the truth of what they say but >> even they can see differences in typography. But in terms of >> abbreviations alone we can't really say what the difference is between >> N and R because you insist their definitions are only abbreviations >> and not their conceptual content. You said there was no way to tell two definitions apart. The >> typographical difference suffices to tell the definitions apart (as you >> just admitted). Responding to Zick is a waste. >Even reading him is a waste. >I found killfiling him to be much more profitable. Which is probably why you keep on responding, Virgil. In the great scheme of things this allows me to talk about your incompetence as if you weren't present which in the great scheme of things you aren't. ~v~~ === Subject: Re: Cantor Confusion > Responding to Zick is a waste. > Even reading him is a waste. > I found killfiling him to be much more profitable. I basically agree. He was almost coherent for a little while there. But, now he seems to have reverted back to making inane comments. He's just a heckler. -- David Marcus === Subject: Re: Cantor Confusion On Sun, 19 Nov 2006 18:32:31 -0500, David Marcus > Responding to Zick is a waste. >> Even reading him is a waste. >> I found killfiling him to be much more profitable. I basically agree. He was almost coherent for a little while there. But, >now he seems to have reverted back to making inane comments. He's just a >heckler. Personally I find inanity closer to sanity in duels with halfwits. ~v~~ === Subject: Re: Cantor Confusion On Sat, 18 Nov 2006 13:31:27 -0500, David Marcus > The usefulness to other fields is demonstrated via the >> scientific method, not by mathematical proof. You mean the empirical method not the scientific method. I think I meant what I said. Fortunately what you say isn't demonstrably true. >> So mathematical axioms are to be empirically demonstrated now? I didn't say mathematical axioms are to be empirically demonstrated. I >said that other fields demonstrate that mathematics is useful to them >using the methods appropriate to those fields. Exactly what I said. The truth of mathematical axioms is empirically demonstrated. >> Are you saying you don't know what the word >> proof means in mathematics? I'm saying you can't prove the truth of whatever you say in or about >> mathematics. But, do you know what the word prove means in Mathematics? It isn't >the same as what it means in English. Hey gimme a break. I'm still trying to biject the set of {proven} with the set of {true}. Not happening. >> A major purpose of axioms is to avoid ambiguity. The main purpose of axioms is to provide assumptions of truth without >> proof. Why do you think that? Because it's true? > Are you going to illustrate the existence of infinites by production > of one or more; or are you going to demonstrate the truth of their > existence by some alternative means? You posit certain properties and > characteristics of things you call infinites but don't show they can > actually be realized in combination with one another. >>Sorry. Don't know what you mean. In particular, I don't know what you >>mean by illustrate the existence, demonstrate the truth of their >>existence, actually be realized. Can you give an example? I should give you examples of the examples of infinites I asked you >> for? I didn't say that. I said you should give an example to show the meaning >of the phrases I quoted. Pick something and illustrate its existence. How about if I pick infinites and you give me an example to prove it's true? >> All you do in modern math is prove theorems from assumptions of truth. >> Not exactly overtaxing intellectually but there it is. If by assumptions of truth you mean axioms, then that is correct. Of course it is. >>Have you read any math books at the junior/senior college level or >>above? Apparently more than you. Could be. Which math books have you read? None. > Now I don't say there aren't cranks out there but there is also > truth out there and you don't have a clue as to how to get at it. >>Why do you think truth is relevant to mathematics? I gave you the citation. I can't imagine why you think truth isn't >> relevant to mathematics. What is the citation? I think I missed it. You miss most everything. >>Anyone who learns math is welcome to call themselves a mathematician. In other words anyone who learns to agree with you is welcome to call >> themselves mathematicians? I didn't say that. You don't say anything that's demonstrably true. >>how it could be that math could be used in science of all types if the >>mathematicians are really as you portray. Lucky guesses. I never said modern mathematikers, quantum empirics, >> and relativists weren't lucky just that they were too lazy or stupid >> to figure out the truth of what they were saying. We should all switch from mathematics to playing the lottery. You already have. >> It isn't enough to learn the word. First, you have to >> understand the concept. This takes work. But you said mathematical definitions are only abbreviations and now >> alluva sudden you expect people to learn concepts instead? How droll. Sure. There is a difference between a thing and its name. Simply being >able to recite a definition doesn't mean you understand it and can use >it properly. But being able to recite an abbreviation does? >>What books on the topic have you read? What courses have you taken? Apparently more than you. Could be. Which books have you read and which courses have you taken? None. >> I can't even get you to discuss the truth of what you say. All I hear >> about are your assumptions of truth in modern math. I never said that. Problem is you don't say anything that's demonstrably true. ~v~~ === Subject: Re: Cantor Confusion > On Sat, 18 Nov 2006 13:31:27 -0500, David Marcus > Are you saying you don't know what the word proof means in mathematics? I'm saying you can't prove the truth of whatever you say in or about >> mathematics. But, do you know what the word prove means in Mathematics? It isn't >the same as what it means in English. Hey gimme a break. I'm still trying to biject the set of {proven} with > the set of {true}. Not happening. I suppose that means you don't know what the word prove means in mathematics. I suggest you refrain from responding to any posts that contain the word. >>Have you read any math books at the junior/senior college level or >>above? Apparently more than you. Could be. Which math books have you read? None. > >>What books on the topic have you read? What courses have you taken? Apparently more than you. Could be. Which books have you read and which courses have you taken? None. Ignorance is bliss? I'm not sure how you could have read more books than me if you've read none. -- David Marcus === Subject: Re: Cantor Confusion On Sun, 19 Nov 2006 20:01:20 -0500, David Marcus >> On Sat, 18 Nov 2006 13:31:27 -0500, David Marcus > Are you saying you don't know what the word proof means in mathematics? I'm saying you can't prove the truth of whatever you say in or about > mathematics. >>But, do you know what the word prove means in Mathematics? It isn't >>the same as what it means in English. Hey gimme a break. I'm still trying to biject the set of {proven} with >> the set of {true}. Not happening. I suppose that means you don't know what the word prove means in >mathematics. I suggest you refrain from responding to any posts that >contain the word. And I suggest you refrain from using the word true. >Have you read any math books at the junior/senior college level or >above? Apparently more than you. >>Could be. Which math books have you read? None. >> >What books on the topic have you read? What courses have you taken? Apparently more than you. >>Could be. Which books have you read and which courses have you taken? None. Ignorance is bliss? Yours would appear to be. >I'm not sure how you could have read more books than me if you've read >none. Simply because I define it so. Just one of the many mysteries of transfinite arithmetic. The books you've read are negative numbers. ~v~~ === Subject: Re: Cantor Confusion > Are you saying you don't know what the word proof means in >> mathematics? I'm saying you can't prove the truth of whatever you say in or about >> mathematics. But, do you know what the word prove means in Mathematics? It isn't >the same as what it means in English. Hey gimme a break. I'm still trying to biject the set of {proven} with > the set of {true}. Not happening. I suppose that means you don't know what the word prove means in > mathematics. I suggest you refrain from responding to any posts that > contain the word. > >>Have you read any math books at the junior/senior college level or >>above? Apparently more than you. Could be. Which math books have you read? None. > >>What books on the topic have you read? What courses have you taken? Apparently more than you. Could be. Which books have you read and which courses have you taken? None. Ignorance is bliss? I'm not sure how you could have read more books than me if you've read > none. Doesn't such a concatenation of claims, having read more books that you and having read none, qualify Zick for a stupid award? === Subject: Re: Cantor Confusion > On Sat, 18 Nov 2006 13:31:27 -0500, David Marcus > On Thu, 16 Nov 2006 02:28:14 -0500, David Marcus > Are you saying you don't know what the word proof means in > mathematics? I'm saying you can't prove the truth of whatever you say in or about > mathematics. >>But, do you know what the word prove means in Mathematics? It isn't >>the same as what it means in English. Hey gimme a break. I'm still trying to biject the set of {proven} with >> the set of {true}. Not happening. I suppose that means you don't know what the word prove means in >> mathematics. I suggest you refrain from responding to any posts that >> contain the word. >> >Have you read any math books at the junior/senior college level or >above? Apparently more than you. >>Could be. Which math books have you read? None. >> >What books on the topic have you read? What courses have you taken? Apparently more than you. >>Could be. Which books have you read and which courses have you taken? None. Ignorance is bliss? I'm not sure how you could have read more books than me if you've read >> none. Doesn't such a concatenation of claims, having read more books that you >and having read none, qualify Zick for a stupid award? It certainly qualifies you for one. ~v~~ === Subject: Re: Cantor Confusion > Are you saying you don't know what the word proof means in >> mathematics? I'm saying you can't prove the truth of whatever you say in or about >> mathematics. But, do you know what the word prove means in Mathematics? It isn't >the same as what it means in English. Hey gimme a break. I'm still trying to biject the set of {proven} with > the set of {true}. Not happening. I suppose that means you don't know what the word prove means in > mathematics. I suggest you refrain from responding to any posts that > contain the word. > >>Have you read any math books at the junior/senior college level or >>above? Apparently more than you. Could be. Which math books have you read? None. > >>What books on the topic have you read? What courses have you taken? Apparently more than you. Could be. Which books have you read and which courses have you taken? None. Ignorance is bliss? I'm not sure how you could have read more books than me if you've read > none. Doesn't such a concatenation of claims, having read more books that you > and having read none, qualify Zick for a stupid award? It certainly would seem to meet the minimum qualifications. -- David Marcus === Subject: Re: Cantor Confusion On Sun, 19 Nov 2006 22:13:46 -0500, David Marcus > Are you saying you don't know what the word proof means in > mathematics? I'm saying you can't prove the truth of whatever you say in or about > mathematics. >>But, do you know what the word prove means in Mathematics? It isn't >>the same as what it means in English. Hey gimme a break. I'm still trying to biject the set of {proven} with >> the set of {true}. Not happening. I suppose that means you don't know what the word prove means in >> mathematics. I suggest you refrain from responding to any posts that >> contain the word. >> >Have you read any math books at the junior/senior college level or >above? Apparently more than you. >>Could be. Which math books have you read? None. >> >What books on the topic have you read? What courses have you taken? Apparently more than you. >>Could be. Which books have you read and which courses have you taken? None. Ignorance is bliss? I'm not sure how you could have read more books than me if you've read >> none. Doesn't such a concatenation of claims, having read more books that you >> and having read none, qualify Zick for a stupid award? It certainly would seem to meet the minimum qualifications. Just as you seem to meet the maximum qualifications. ~v~~ === Subject: Re: Cantor Confusion On Sat, 18 Nov 2006 13:13:56 -0500, David Marcus >> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus > Provability of what pray tell? If you're not concerned with proving > the truth of what you say in mathematics exactly when are you not > discussing philosophy every time you say anything in mathematics? >>Do you really not know the mathematical meaning of the word prove? If >>so, I (and others) could try to explain it to you. But, if you are just >>being argumentative, we won't bother. What is it you think you're proving? Does that mean you don't know the mathematical meaning of the word >prove? It isn't the same as the English meaning. Apparently it isn't also the same as true either. >>Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. Are you sure that is what Stephen thinks? What difference does that make if what Stephen thinks isn't true? >A set isn't a proper subset of itself. Sure it is if you define it so. > If we have two sets A and B, then >we say that A is a proper subset of B if every element in A is also in B > and > there is some element in B that is not in A If we take B = A, then every element in B is also in A. So, A is not a >proper subset of itself. So what? You say a lot of stuff but nothing you say is demonstrably true. ~v~~ === Subject: Re: Cantor Confusion On Sat, 18 Nov 2006 13:35:27 -0500, David Marcus > Don't know what you mean by actual 'truth'. Can you give an example? Oh dear, oh dear, you have not, I detect, being paying attention. Probably not. I've been skipping over most of Lester's posts, since they >haven't been too interesting. Uninteresting but true. >> Lester tells us bits of Truth all the Time, in particular our old >> favourite ~v~~. This is Truth, baby, pure, 100% fat-free Truth. (I >> mean, you can read it, right? Not-or-not-not. Oh. So that's what that is. True. >> Just search the archives for Lester's explanation, Ah, so much to read, so little time. Ah so much to read so little that's true. >> and no, don't ask me, or anyone >> but Lester, not that that is likely to get you any further than it got >> everyone else.) I'm sure you are correct. But not true? >> Zick: >> go yourself. Ah, well. It seems you aren't interested in really learning anything, >> after all. Hmm, perhaps you are catching on, after all. Despite lofty assertions, >> it seems Lester's ultimate regression is always to foul language. Deep-seated anger? Deep seated truth. ~v~~ === Subject: Re: Cantor Confusion On Sat, 18 Nov 2006 12:50:30 -0500, David Marcus >> On Thu, 16 Nov 2006 01:35:12 -0500, David Marcus >It is difficult to answer this question, because the expression set > is occupied in modern mathematics by collections of elements which are > actually there (you don't know what that means, imagine just a set as > you know it). Such infinite sets do not exist. While infinite collections in any physical sense are not possible, why > are imaginary infinities, such as sets of numbers must be, unimaginable? Why are square circles unimaginable? Depends on what you mean by unimaginable. Not my term, slick. Ask whatsitsface. >>For that matter, we can always switch from Platonism to formalism and >>declare the question of whether sets really exist to be a philosophical >>question. So is the switch from platonism to formalism a philosophical question? Yes. Platonism and formalism are philosophies of mathematics. Untrue. They're philosophies of daydreaming. > Regardless >of which you prefer (or if you prefer something else), it doesn't change >which theorems are provable in which axiom systems. Fortunately it also doesn't change which theorems are demonstrably true and which are not demonstrably true except by assumptions of truth in modern mathematics. ~v~~ === Subject: Re: Cantor Confusion >>Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. Are you sure that is what Stephen thinks? I see Lester has resorted to lying. This is all part of his > standard pattern. He really is pathetic. It is amusing to see > how increasingly pathetic he becomes. > And, such a silly lie. What could he hope to gain? I am not sure if it is really even just a lie. Of course not. You're not really sure of anything. > Honestly I do not >think Lester is capable of comprehending anything mathematical, >so when he tries to restate what anybody else has said he >gets it wrong not just because of dishonesty (he has >clearly demonstrated a decided dishonest streak), but because of >ineptitude. Of course. Just not quite the degree of ineptitude as those who think dr=v. > When he reads something like > A set is infinite if there exists a bijection between itself > and a proper subset. >all that comes through is > * set is infinite ** ***** ****** * ********* ****** itself > *** * proper subset. Fortunately it doesn't really matter how it comes through because it isn't really true to a modern mathematiker in any event. >and that is somehow boiled down into An infinite set is a proper subset >of itself. It reminds me of this: dr=v perchance? ~v~~ === Subject: Re: Cantor Confusion On Sat, 18 Nov 2006 15:41:13 -0500, David Marcus >Please give a specific example of something that you think is absurd or >a contradiction. I don't know what you mean by containment of sets and >subsets. Well as I recollect Stephen seems to think infinite sets are proper > subsets of themselves. Are you sure that is what Stephen thinks? I see Lester has resorted to lying. This is all part of his >> standard pattern. He really is pathetic. It is amusing to see >> how increasingly pathetic he becomes. And, such a silly lie. As opposed to a silly truth? > What could he hope to gain? Some insight into demonstrable truth no doubt. ~v~~ === Subject: Re: Cantor Confusion > On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus >> Provability of what pray tell? If you're not concerned with proving >> the truth of what you say in mathematics exactly when are you not >> discussing philosophy every time you say anything in mathematics? >>Do you really not know the mathematical meaning of the word prove? If >so, I (and others) could try to explain it to you. But, if you are just >being argumentative, we won't bother. What is it you think you're proving? > Does that mean you don't know the mathematical meaning of the word >> prove? It isn't the same as the English meaning. >>Please give a specific example of something that you think is absurd or >a contradiction. I don't know what you mean by containment of sets and >subsets. Well as I recollect Stephen seems to think infinite sets are proper > subsets of themselves. > Are you sure that is what Stephen thinks? I see Lester has resorted to lying. Ever since you mathematically demonstrated that dr=v Lester has just resorted to modern mathematics. > This is all part of his >standard pattern. He really is pathetic. It is amusing to see >how increasingly pathetic he becomes. Fortunately for me nothing you say is demonstrably true, Stephen. ~v~~ === Subject: Re: Cantor Confusion <1o7cl2phksk2q7ng3dibc0ogfg25vf04v1@4ax.com> <3dcpl2phob3045cki7eobs4tjspd7cnh9p@4ax.com> <7kk1m21fhoaoqud7a2ladfr2lmhqgf5e7c@4ax.com > On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus >> Provability of what pray tell? If you're not concerned with proving >> the truth of what you say in mathematics exactly when are you not >> discussing philosophy every time you say anything in mathematics? >>Do you really not know the mathematical meaning of the word prove? If >so, I (and others) could try to explain it to you. But, if you are just >being argumentative, we won't bother. >> What is it you think you're proving? > Does that mean you don't know the mathematical meaning of the word >> prove? It isn't the same as the English meaning. >>Please give a specific example of something that you think is absurd or >a contradiction. I don't know what you mean by containment of sets and >subsets. >> Well as I recollect Stephen seems to think infinite sets are proper > subsets of themselves. > Are you sure that is what Stephen thinks? I see Lester has resorted to lying. That was Stephen... I must say I've always been puzzled by accusations of lying in this group. In particular, how can a claim to seem to recollect.. be lying? Of course, one may be annoyed by people who jumble up what one says, but... > Ever since you mathematically demonstrated that dr=v Lester has just > resorted to modern mathematics. That was Lester, with just one of his endearing habits, viz referring to himself in the third person. (Or was that Eldon? Or do all cranks do it?) And of course, another of his endearing habits, viz jumbling up what Stephen said. But when your dog hears **** ****** **** ******** **** ****** Fido ***** **** ***** dr *** ***** **** **** Fido ****** *** dr ***v *** ****! how can woof be lying? Beats me. > ~v~~ Arf arf! Brian Chandler http://imaginatorium.org === Subject: Re: Cantor Confusion >> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus Provability of what pray tell? If you're not concerned with proving the truth of what you say in mathematics exactly when are you not discussing philosophy every time you say anything in mathematics? >>Do you really not know the mathematical meaning of the word prove? If >>so, I (and others) could try to explain it to you. But, if you are just >>being argumentative, we won't bother. >> What is it you think you're proving? > Does that mean you don't know the mathematical meaning of the word > prove? It isn't the same as the English meaning. >>Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. >> Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. > Are you sure that is what Stephen thinks? >>I see Lester has resorted to lying. That was Stephen... I must say I've always been puzzled by accusations >of lying in this group. In particular, how can a claim to seem to >recollect.. be lying? Of course, one may be annoyed by people who >jumble up what one says, but... > Ever since you mathematically demonstrated that dr=v Lester has just >> resorted to modern mathematics. That was Lester, with just one of his endearing habits, viz referring >to himself in the third person. I only respond in the third person to those who refer to me in the third person. Do you think dogs appreciate being referred to in the third person in their presence? > (Or was that Eldon? Or do all cranks do >it?) This crank does. > And of course, another of his endearing habits, viz jumbling up >what Stephen said. Does it really matter what Stephen said? I mean in the great scheme of things is there really any repository of faith for sayings of Stephen? > But when your dog hears **** ****** **** ******** >**** ****** Fido ***** **** ***** dr *** ***** **** **** Fido ****** >*** dr ***v *** ****! how can woof be lying? In the interests of truth Lester tends to filter out what is not true. Blessed be the unconscious for they shall inherit the truth. >Beats me. > ~v~~ Arf arf! Well, Brian, at least you've learn to become a little charming if not particularly insightful at mathematical finishing school. Now if you could just learn to bark out technically on occasion I rather imagine we could all bark along to the same tune once in a while. ~v~~ === Subject: Re: Cantor Confusion > On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus >Please give a specific example of something that you think is absurd or >a contradiction. I don't know what you mean by containment of sets and >subsets. >> Well as I recollect Stephen seems to think infinite sets are proper > subsets of themselves. > Are you sure that is what Stephen thinks? I see Lester has resorted to lying. That was Stephen... I must say I've always been puzzled by accusations > of lying in this group. In particular, how can a claim to seem to > recollect.. be lying? Depends on whether the person's memory is really hazy or they are being disingenuous. > Of course, one may be annoyed by people who > jumble up what one says, but... Or, people who don't listen. Or, people who don't learn. -- David Marcus === Subject: Re: Cantor Confusion On Mon, 20 Nov 2006 01:39:59 -0500, David Marcus >> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus >Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. >> Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. > Are you sure that is what Stephen thinks? >>I see Lester has resorted to lying. That was Stephen... I must say I've always been puzzled by accusations >> of lying in this group. In particular, how can a claim to seem to >> recollect.. be lying? Depends on whether the person's memory is really hazy or they are being >disingenuous. I seem to recollect being disingenuous on occasion. >> Of course, one may be annoyed by people who >> jumble up what one says, but... Or, people who don't listen. Or, people who don't learn. Or people who don't teach. Or people who can't speak the truth just a little even once in a while. Or people who have no conception of truth and just pretend to speak, teach, and do mathematics anyway. Or people who are annoyed with demonstrations of truth. Or people who get doctorates in philosophy because they're too lazy or stupid to figure out the truth of what they're talking about but not too lazy or stupid to talk about it anyway. (Technically one wonders if Brian rather enjoys all this on a certain disingenuous masochistic level.) ~v~~ === Subject: Re: Cantor Confusion .com> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus >Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. >> Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. > Are you sure that is what Stephen thinks? >>I see Lester has resorted to lying. >> That was Stephen... I must say I've always been puzzled by accusations >> of lying in this group. In particular, how can a claim to seem to >> recollect.. be lying? Depends on whether the person's memory is really hazy or they are being >disingenuous. I seem to recollect being disingenuous on occasion. > Of course, one may be annoyed by people who >> jumble up what one says, but... Or, people who don't listen. Or, people who don't learn. Or people *** ***'* *****. Or people *** ***'* ***** *** ***** **** * > ****** **** **** ** * *****. Or ****** *** **** ** ********** ** ***** > *** **** ******* ***** *** ** *********** ******. Or ****** > *** *** ******* **** demonstrations of truth. Or ****** *** *** > ********** ** ********** ******* ****** *** **** ** ****** ** ****** > *** *** ***** ** *** ****** ******** ***** *** *** *** **** ** ****** > ** **** ***** ** ******. (Technically *** ******* ** Brian ****** ****** *** **** ** * ******* > dizinzzzzzzz yyyyyyyyyyyy zzzzz.) You spoke, Lester? Very good. Very good indeed. Actually somewhat more intellectually challenging than your usual level. Very even-handed, neigh, calm. > ~v~~ Arf arf! Brian Chandler http://imaginatorium.org === Subject: Re: Cantor Confusion > On Mon, 20 Nov 2006 01:39:59 -0500, David Marcus On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus Please give a specific example of something that you think is absurd or a contradiction. I don't know what you mean by containment of sets and subsets. > Well as I recollect Stephen seems to think infinite sets are proper subsets of themselves. Are you sure that is what Stephen thinks? >>I see Lester has resorted to lying. >> That was Stephen... I must say I've always been puzzled by accusations > of lying in this group. In particular, how can a claim to seem to > recollect.. be lying? >>Depends on whether the person's memory is really hazy or they are being >>disingenuous. >> I seem to recollect being disingenuous on occasion. > Of course, one may be annoyed by people who > jumble up what one says, but... >>Or, people who don't listen. Or, people who don't learn. >> Or people *** ***'* *****. Or people *** ***'* ***** *** ***** **** * >> ****** **** **** ** * *****. Or ****** *** **** ** ********** ** ***** >> *** **** ******* ***** *** ** *********** ******. Or ****** >> *** *** ******* **** demonstrations of truth. Or ****** *** *** >> ********** ** ********** ******* ****** *** **** ** ****** ** ****** >> *** *** ***** ** *** ****** ******** ***** *** *** *** **** ** ****** >> ** **** ***** ** ******. >> (Technically *** ******* ** Brian ****** ****** *** **** ** * ******* >> dizinzzzzzzz yyyyyyyyyyyy zzzzz.) You spoke, Lester? Very good. Very good indeed. Actually somewhat more >intellectually challenging than your usual level. Very even-handed, >neigh, calm. Except, Brian, you might have used ~v~~ or even ~ instead of *. Would have indicated a little more ingenuity on your part. You know a little more model schmodel kind of originality. >> ~v~~ Arf arf! Bark on, Brian. You know, the Mystery of the Dog who Didn't Bark in the Night? ~v~~ === Subject: Re: Cantor Confusion .com> On Mon, 20 Nov 2006 01:39:59 -0500, David Marcus On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus ~v~~ Arf arf! Bark on, Brian. You know, the Mystery of the Dog who Didn't Bark in > the Night? I've read the novel. Is that what you mean? Brian Chandler http://imaginatorium.org === Subject: Re: Cantor Confusion > On Mon, 20 Nov 2006 01:39:59 -0500, David Marcus >> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus ~v~~ >>Arf arf! >> Bark on, Brian. You know, the Mystery of the Dog who Didn't Bark in >> the Night? I've read the novel. Is that what you mean? Whaddya mean is that what I mean? I mean you're a fourth order philosophy whore playing grabass with the truth and calling it mathematics. Otherwise known in polite society as a bull artist and lying sack of . ~v~~ === Subject: Re: Cantor Confusion >> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus >>Please give a specific example of something that you think is absurd or >>a contradiction. I don't know what you mean by containment of sets and >>subsets. >> Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. > Are you sure that is what Stephen thinks? >>I see Lester has resorted to lying. That was Stephen... I must say I've always been puzzled by accusations >> of lying in this group. In particular, how can a claim to seem to >> recollect.. be lying? > Depends on whether the person's memory is really hazy or they are being > disingenuous. a hazy memory. Stephen === Subject: Re: Cantor Confusion On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus >>Please give a specific example of something that you think is absurd or a contradiction. I don't know what you mean by containment of sets and subsets. > Well as I recollect Stephen seems to think infinite sets are proper subsets of themselves. Are you sure that is what Stephen thinks? >>I see Lester has resorted to lying. That was Stephen... I must say I've always been puzzled by accusations > of lying in this group. In particular, how can a claim to seem to > recollect.. be lying? > Depends on whether the person's memory is really hazy or they are being >> disingenuous. a hazy memory. ~v~~ === Subject: Re: Cantor Confusion On Sat, 18 Nov 2006 13:37:07 -0500, David Marcus >> What is it modern zen mathematikers do instead of thinking about the >> truth of what they say? Sit around all day massaging their middle >> legs? I mean really what is it they expect they get paid for? Proving theorems, of course. Fess up: you really knew that, didn't you? I already knew they don't prove the truth of their theorems. ~v~~ === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net > 1 > 2 > 3 > ... > n <--> 1,2,3,...n Please distinguish: > iia: There is no number counting the elements of N. > iib: There is a number omega counting the elements of N. > No. You are trying to show that assuming case iia leads > to a contradiction. No. Case iia does not lead to a contradiction. > To do this you need to assume case iia. In > particular > you are assuming - the set, N, of all natural numbers exists > - the set N is infinite. > - N has no last element Yes. The diagonal contains all the d_nn. No line contains all the d_nn. > Therefore the diagonal is longer than every line. The diagonal consists of line indexes, i.e., of the line ends. Therefore it is a subset of the line indexes. The set of all lines contains every d_nn. No single line > contains every d_nn. The diagonal contains all d_nn. > The diagonal is longer than every line. The diagonal consists of line indexes, i.e., of the line ends. Therefore it is a subset of the line indexes. > So we have two results: > 1) The diagonal must be longer than every line. > 2) The diagonal cannot be longer than every line. No. The diagonal contains all the d_nn. No line contains > all the d_nn. The diagonal is longer than every line. The diagonal consists of line indexes, i.e., of the line ends. Therefore it is a subset of the line indexes. Only if the complete column does exist. But just that is wrong. > If we assume case iia the complete column does exist.. The assumption leads to a contradiction. Therefore the assumption is false. adding one element to every line does not add a column. > So if we start with the same number of lines and columns > we do not end with the same number of lines and columns. That should show you that the assertion of an actually infinite set of > finite numbers is a self contradiction. No. it just shows that the assertion of an actually infinite set of > finite > numbers leads to results that you do not like. However, these > are not contradictory results. These are: 1) The diagonal must be longer than every line. 2) The diagonal cannot be longer than every line. That is in my opinion a contradiction. > Why does addition of one element yield different results for columns, > diagonal and lines? The columns and the diagonal both contain an infinite initial > segment. No line contains an infinite initial segment. I know that. But you should recognize that this is a contradiction. If not, try to transpose the matrix. It does (the set of finite segments). Look at the introductory sketch. > The set of finite segments is the same. There is one segment that > is not finite. The set of initial segments does not correspond to > the set of lines. That is true. Therefore there is no infinite set of finite segments, contrary to the assumption. The bijection is only valid for finitely many segments (finitely many > numbers) > The segments count them selves. Finite and finitely many are the same as the matrix shows. > No. The bijection is only valid for finite (not finitely many) segments. > By iia there are an infinitely many finite sements. and lines with finite index (finite numbers). By iia there are an infinite number of finite numbers. We cannot give > an upper bound, but we can exclude an infinite ordinal and cardinal. > Not if you assume the set of all natural numbers, N, exists. The assumption has been assumed. The assumption leads to a contradiction. Therefore the assumption is false. === Subject: Re: Cantor Confusion > 1 > 2 > 3 > ... > n <--> 1,2,3,...n Please distinguish: > iia: There is no number counting the elements of N. > iib: There is a number omega counting the elements of N. > No. You are trying to show that assuming case iia leads > to a contradiction. No. Case iia does not lead to a contradiction. That depends on (1) the axiom system in which one is operating and (2) the definition of number one is using. To do this you need to assume case iia. In > particular > you are assuming - the set, N, of all natural numbers exists > - the set N is infinite. > - N has no last element Yes. > The diagonal contains all the d_nn. No line contains all the d_nn. > Therefore the diagonal is longer than every line. The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. But no single line contains all the line indices, so that is totally irrelevant when comparing single lines to the diagonal. No matter how you squirm, WM, you are wrong here! > The set of all lines contains every d_nn. No single line > contains every d_nn. The diagonal contains all d_nn. > The diagonal is longer than every line. The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. But no single line contains all the line indices, so that is totally irrelevant when comparing single lines to the diagonal. No matter how you squirm, WM, you are wrong here! > > So we have two results: > 1) The diagonal must be longer than every line. > 2) The diagonal cannot be longer than every line. No. The diagonal contains all the d_nn. No line contains > all the d_nn. The diagonal is longer than every line. The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. But no single line contains all the line indices, so that is totally irrelevant when comparing single lines to the diagonal. No matter how you squirm, WM, you are wrong here! Only if the complete column does exist. But just that is wrong. > If we assume case iia the complete column does exist.. The assumption leads to a contradiction. Therefore the assumption is > false. Or some other assumption necessary for existence of the alleged contradiction arises is false. Very few assumptions are sufficient in isolation to produce contradictions, they usually have to contradict something else. In which case, it can be any of the several assumptions leading to that contradiction which one may wish to reject. No. it just shows that the assertion of an actually infinite set of > finite > numbers leads to results that you do not like. However, these > are not contradictory results. These are: > 1) The diagonal must be longer than every line. > 2) The diagonal cannot be longer than every line. (2) is an unprovable assumption. WM's attempt to demonstrate it are all trivially invalid. > That is in my opinion a contradiction. The contradiction is not in the axiom but in WM's assumptinos over and above the axioms.. Why does addition of one element yield different results for columns, > diagonal and lines? The columns and the diagonal both contain an infinite initial > segment. No line contains an infinite initial segment. I know that. But you should recognize that this is a contradiction. As all that is contradicted are WM's beliefs, that is WM's problem, not ours. > If > not, try to transpose the matrix. Easy enough, just list each line vertically downwards and align the lines horizontally with first members in a row instead of a column. The assumption has been assumed. The assumption leads to a > contradiction. Therefore the assumption is false. AS it is only WM's assumption that is false, everyone else can be happy. > === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net 1 > 2 > 3 > ... > n <--> 1,2,3,...n Please distinguish: > iia: There is no number counting the elements of N. > iib: There is a number omega counting the elements of N. > No. You are trying to show that assuming case iia leads > to a contradiction. No. Case iia does not lead to a contradiction. To do this you need to assume case iia. In > particular > you are assuming - the set, N, of all natural numbers exists > - the set N is infinite. > - N has no last element Yes. > The diagonal contains all the d_nn. No line contains all the d_nn. > Therefore the diagonal is longer than every line. The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. > Yes, the diagonal is the set of line indexes (a subset, but not a proper subset). We need more than this. Consider the set X={1,2} and the two finite sequences A and B A= {1,2,1,2} B={1,2} A and B both consist of elements of the set X, but A is longer than B. So to say that both the diagonal and the lines are subsets of the line indexes is not enough to show that A cannot be longer than B. So you must have something more in mind when you claim that the fact that every element of the diagonal is a line index means that the diagonal cannot be longer than every line. What is this? Recall: every line is shorter than the diagonal; there is no longest line. - William Hughes === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. > Yes, the diagonal is the set of line indexes (a subset, but > not a proper subset). Every line is a subset of the diagonal. And the diagonal is a subset of the line ends. We need more than this. Consider the set > X={1,2} and the two finite sequences A and B A= {1,2,1,2} B={1,2} A and B both consist of elements of the set X, but A is > longer than B. So to say that both the diagonal and the > lines are subsets of the line indexes is not enough > to show that A cannot be longer than B. So you must have something more in mind when > you claim that the fact that every element of the diagonal > is a line index means that the diagonal cannot > be longer than every line. What is this? Of course there are more conditions. They follow from the EIT. Therefore I chose it. 1 12 123 ... Every line has only different indexes (there are not two equal indexes in one line). Line n has all indexes from 1 to n. Every initial segment of line ends is a subset of a line. Every initial segment of line ends is a subset of the diagonal. Every initial segment of the diagonal is a line. === Subject: Re: Cantor Confusion > The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. > Yes, the diagonal is the set of line indexes (a subset, but > not a proper subset). Every line is a subset of the diagonal. Only if all the characters in all the lines are the same will that be necessary, but then the diagonal is like the set of all naturals, an infinite set of finite members. > And the diagonal is a subset of > the line ends. > So you must have something more in mind when > you claim that the fact that every element of the diagonal > is a line index means that the diagonal cannot > be longer than every line. What is this? Of course there are more conditions. They follow from the EIT. > Therefore I chose it. > 1 > 12 > 123 > ... Every line has only different indexes (there are not two equal indexes > in one line). > Line n has all indexes from 1 to n. > Every initial segment of line ends is a subset of a line. > Every initial segment of line ends is a subset of the diagonal. > Every initial segment of the diagonal is a line. Every initial segment is finite, but the diagonal is not. WM has not found any internal contradictions within ZF, what he finds is contradictions between his own assumptions and those of ZF. === Subject: Re: Cantor Confusion <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. > Yes, the diagonal is the set of line indexes (a subset, but > not a proper subset). Every line is a subset of the diagonal. And the diagonal is a subset of > the line ends. We need more than this. Consider the set > X={1,2} and the two finite sequences A and B A= {1,2,1,2} B={1,2} A and B both consist of elements of the set X, but A is > longer than B. So to say that both the diagonal and the > lines are subsets of the line indexes is not enough > to show that A cannot be longer than B. So you must have something more in mind when > you claim that the fact that every element of the diagonal > is a line index means that the diagonal cannot > be longer than every line. What is this? Of course there are more conditions. They follow from the EIT. > Therefore I chose it. > 1 > 12 > 123 > ... Every line has only different indexes (there are not two equal indexes > in one line). Yes > Line n has all indexes from 1 to n. Yes > Every initial segment of line ends is a subset of a line. Yes > Every initial segment of line ends is a subset of the diagonal. Yes > Every initial segment of the diagonal is a line. No. There are two types of initial segments Initial segments with a largest element Initial segments without a largest element. There is an initial segment of the diagonal that does not have a largest element. (Recall: the diagonal has a largest element if and only if there is a last line. There is no last line.) Every line has a largest element. Thus there is an initial segment of the diagonal that is not a line. Assume that a sequence S is longer than every initial segement of S with a largest element. Then since S is an initial segment of itself, it follows that S does not have a largest element. You are trying to argue The diagonal cannot be longer than every initial segment with a largest element because there is no set of lines without a largest line. There is no set of lines without a largest line because the diagonal cannot be longer than every initial segment with a largest element. This is circular. - William Hughes === Subject: Re: Cantor Confusion <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net > You are trying to argue The diagonal cannot be longer than every > initial segment with a largest element because > there is no set of lines without a largest line. No. The diagonal cannot be longer than every line because it consists of line-ends only. There is no set of lines without a largest line > because the diagonal cannot be longer than > every initial segment with a largest element. This is circular. > No. Again the arguing is: The diagonal cannot be longer than every line because it consists of line-ends only. If the diagonal has omega elements, then a line must have omega elements. if no line has omega elements, then the diagonal cannot have omega elements. This is not circular but a bijection, the same bijection as between first column and diagonal. The bijection between lines and initial segments of the first column is brought about by the diagonal. The reason for this bijection is the fact that natural numbers count themselves. 1 2 3 ... <--> 1,2,3...n This bijection fails in case of the full column with omega elements but no line with omega elements. That is so obvious that I cannot understand how one can miss it. === Subject: Re: Cantor Confusion > You are trying to argue The diagonal cannot be longer than every > initial segment with a largest element because > there is no set of lines without a largest line. No. The diagonal cannot be longer than every line because it consists > of line-ends only. The diagonal is clearly and unabiguously longer than each line for which there is a next line. Does WM declare that within ZF there is a line for which there is no next line? > The diagonal cannot be longer than every line > because it consists of line-ends only. Non-sequitur. there is nothing in it consists of line-ends only which requires the diagonal cannot be longer than every line. If the diagonal has omega elements, then a line must have omega > elements. if no line has omega elements, then the diagonal cannot have > omega elements. Not in ZF. WM keeps assuming things which are false in ZF, and not provable without being assumed. So WM is assuming a set of axioms different from ZF, and what WM claims for his system is irrelevant for those using ZF. This is not circular but a bijection, the same bijection as between > first column and diagonal. The bijection between lines and initial > segments of the first column is brought about by the diagonal. The > reason for this bijection is the fact that natural numbers count > themselves. 1 > 2 > 3 > ... > <--> 1,2,3...n This bijection fails in case of the full column with omega elements but > no line with omega elements. That is so obvious that I cannot understand how one can miss it. Just lucky, I guess. === Subject: Re: Cantor Confusion <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net> < removal of the arg > You are trying to argue The diagonal cannot be longer than every > initial segment with a largest element because > there is no set of lines without a largest line. No. The diagonal cannot be longer than every line because it consists > of line-ends only. There is no set of lines without a largest line > because the diagonal cannot be longer than > every initial segment with a largest element. This is circular. No. Again the arguing is: The diagonal cannot be longer than every line > because it consists of line-ends only. Let's restore the bit you snipped. > Of course there are more conditions. They follow from the EIT. > Therefore I chose it. > 1 > 12 > 123 > ... > Every line has only different indexes (there are not two equal indexes > in one line). Yes > Line n has all indexes from 1 to n. Yes > Every initial segment of line ends is a subset of a line. Yes > Every initial segment of line ends is a subset of the diagonal. Yes > Every initial segment of the diagonal is a line. No. There are two types of initial segments Initial segments with a largest element Initial segments without a largest element. There is an initial segment of the diagonal that does not have a largest element. (Recall: the diagonal has a largest element if and only if there is a last line. There is no last line.) Every line has a largest element. Thus there is an initial segment of the diagonal that is not a line. Therefore you cannot say: The diagonal cannot be longer than every line because it consists of line-ends only. Do you intend to respond to this? - William Hughes === Subject: Re: Cantor Confusion <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net There are two types of initial segments Initial segments with a largest element > Initial segments without a largest element. There is an initial segment of the diagonal that does > not have a largest element. > (Recall: the diagonal has a largest element if and only > if there is a last line. There is no last line.) > Every line has a largest element. Thus there is an initial > segment of the diagonal that is not a line. The diagonal has only finite indexes n. Therefore it consists of line ends only. Every line end is the end of a line, no? Therefore you cannot say: The diagonal cannot be longer than every line > because it consists of line-ends only. Of course that must be true. The diagonal has only finite indexes n. Every line end is the end of a finite line. As it cannot be infinite, the complete diagonal does not exist. > If element d_nn of the diagonal is a member of line n, then all elements d_mm wit m =< n of the diagonal are members of the same line n. In other words: There is never more than one single line required to establish a bijection with all the elements of the diagonal d_mm with m =< n. As the diagonal has only finite indexes n, there is never more than one line required to establish a bijection with all elements of the diagonal. === Subject: Re: Cantor Confusion > There are two types of initial segments Initial segments with a largest element > Initial segments without a largest element. There is an initial segment of the diagonal that does > not have a largest element. > (Recall: the diagonal has a largest element if and only > if there is a last line. There is no last line.) > Every line has a largest element. Thus there is an initial > segment of the diagonal that is not a line. The diagonal has only finite indexes n. But infinitely many of them. > Therefore it consists of line > ends only. Every line end is the end of a line, no? So what? Therefore you cannot say: The diagonal cannot be longer than every line > because it consists of line-ends only. Of course that must be true. It is equally obviously false. More obviously to those with their eyes open. > The diagonal has only finite indexes n. > Every line end is the end of a finite line. > As it cannot be infinite, the complete diagonal does not exist. WM wants to prove that N does not exist, so that he keeps assuming things which he cannot prove to get that result. In ZF: For each n in N, let N_n = {m in N: m <= N} Note that for every n in N, N_n is a PROPER subset of N. Let L_n : N_n --> N : m |--> m Let D: N --> N : m |--> m Then L_n is the nth line and D is the diagonal. For m <= n, L_n(m) = m is the m'th term of line n For m <= n, L_n(m) = D(m) = m is the m'th term of D For m > n, L_n(m) does not exist, but D(m) = m. For every n in M there is m > n , for example m = n+1, such that L_n(m) does not exist, but D(m) does. So for every n in N, D is longer than L_n. === Subject: Re: Cantor Confusion > There are two types of initial segments Initial segments with a largest element > Initial segments without a largest element. There is an initial segment of the diagonal that does > not have a largest element. > (Recall: the diagonal has a largest element if and only > if there is a last line. There is no last line.) > Every line has a largest element. Thus there is an initial > segment of the diagonal that is not a line. The diagonal has only finite indexes n. Therefore it consists of line > ends only. Every line end is the end of a line, no? A classic case of Person 1: P_1 is true. P_1 implies P_2 Therefore P_2 is true Person 2: No P_1 is true However, P_1 does not imply P_2 P_2 is false Person 1: Clearly you do not understand. Here is another reason why P_1 must be true. Yes. P_1: The diagonal consists of the ends of lines. No. The fact that P_1: the diagonal consists of the ends of lines does not imply that P_2: the diagonal is not longer than every line. Therefore you cannot say: The diagonal cannot be longer than every line > because it consists of line-ends only. Of course that must be true. The diagonal has only finite indexes n. > Every line end is the end of a finite line. > As it cannot be infinite, the complete diagonal does not exist. > If element d_nn of the diagonal is a member of line n, then all > elements d_mm wit m =< n of the diagonal are members of the same line > n. In other words: There is never more than one single line required to > establish a bijection with all the elements of the diagonal d_mm with m > =< n. Yes. However, since there is always an element (n+1), no single line can establish a bijection with *all* the elements of the diagonal (both those <=n and those > n). > As the diagonal has only finite indexes n, there is never more > than one line required to establish a bijection with all elements of > the diagonal. No. No line with finite index n can establish a bijection with all elements of the diagonal. Every line has a finite index. Therefore no line can establish a bijection with all elements of the diagonal - William Hughes. === Subject: Re: Cantor Confusion > There are two types of initial segments Initial segments with a largest element > Initial segments without a largest element. There is an initial segment of the diagonal that does > not have a largest element. > (Recall: the diagonal has a largest element if and only > if there is a last line. There is no last line.) > Every line has a largest element. Thus there is an initial > segment of the diagonal that is not a line. The diagonal has only finite indexes n. Therefore it consists of line > ends only. Every line end is the end of a line, no? > A classic case of > Person 1: P_1 is true. Therefore P_2 is true Person 2: No > P_1 is true > However, P_1 does not imply P_2 not imply P_2 P_2 is false > Person 1: Clearly you do not understand. > Here is another reason why P_1 must be true. > Yes. P_1: The diagonal consists of the ends of lines. No. The fact that > P_1: the diagonal consists of the ends of lines > does not imply that > P_2: the diagonal is not longer than every line. Therefore you cannot say: The diagonal cannot be longer than every line > because it consists of line-ends only. Of course that must be true. The diagonal has only finite indexes n. > Every line end is the end of a finite line. > As it cannot be infinite, the complete diagonal does not exist. > If element d_nn of the diagonal is a member of line n, then all > elements d_mm wit m =< n of the diagonal are members of the same line > n. In other words: There is never more than one single line required to > establish a bijection with all the elements of the diagonal d_mm with m > =< n. > Yes. And the diagonal consists of only such elements. However, since there is always an element (n+1), no single > line can establish a bijection with *all* the elements of the diagonal > (both those <=n and those > n). Together with the proof above this shows that not *all* the elements of the diagonal exist. > As the diagonal has only finite indexes n, there is never more > than one line required to establish a bijection with all elements of > the diagonal. No. No line with finite index n can establish a bijection with > all elements of the diagonal. Every line has a finite index. > Therefore no line can establish a bijection with all elements of the > diagonal If there is a bijection with all finite line ends, then there is a bijection with one line. Why do you prefer one of them? === Subject: Re: Cantor Confusion There are two types of initial segments Initial segments with a largest element > Initial segments without a largest element. There is an initial segment of the diagonal that does > not have a largest element. > (Recall: the diagonal has a largest element if and only > if there is a last line. There is no last line.) > Every line has a largest element. Thus there is an initial > segment of the diagonal that is not a line. The diagonal has only finite indexes n. Therefore it consists of line > ends only. Every line end is the end of a line, no? > A classic case of > Person 1: P_1 is true. Therefore P_2 is true Person 2: No > P_1 is true > However, P_1 does not imply P_2 not imply P_2 P_2 is false > Person 1: Clearly you do not understand. > Here is another reason why P_1 must be true. > Yes. P_1: The diagonal consists of the ends of lines. No. The fact that > P_1: the diagonal consists of the ends of lines > does not imply that > P_2: the diagonal is not longer than every line. Therefore you cannot say: The diagonal cannot be longer than every line > because it consists of line-ends only. Of course that must be true. The diagonal has only finite indexes n. > Every line end is the end of a finite line. > As it cannot be infinite, the complete diagonal does not exist. > If element d_nn of the diagonal is a member of line n, then all > elements d_mm wit m =< n of the diagonal are members of the same line > n. In other words: There is never more than one single line required to > establish a bijection with all the elements of the diagonal d_mm with m > =< n. > Yes. And the diagonal consists of only such elements. However, since there is always an element (n+1), no single > line can establish a bijection with *all* the elements of the diagonal > (both those <=n and those > n). Together with the proof above this shows that not *all* the elements > of the diagonal exist. > As the diagonal has only finite indexes n, there is never more > than one line required to establish a bijection with all elements of > the diagonal. No. No line with finite index n can establish a bijection with > all elements of the diagonal. Every line has a finite index. > Therefore no line can establish a bijection with all elements of the > diagonal If there is a bijection with all finite line ends, then there is a > bijection with one line. No. The diagonal contains only finite line ends. Every element d_nn, of the diagonal, is the end of the line which ends at the natural number n. The bijection with all finite line ends is: The line which ends in n maps to the element d_nn. There is no bijection with one line. Such a bijection would imply that there is a largest d_nn. - William Hughes === Subject: Re: Cantor Confusion If there is a bijection with all finite line ends, then there is a > bijection with one line. No. > The diagonal contains only finite line ends. Every element > d_nn, of the diagonal, is the end of the line which ends > at the natural number n. The bijection with all finite line ends is: The line which ends > in n > maps to the element d_nn. There is no bijection with one line. Such a bijection would > imply that there is a largest d_nn. No. It implies only that all d_nn are finite. If you disagree, please give an example for two elements of the diagonal which cannot be found in one single line. === Subject: Re: Cantor Confusion If there is a bijection with all finite line ends, then there is a > bijection with one line. No. > The diagonal contains only finite line ends. Every element > d_nn, of the diagonal, is the end of the line which ends > at the natural number n. The bijection with all finite line ends is: The line which ends > in n > maps to the element d_nn. There is no bijection with one line. Such a bijection would > imply that there is a largest d_nn. No. It implies only that all d_nn are finite. If you disagree, please > give an example for two elements of the diagonal which cannot be found > in one single line. If you agree, please give an example of 2 = 1. makes as much sense as WM's request. For every line, there is a finite length, say n, so that the diagonal elements beying n, and there are infinitely many of them, are all not in that line. WM's blind spot seems to be growing. === Subject: Re: Cantor Confusion If there is a bijection with all finite line ends, then there is a > bijection with one line. No. > The diagonal contains only finite line ends. Every element > d_nn, of the diagonal, is the end of the line which ends > at the natural number n. The bijection with all finite line ends is: The line which ends > in n > maps to the element d_nn. There is no bijection with one line. Such a bijection would > imply that there is a largest d_nn. No. It implies only that all d_nn are finite. If you disagree, please > give an example for two elements of the diagonal which cannot be found > in one single line. > It is trivially true that given any two elements of the diagonal that a line exists that contains both elements. It would be quantifier dyslexia to claim that there exists a line that contains any two elements of the diagonal. [two can of course be replace by any natural number.] Your claim is that there is a bijection with one line. Call this line Kumquat. Kumquat has a largest element. Since there is a bijection between the elements of Kumquat and the d_nn, there must be a largest d_nn. - William Hughes === Subject: Re: Cantor Confusion > It is trivially true that given any two elements of the diagonal > that a line exists that contains both elements. Fine. It would be quantifier dyslexia to claim that there exists > a line that contains any two elements of the diagonal. So these any two elements are different from those any two which can be in a line? How do you distinguish the first any two from the second any two? [two can of course be replace by any natural number.] No. Your quantifier magic may apply to sets of men and women and dancers and what else you may like, but not to linearly ordered sets of finite elements. In case of the finite lines your assertion is simply absurd. This implies that the diagonal has not omega elements. > Your claim is that there is a bijection with one line. > Call this line Kumquat. Kumquat has a largest element. > Since there is a bijection between the elements of Kumquat and > the d_nn, there must be a largest d_nn. > If there is no bijection with one line, then there must be an element of the diagonal outside of every line. (Because what can be done with several *finite* lines in their linear order, can be done with one line). This assumption is 2^Kumquat. But if you want to entertain that idea, do it. There is no reason to further discuss this aberration of mind and, above all, no hope to rectify it. === Subject: Re: Cantor Confusion > If there is no bijection with one line, then there must be an element > of the diagonal outside of every line. No. Not unless the one line contains every element from every line. > (Because what can be done with > several *finite* lines in their linear order, can be done with one > line). What can be done with all lines in their linear order cannot be done with one line. No, you cannot use the fact that a bijection with one line must exist to prove that the set of all lines doesn't exist, then turn around and use the fact the the set of all lines doesn't exist to prove that a bijection with one line must exist - William Hughes === Subject: Re: Cantor Confusion > If there is no bijection with one line, then there must be an element > of the diagonal outside of every line. No. Not unless the one line > contains every element from every line. This holds for every finite line. Every finite line contains every element from every preceding line. There are only finite lines. (Because what can be done with > several *finite* lines in their linear order, can be done with one > line). > There are only finite lines. > What can be done with all lines in their linear order > cannot be done with one line. There are only finite lines. Each has as many elements as to count its position. Everything is finite. Fine. No, you cannot use the fact that a bijection with one line > must exist to prove that the set of all lines doesn't exist, Of course I can. > then turn around and use the fact the the set of all lines > doesn't exist to prove that a bijection with one line must exist > Drop simply worshipping the infinite. === Subject: Re: Cantor Confusion If there is no bijection with one line, then there must be an element > of the diagonal outside of every line. No. Not unless the one line > contains every element from every line. This holds for every finite line. Every finite line contains every > element from every preceding line. There are only finite lines. I claim There does not exists a line L_1 such that L_1 contains every element from every line. You counter with For every line L_1, L_1 contains every element from every line preceding L_1. However, these two statments are not contradictory. every element from every line is not the same thing as every element from every line preceding L_1 Yes it is true that For any element of the diagonal, d_nn, there exists a line L_2, such that L_2 contains d_nn. However, the statement There exists a line L_1, such that L_1 contains every element from every line preceding L_1. is not the same as the statement There exists a line L_1, such that L_1 contains every element from every line preceding L_2. So we cannot use There exists a line L_1, such that L_1 contains every element from every line preceding L_1. to show that There exists a line L_1, such that L_1 contains every element from every line - William Hughes === Subject: Re: Cantor Confusion If there is no bijection with one line, then there must be an element > of the diagonal outside of every line. No. Not unless the one line > contains every element from every line. This holds for every finite line. Every finite line contains every > element from every preceding line. There are only finite lines. > I claim There does not exists a line L_1 such that L_1 contains every > element from every line. You counter with For every line L_1, L_1 contains every element from every line > preceding L_1. However, these two statments are not contradictory. every element from every line is not the same thing as every element from every line preceding L_1 > Yes it is true that For any element of the diagonal, d_nn, there exists a line > L_2, such that L_2 contains d_nn. However, the statement There exists a line L_1, such that > L_1 contains every element from every line > preceding L_1. is not the same as the statement There exists a line L_1, such that > L_1 contains every element from every line > preceding L_2. So we cannot use There exists a line L_1, such that > L_1 contains every element from every line > preceding L_1. to show that There exists a line L_1, such that L_1 > contains every element from every line For the lines, each of with has a finite number of elements, this *is the same*. If you have a linearly ordered set with a finite number of elements, then there is a maximum. Every line has a finite number of elements. Further the elements are linearly ordered. Therefore the above requirement is satisfied. There is a line which contains all the elements off all lines - unless there were an infinite number of elements. But that would not represent a natural number. Therefore you cannot counter with the usual argument that the elements of any line are all finite but that there are infinitely many of them. This paradigm of uncritical belief does *not* work here. === Subject: Re: Cantor Confusion > If there is no bijection with one line, then there must be an element > of the diagonal outside of every line. No. Not unless the one line > contains every element from every line. This holds for every finite line. Every finite line contains every > element from every preceding line. There are only finite lines. > I claim There does not exists a line L_1 such that L_1 contains every > element from every line. You counter with For every line L_1, L_1 contains every element from every line > preceding L_1. However, these two statments are not contradictory. every element from every line is not the same thing as every element from every line preceding L_1 > Yes it is true that For any element of the diagonal, d_nn, there exists a line > L_2, such that L_2 contains d_nn. However, the statement There exists a line L_1, such that > L_1 contains every element from every line > preceding L_1. is not the same as the statement There exists a line L_1, such that > L_1 contains every element from every line > preceding L_2. So we cannot use There exists a line L_1, such that > L_1 contains every element from every line > preceding L_1. to show that There exists a line L_1, such that L_1 > contains every element from every line For the lines, each of with has a finite number of elements, this *is > the same*. But in EIT, for every line L_1 there is a successor line containing an element not in L_1. If you have a linearly ordered set with a finite number of elements, > then there is a maximum. But this does not happen in EIT. > Therefore you cannot counter with the usual argument that the elements > of any line are all finite but that there are infinitely many of them. In EIT one can. It is your model which h disproves your own claims. === Subject: Re: Cantor Confusion > If there is no bijection with one line, then there must be an element > of the diagonal outside of every line. No. Not unless the one line > contains every element from every line. This holds for every finite line. Every finite line contains every > element from every preceding line. There are only finite lines. > I claim There does not exists a line L_1 such that L_1 contains every > element from every line. You counter with For every line L_1, L_1 contains every element from every line > preceding L_1. However, these two statments are not contradictory. every element from every line is not the same thing as every element from every line preceding L_1 > Yes it is true that For any element of the diagonal, d_nn, there exists a line > L_2, such that L_2 contains d_nn. However, the statement There exists a line L_1, such that > L_1 contains every element from every line > preceding L_1. is not the same as the statement There exists a line L_1, such that > L_1 contains every element from every line > preceding L_2. So we cannot use There exists a line L_1, such that > L_1 contains every element from every line > preceding L_1. to show that There exists a line L_1, such that L_1 > contains every element from every line For the lines, each of with has a finite number of elements, this *is > the same*. > It is very important to keep the two concepts, the set of elements of an arbitray line l the set of all elements from all lines L distinct. Both are composed of finite elements. l has a finite number of elements. L has an infinite number of elements. > If you have a linearly ordered set with a finite number of elements, > then there is a maximum. Yes > Every line has a finite number of elements. Further the elements are > linearly ordered. >Therefore the above requirement is satisfied. Yes, for any set of elements of an arbitrary line l, l has a maximum element. Now consider L. L is linearly ordered, but does not have a finite number of elements > There > is a line which contains all the elements off all lines - unless there > were an infinite number of elements. But that would not represent a > natural number. Here you confuse an arbitrary line with the set of all lines. Recall l and L are not the same thing . Therefore you cannot counter with the usual argument that the elements > of any line are all finite but that there are infinitely many of them. Your phasing makes the antecedent of them, the elements of any line. However, I do not claim that this set is infinite. I do claim that both the set of lines and the set of every element of every lineare infinite. Rephrasing The elements of any line are all finite, futher there are only a finite number of elements in any one line. However, there are an infinite number of lines. If we let L be the set of every element from every line, then the set L is inifinite. - William Hughes === Subject: Re: Cantor Confusion > > There are two types of initial segments Initial segments with a largest element > Initial segments without a largest element. There is an initial segment of the diagonal that does > not have a largest element. > (Recall: the diagonal has a largest element if and only > if there is a last line. There is no last line.) > Every line has a largest element. Thus there is an initial > segment of the diagonal that is not a line. The diagonal has only finite indexes n. Therefore it consists of line > ends only. Every line end is the end of a line, no? > A classic case of > Person 1: P_1 is true. Therefore P_2 is true Person 2: No > P_1 is true > However, P_1 does not imply P_2 not imply P_2 P_2 is false > Person 1: Clearly you do not understand. > Here is another reason why P_1 must be true. > Yes. P_1: The diagonal consists of the ends of lines. No. The fact that > P_1: the diagonal consists of the ends of lines > does not imply that > P_2: the diagonal is not longer than every line. Therefore you cannot say: The diagonal cannot be longer than every line > because it consists of line-ends only. Of course that must be true. The diagonal has only finite indexes n. > Every line end is the end of a finite line. > As it cannot be infinite, the complete diagonal does not exist. > If element d_nn of the diagonal is a member of line n, then all > elements d_mm wit m =< n of the diagonal are members of the same line > n. In other words: There is never more than one single line required to > establish a bijection with all the elements of the diagonal d_mm with m > =< n. > Yes. And the diagonal consists of only such elements. Can WM deny that the diagonal contains for every line at least one element not in that line. However, since there is always an element (n+1), no single > line can establish a bijection with *all* the elements of the diagonal > (both those <=n and those > n). Together with the proof above this shows that not *all* the elements > of the diagonal exist. Not at all! WM is going though all sorts of contortions to maintain his falsehood, but nevertheless it is a falsehood, at least in ZF and NBG. And WM has no stated system of his own. > As the diagonal has only finite indexes n, there is never more > than one line required to establish a bijection with all elements of > the diagonal. No. No line with finite index n can establish a bijection with > all elements of the diagonal. Every line has a finite index. > Therefore no line can establish a bijection with all elements of the > diagonal If there is a bijection with all finite line ends, then there is a > bijection with one line. Why do you prefer one of them? It is WM who is insisting on a bijection from the diagonal to one of the lines whereas we say to none of them. There is a bijection between the set of positions in the diagonal and the set of ALL lines, but not with the set of positions of any single line. === Subject: Re: Cantor Confusion <455aeccb$0$97262$892e7fe2@authen.yellow.readfreenews.net> <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net [...] >> And there are lines as long as the diagonal is. Name one. The elements of the diagonal are a subset of the line ends. === Subject: Re: Cantor Confusion > > [...] >> And there are lines as long as the diagonal is. Name one. The elements of the diagonal are a subset of the line ends. The SET of elements of the diagonal equals the set of all line ends, but that does not name any line as being as long as the diagonal. It is easy to see that for every line in WM's list the diagaonal must contain at least one more character than that line. === Subject: Re: Cantor Confusion <455aeccb$0$97262$892e7fe2@authen.yellow.readfreenews.net> <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> > And there are lines as long as the diagonal is. Name one. The elements of the diagonal are a subset of the line ends. The SET of elements of the diagonal equals the set of all line ends, but > that does not name any line as being as long as the diagonal. > It is easy to see that for every line in WM's list the diagaonal must > contain at least one more character than that line. Yes. And it is as easy to see that for any character of the diagonal there exists a line which contains this one and the next one. Why do you see only the one side of the medal and dispel that the other side exists too ? Is it because of the joke with the at least one cow which is black at least at one side? === Subject: Re: Cantor Confusion > > [...] >> And there are lines as long as the diagonal is. Name one. The elements of the diagonal are a subset of the line ends. The SET of elements of the diagonal equals the set of all line ends, but > that does not name any line as being as long as the diagonal. It is easy to see that for every line in WM's list the diagaonal must > contain at least one more character than that line. Yes. And it is as easy to see that for any character of the diagonal > there exists a line which contains this one and the next one. Why do you see only the one side of the medal and dispel that the other > side exists too ? Is it because of the joke with the at least one cow > which is black at least at one side? WM can't even get the joke right. It was a sheep. When WM claims that there exists some line having a property, and I show that every line, without exception, fails to have that property, I have disproved WM's claim. WM claims existence of a line containing every element of the diagonal. I show that for any and every line, there is at least one element of the diagonal not in that line. I have disproved WM's claim. At least in any system governed by standard logic. What sort of logic, if any, governs WM's arguments is no longer clear. === Subject: Re: Cantor Confusion >> [...] > And there are lines as long as the diagonal is. >> Name one. The elements of the diagonal are a subset of the line ends. Though Virgil posed this question: Name one single _line_ which is as long as the diagonal ist. F. N. -- xyz === Subject: Re: Cantor Confusion <455aeccb$0$97262$892e7fe2@authen.yellow.readfreenews.net> <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net >> [...] > And there are lines as long as the diagonal is. >> Name one. The elements of the diagonal are a subset of the line ends. Though Virgil posed this question: Name one single _line_ which is as > long as the diagonal ist. Name one single element of the diagonal which is not contained in a line (which contains this and all preceding elements). This is what I call the one-eyedness of set theory: Its proponents see that for every line, there is a diagonal element not contained in this and all preceding lines. But they don't see, or at least dispel it, that there is no element of the diagonal which is outside of any line. The first observation leads to the theorem: The diagonal is superset of all lines. The second observation (together with the fact that every line is a superset of all preceding lines) leads to the theorem: There is at least one line which is superset of the diagonal. As there is no line with omega elements, there can be no diagonal with omega elements. === Subject: Re: Cantor Confusion [...] > And there are lines as long as the diagonal is. >> Name one. The elements of the diagonal are a subset of the line ends. Though Virgil posed this question: Name one single _line_ which is as > long as the diagonal ist. Name one single element of the diagonal which is not contained in a > line (which contains this and all preceding elements). This is what I call the one-eyedness of set theory: Its proponents see > that for every line, there is a diagonal element not contained in this > and all preceding lines. But they don't see, or at least dispel it, > that there is no element of the diagonal which is outside of any line. Every element except the first is outside at least one line. While it is true that there is no diagonal element that is outside of EVERY line, that is quite a different issue. The first observation leads to the theorem: The diagonal is superset of > all lines. The second observation (together with the fact that every > line is a superset of all preceding lines) leads to the theorem: There > is at least one line which is superset of the diagonal. Not in ZF. If L were such a line, then it would have to have a last element (as every line has a last element), but every line has a successor line with one more element, and that one more element is now in the diagonal but not in the line alleged to be a superset of the diagonal. In ZF, statements with counterexamples are not theorems. === Subject: Re: Cantor Confusion <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net> EVERY line, that is quite a different issue. > No, just this is the decisive issue. In ZF, statements with counterexamples are not theorems. One counter example contradicts ZFC: There is not one single element of the diagonal which is not contained in a line. This line contains this and all preceding elements. === Subject: Re: Cantor Confusion > Every element except the first is outside at least one line. While it is true that there is no diagonal element that is outside of > EVERY line, that is quite a different issue. > No, just this is the decisive issue. Only to those with quantifier dyslexia. In ZF, statements with counterexamples are not theorems. One counter example contradicts ZFC: > There is not one single element of the diagonal which is not contained > in a line. This line contains this and all preceding elements. While true, it is irrelevant to the issue of whether there is one line containing every element of the diagonal, which there is not. WM conflates every element of the diagonal is in SOME line, which is is false. === Subject: Re: Cantor Confusion <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net> There is not one single element of the diagonal which is not contained > in a line. This line contains this and all preceding elements. While true, it is irrelevant to the issue of whether there is one line > containing every element of the diagonal, which there is not. WM conflates every element of the diagonal is in SOME line, which is > is false. Tha means we need at least two lines for the elements o the diagonal? Please give an example which requires that at least two different lines are needed to contain two elements of the diagonal. === Subject: Re: Cantor Confusion One counter example contradicts ZFC: > There is not one single element of the diagonal which is not contained > in a line. This line contains this and all preceding elements. While true, it is irrelevant to the issue of whether there is one line > containing every element of the diagonal, which there is not. WM conflates every element of the diagonal is in SOME line, which is > is false. Tha means we need at least two lines for the elements o the diagonal? Not to anyone who understands logic. What it does mean is that we need infinitely many finite lines to get all of the infinitely many members of the diagonal. At least to people of any sense. > Please give an example which requires that at least two different lines > are needed to contain two elements of the diagonal. Please give any line for which every element of the diagonal is in THAT line. === Subject: Re: Cantor Confusion <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net> There is not one single element of the diagonal which is not contained > in a line. This line contains this and all preceding elements. While true, it is irrelevant to the issue of whether there is one line > containing every element of the diagonal, which there is not. WM conflates every element of the diagonal is in SOME line, which is > is false. That means we need at least two lines for the elements o the diagonal? Not to anyone who understands logic. What is the opposite of at least two? What it does mean is that we need infinitely many finite lines to get > all of the infinitely many members of the diagonal. So infinite is less than two? At least to people of any sense. Look, what is the value of the statement of a fool that his companion is not a fool? I think, there is no value at all. Please give an example which requires that at least two different lines > are needed to contain two elements of the diagonal. Please give any line for which every element of the diagonal is in THAT > line. Project the diagonal in horizonat direction. Then you have it. === Subject: Re: Cantor Confusion > > One counter example contradicts ZFC: > There is not one single element of the diagonal which is not contained > in a line. This line contains this and all preceding elements. While true, it is irrelevant to the issue of whether there is one line > containing every element of the diagonal, which there is not. WM conflates every element of the diagonal is in SOME line, which is > is false. That means we need at least two lines for the elements o the diagonal? Not to anyone who understands logic. What is the opposite of at least two? What we need is infinitely many lines, at least two is necessary but not sufficient. What it does mean is that we need infinitely many finite lines to get > all of the infinitely many members of the diagonal. So infinite is less than two? What we need is infinitely many lines, at least two is necessary but not sufficient. At least to people of any sense. Look, what is the value of the statement of a fool that his companion > is not a fool? I think, there is no value at all. Then WM makes a fool of himself calling others foolish. Please give an example which requires that at least two different lines > are needed to contain two elements of the diagonal. Please give any line for which every element of the diagonal is in THAT > line. Project the diagonal in horizonat direction. Then you have it. WM seems disoriented. In WM's own model, that forms a column, not a line. === Subject: Re: Cantor Confusion <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net> EVERY line, that is quite a different issue. > No, just this is the decisive issue. In ZF, statements with counterexamples are not theorems. One counter example contradicts ZFC: > There is not one single element of the diagonal which is not contained > in a line. This line contains this and all preceding elements. Quantifier dyslexia again. For every n there exists an line L(n) such that L(n) contains all elements m <=n Does not imply There exists a line L such that for every n, L contains all elements m <=n -William Hughes === Subject: Re: Cantor Confusion [...] >> And there are lines as long as the diagonal is. >> Name one. >> The elements of the diagonal are a subset of the line ends. >> Though Virgil posed this question: Name one single _line_ which is as >> long as the diagonal ist. Name one single element of the diagonal which is not contained in a > line (which contains this and all preceding elements). 1. Burden of proof weighs upon you who claimed And there are lines as long as the diagonal is.. The question remains: Which lines? 2. Since neither Virgil nor I did posit any claim which needs support, not even by naming one single element of the diagonal ..., we don't have to. > This is what I call the one-eyedness of set theory: Its proponents see > that for every line, there is a diagonal element not contained in this > and all preceding lines. This is your wording not mine and very likely not Virgils, too. As you know from my matrix-view of your list every diagonal element d_nn is necessarily located in row n and column n. Where else? Outside of this matrix there are no diagonal elements. > But they don't see, or at least dispel it, that there is no element of > the diagonal which is outside of any line. The sequence of diagonal elements is embedded in the matrix. There are neither n of the index set which have no d_nn nor are there d_mm which are not of the index set of the matrix. > The first observation leads to the theorem: The diagonal is superset > of all lines. Only sets have supersets. Define all lines please. > The second observation (together with the fact that > every line is a superset of all preceding lines) leads to the theorem: > There is at least one line which is superset of the diagonal. Please prove. > As there is no line with omega elements, there can be no diagonal with > omega elements. Non sequitur (can be replaced by exists). F. N. -- xyz === Subject: Re: Cantor Confusion <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net> <4561c6b4$0$97257$892e7fe2@authen.yellow.readfreenews.net > Name one single element of the diagonal which is not contained in a > line (which contains this and all preceding elements). 1. Burden of proof weighs upon you who claimed And there are lines as > long as the diagonal is.. The question remains: Which lines? The diagonal D is a subset of the union U of lines L_n. Therefore D - UL_n = empty set. 2. Since neither Virgil nor I did posit any claim which needs support, > not even by naming one single element of the diagonal ..., we don't > have to. Virgil claimed the diagonal be longer than any line. That needs support. Therefore he has to name an element which is not member of a line. This is what I call the one-eyedness of set theory: Its proponents see > that for every line, there is a diagonal element not contained in this > and all preceding lines. This is your wording not mine and very likely not Virgils, too. As you > know from my matrix-view of your list every diagonal element d_nn is > necessarily located in row n and column n. Where else? Outside of this > matrix there are no diagonal elements. That's just my arguing. But with set theorists you never can be sure that not the foolishest claims are supported. But they don't see, or at least dispel it, that there is no element of > the diagonal which is outside of any line. The sequence of diagonal elements is embedded in the matrix. There are > neither n of the index set which have no d_nn nor are there d_mm which > are not of the index set of the matrix. The first observation leads to the theorem: The diagonal is superset > of all lines. Only sets have supersets. Define all lines please. The diagonal is a sequence and as such an ordered set. Every coumn/line is a sequence and as such an ordered set. I don't define all. Look it up in a dictionary of mathematical expressions. The second observation (together with the fact that > every line is a superset of all preceding lines) leads to the theorem: > There is at least one line which is superset of the diagonal. Please prove. I leave it to you as an exercise. Hint: If element d_nn of the diagonal is a member of line n, then all elements d_mm wit m =< n of the diagonal are members of the same line n. In other words: There is never more than one single line required to establish a bijection with all the elements of the diagonal d_mm with m =< n. As the diagonal has only finite indexes n, there is never more than one line required to establish a bijection with all elements of the diagonal. > === Subject: Re: Cantor Confusion Name one single element of the diagonal which is not contained in a > line (which contains this and all preceding elements). 1. Burden of proof weighs upon you who claimed And there are lines as > long as the diagonal is.. The question remains: Which lines? The diagonal D is a subset of the union U of lines L_n. Therefore D - > UL_n = empty set. 2. Since neither Virgil nor I did posit any claim which needs support, > not even by naming one single element of the diagonal ..., we don't > have to. Virgil claimed the diagonal be longer than any line. That needs > support. Therefore he has to name an element which is not member of a > line. I have supported it many times, but apparently WM has a form of selective blindness which prevents him from seeing anything that conflicts with his beliefs. For ANY line n, d_{n+1,n+1} is not a member of that line. Thus for every n in N, the diagonal is longer than line n. To refute this proof, WM must find some line, and therefore some n in N numbering that line, containing every term of the diagonal. > This is what I call the one-eyedness of set theory: Its proponents see > that for every line, there is a diagonal element not contained in this > and all preceding lines. This is your wording not mine and very likely not Virgils, too. As you > know from my matrix-view of your list every diagonal element d_nn is > necessarily located in row n and column n. Where else? Outside of this > matrix there are no diagonal elements. That's just my arguing. But with set theorists you never can be sure > that not the foolishest claims are supported. With anti-set-theorists, such as WM, claiming that the diagonal (a set that contains the successor of every element in it) can be contained within a proper subset ( a line, bounded initial segment with an element having no successor in the set), the are no in a position to call the kettle black. But they don't see, or at least dispel it, that there is no element of > the diagonal which is outside of any line. The sequence of diagonal elements is embedded in the matrix. There are > neither n of the index set which have no d_nn nor are there d_mm which > are not of the index set of the matrix. The first observation leads to the theorem: The diagonal is superset > of all lines. Only sets have supersets. Define all lines please. The diagonal is a sequence and as such an ordered set. Every coumn/line > is a sequence and as such an ordered set. I don't define all. Look it > up in a dictionary of mathematical expressions. Regarding the diagonal and each line as merely sets, the diagonal is a superset of every line, and a proper superset of every line into the bargain. The second observation (together with the fact that > every line is a superset of all preceding lines) leads to the theorem: > There is at least one line which is superset of the diagonal. Please prove. I leave it to you as an exercise. As we have exercised our disproofs of that non-theorem several times now ( see above for example) with no adequate response, we find that WM need exercise a good deal more than we do. Hint: If element d_nn of the diagonal is a member of line n, then all > elements d_mm wit m =< n of the diagonal are members of the same line > n. In other words: There is never more than one single line required to > establish a bijection with all the elements of the diagonal d_mm with m > =< n. As the diagonal has only finite indexes n, there is never more > than one line required to establish a bijection with all elements of > the diagonal. WM is being silly again by ignoring that there is no n in N for which d_nn is the last term but for every line there is such an n. Alternately: For every line L, there is an n in N for which there is no nth term in that line, no L_n, L_n does not exist,but d_nn does. === Subject: Re: Cantor Confusion >> Name one single element of the diagonal which is not contained in a >> line (which contains this and all preceding elements). >> 1. Burden of proof weighs upon you who claimed And there are lines >> as long as the diagonal is.. The question remains: Which lines? The diagonal D is a subset of the union U of lines L_n. > Therefore D - UL_n = empty set. Do I miscomprehend your sentence And there are lines as long as the diagonal is.? Condider the following scenario: state 1: Given a parking space with *two* cars having their wheels mounted. (each car has 4 wheels) state 2: Unmount all the wheels of both cars and put them on one stack of wheels. (stack of wheels has 8 wheels) Proposition 1: There are cars in state 1 which have as many wheels as the stack of wheels in state 2 has. Proposition 2: There is *a* car in state 1 which has as many wheels as the stack of wheels in state 2 has. Now my interpretation: Claim 1 states that there are at least two (due to the plural form cars and due to the are instead of is) cars each of which have the property of having as many wheels as the stack of wheels has (8). That is wrong in my view. Claim 2 states that there is (at least) one car which has as many wheels as the stack of wheels has (8). Which is wrong, too. Question 1: Do you agree with my interpretation? We frequently use formulas like Ex(p(x)) which is translated as There is an x having property p. Question 2: Is it correct that you mean by writing And there are lines as long as the diagonal is. you mean There is a line as long as the diagonal is? I. e. in the sense of Ex (p (x)) with x for line and p(x) lenght of line equal length of diagonal? >> 2. Since neither Virgil nor I did posit any claim which needs >> support, not even by naming one single element of the diagonal ..., >> we don't have to. Virgil claimed the diagonal be longer than any line. That needs > support. Of course. It is proven that for every single line l |l| < |diagonal|. > Therefore he has to name an element which is not member of a > line. No Way. You reversed the quantifiers. >> This is what I call the one-eyedness of set theory: Its proponents >> see that for every line, there is a diagonal element not contained >> in this and all preceding lines. >> This is your wording not mine and very likely not Virgils, too. As >> you know from my matrix-view of your list every diagonal element d_nn >> is necessarily located in row n and column n. Where else? Outside of >> this matrix there are no diagonal elements. That's just my arguing. But with set theorists you never can be sure > that not the foolishest claims are supported. > But they don't see, or at least dispel it, that there is no element >> of the diagonal which is outside of any line. >> The sequence of diagonal elements is embedded in the matrix. There >> are neither n of the index set which have no d_nn nor are there d_mm >> which are not of the index set of the matrix. >> The first observation leads to the theorem: The diagonal is >> superset of all lines. >> Only sets have supersets. Define all lines please. The diagonal is a sequence and as such an ordered set. Every > coumn/line is a sequence and as such an ordered set. I don't define > all. Look it up in a dictionary of mathematical expressions. I did not ask you to define all but all lines. Do you mean A line (p(line)) or do you mean p(set of all lines)? >> The second observation (together with the fact that >> every line is a superset of all preceding lines) leads to the >> theorem: There is at least one line which is superset of the >> diagonal. >> Please prove. I leave it to you as an exercise. I see no superset and no proof. > Hint: If element d_nn of the diagonal is a member of line n, then all > elements d_mm wit m =< n of the diagonal are members of the same line > n. That is not true in the sense that the double index mm with m != n is not a location in line n but in line m. It may be true that the value of d_mm is equal to d_nm. But this is an issue of occupancy. > In other words: There is never more than one single line required > to establish a bijection with all the elements of the diagonal d_mm > with m =< n. I don't understand what you mean. > As the diagonal has only finite indexes n, Finite valued indexes n e omega, but not finitely many of them. > there is never more than one line required to establish a bijection > with all elements of the diagonal. Do you want to posit, that there is some line identical to the diagonal? That is not (yet) proven. F. N. -- xyz === Subject: Re: Cantor Confusion <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net> <4561c6b4$0$97257$892e7fe2@authen.yellow.readfreenews.net> <4562fc49$0$97261$892e7fe2@authen.yellow.readfreenews.net > Do you want to posit, that there is some line identical to the diagonal? > That is not (yet) proven. The number of elements of the diagonal is larger than all n: E omega A n : omega > n. The lines have only finitely many elements. In a linear order of finitely many elements n exactly the following is implied: If for every n there exists an line L(n) such that L(n) contains all elements m <=n then there exists a line L such that for every n, L contains all elements m <=n . This is valid for the finitely many elements of every line. (How many lines there are is irrelevant.) All elements of the diagonal are elements of the lines UL(n). Therefore all elements of the diagonal are elements of a finite line. Therefore: E omega A n : omega > n is false. === Subject: Re: Cantor Confusion Do you want to posit, that there is some line identical to the diagonal? > That is not (yet) proven. > The number of elements of the diagonal is larger than all n: > E omega A n : omega > n. The lines have only finitely many elements. In a linear order of finitely many elements n exactly the following is > implied: > If for every n there exists an line L(n) such that L(n) contains all > elements m <=n > then there exists a line L such that for every n, L contains all > elements m <=n . This is valid for the finitely many elements of every line. (How many > lines there are is irrelevant.) As long as that number is finite. All elements of the diagonal are elements of the lines UL(n). > Therefore all elements of the diagonal are elements of a finite line. Consider For all m in N there is an n in N such that D(m) in L_n and There is an n in N such that for all m in N D(m) in L_n. The former is trivial, the latter is false. This can more easily be seen in comparing For all m in N there is n in N such that n > m which is true with There is an n in N such that for all m in N, n > m which is false. Those who claim the former establishes the latter, or who fail to distinguish between the two, exhibit quantifier dyslexia. Some do it deliberately, knowing it to be wrong, others just do not know any better. In which category does WM belong? === Subject: Re: Cantor Confusion <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net> <4561c6b4$0$97257$892e7fe2@authen.yellow.readfreenews.net> <4562fc49$0$97261$892e7fe2@authen.yellow.readfreenews.net Consider > For all m in N there is an n in N such that D(m) in L_n > and > There is an n in N such that for all m in N D(m) in L_n. The former is trivial, the latter is false. This can more easily be seen in comparing > For all m in N there is n in N such that n > m which is true > with > There is an n in N such that for all m in N, n > m which is false. This is false. And by means of the EIT we can conclude that this falsehood implies the falsehood of There is an omega such that for all n in N : omega > n. If you do not see it, then try to find two elements of the diagonal which cannot belong to one single line. If you seek long enough, perhaps your eyes will be opened. === Subject: Re: Cantor Confusion Consider > For all m in N there is an n in N such that D(m) in L_n > and > There is an n in N such that for all m in N D(m) in L_n. The former is trivial, the latter is false. This can more easily be seen in comparing > For all m in N there is n in N such that n > m which is true > with > There is an n in N such that for all m in N, n > m which is false. This is false. In XF and NBG, 'Am e N, En e N, n > m' is true, 'En e N, Am e N, n > m' is false. WM has yet to produce a system which differs. > And by means of the EIT we can conclude WM concluding' something does not make it so. > If you do not see it, then try to find two elements of the diagonal > which cannot belong to one single line. There are no two elements that can both be the last element of any one line. > If you seek long enough, > perhaps your eyes will be opened. WM's eyes are so far open that he keeps seeing things which are not there. I have no wish to have mine opened that far. === Subject: Re: Cantor Confusion You have not commented on the first part of my posting: ,----[ <4562fc49$0$97261$892e7fe2@authen.yellow.readfreenews.net> ] | | >> Name one single element of the diagonal which is not contained in | >> a line (which contains this and all preceding elements). | >> | >> 1. Burden of proof weighs upon you who claimed And there are lines | >> as long as the diagonal is.. The question remains: Which lines? | > | > The diagonal D is a subset of the union U of lines L_n. | > Therefore D - UL_n = empty set. | | Do I miscomprehend your sentence | | And there are lines as long as the diagonal is.? | | Condider the following scenario: | | state 1: Given a parking space with two cars having their | wheels mounted. (each car has 4 wheels) | | state 2: Unmount all the wheels of both cars and put them on one | stack of wheels. (stack of wheels has 8 wheels) | | Proposition 1: There are cars in state 1 which have as many wheels | as the stack of wheels in state 2 has. | | Proposition 2: There is a car in state 1 which has as many | wheels as the stack of wheels in state 2 has. | | Now my interpretation: Claim 1 states that there are at least two (due | to the plural form cars and due to the are instead of is) cars | each of which have the property of having as many wheels as the stack | of wheels has (8). That is wrong in my view. | | Claim 2 states that there is (at least) one car which has as many | wheels as the stack of wheels has (8). Which is wrong, too. | | Question 1: Do you agree with my interpretation? | | We frequently use formulas like Ex(p(x)) which is translated as There | is an x having property p. | | Question 2: Is it correct that you mean by writing | And there are lines as long as the diagonal is. you mean | There is a line as long as the diagonal is? I. e. in the sense of | Ex (p (x)) with x for line and p(x) lenght of line equal length of | diagonal? `---- >> Do you want to posit, that there is some line identical to the >> diagonal? That is not (yet) proven. The number of elements of the diagonal is larger than all n: > E omega A n : omega > n. 1. n is not suitably constrained. For any n > omega the formula is omega > n is wrong hence the formula is wrong. 2. omega is asserted to exist so the existencial quantification is redundant. What you probably mean is An (n e (What set do you have in mind?) -> omega > n) Please correct. > The lines have only finitely many elements. In a linear order of finitely many elements n exactly the following is > implied: > If for every n there exists an line L(n) such that L(n) contains all > elements m <=n > then there exists a line L such that for every n, L contains all > elements m <=n . The constraint on n is missing here, too. > This is valid for the finitely many elements of every line. (How many > lines there are is irrelevant.) Proof? > All elements of the diagonal are elements of the lines UL(n). > Therefore all elements of the diagonal are elements of a finite line. Therefore: E omega A n : omega > n is false. E omega (A n (omega > n)) is not (yet) a theorem in ZF. F. N. -- xyz === Subject: Re: Cantor Confusion <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455f71bb$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605cf3$0$97253$892e7fe2@authen.yellow.readfreenews.net> <4561c6b4$0$97257$892e7fe2@authen.yellow.readfreenews.net> <4562fc49$0$97261$892e7fe2@authen.yellow.readfreenews.net> <456371d7$0$97219$892e7fe2@authen.yellow.readfreenews.net You have not commented on the first part of my posting: No. >> Do you want to posit, that there is some line identical to the >> diagonal? That is not (yet) proven. The number of elements of the diagonal is larger than all n: > E omega A n : omega > n. 1. n is not suitably constrained. For any n > omega the formula > is omega > n is wrong hence the formula is wrong. Please read yourself. In future I am not willing to explain everything twice to you. We discuss a diagonal with omega natural elements. The lines have only finitely many elements. In a linear order of finitely many elements n exactly the following is > implied: > If for every n there exists an line L(n) such that L(n) contains all > elements m <=n > then there exists a line L such that for every n, L contains all > elements m <=n . This is valid for the finitely many elements of every line. (How many > lines there are is irrelevant.) Proof? It is a self evident truth for finitely many elements like the elements of a line, because every finite set has a largest element. There exists no counter example, and it is impossible to construct a counter example. === Subject: Re: Cantor Confusion >> You have not commented on the first part of my posting: > No. So you don't want to tell me whether I miscomprehended you or not? ,----[ <4562fc49$0$97261$892e7fe2@authen.yellow.readfreenews.net> ] | | >> Name one single element of the diagonal which is not contained in | >> a line (which contains this and all preceding elements). | >> | >> 1. Burden of proof weighs upon you who claimed And there are lines | >> as long as the diagonal is.. The question remains: Which lines? | > | > The diagonal D is a subset of the union U of lines L_n. | > Therefore D - UL_n = empty set. | | Do I miscomprehend your sentence | | And there are lines as long as the diagonal is.? | | Condider the following scenario: | | state 1: Given a parking space with two cars having their | wheels mounted. (each car has 4 wheels) | | state 2: Unmount all the wheels of both cars and put them on one | stack of wheels. (stack of wheels has 8 wheels) | | Proposition 1: There are cars in state 1 which have as many wheels | as the stack of wheels in state 2 has. | | Proposition 2: There is a car in state 1 which has as many | wheels as the stack of wheels in state 2 has. | | Now my interpretation: Claim 1 states that there are at least two (due | to the plural form cars and due to the are instead of is) cars | each of which have the property of having as many wheels as the stack | of wheels has (8). That is wrong in my view. | | Claim 2 states that there is (at least) one car which has as many | wheels as the stack of wheels has (8). Which is wrong, too. | | Question 1: Do you agree with my interpretation? | | We frequently use formulas like Ex(p(x)) which is translated as There | is an x having property p. | | Question 2: Is it correct that you mean by writing | And there are lines as long as the diagonal is. you mean | There is a line as long as the diagonal is? I. e. in the sense of | Ex (p (x)) with x for line and p(x) lenght of line equal length of | diagonal? `---- > Do you want to posit, that there is some line identical to the > diagonal? That is not (yet) proven. >> The number of elements of the diagonal is larger than all n: >> E omega A n : omega > n. >> 1. n is not suitably constrained. For any n > omega the formula >> is omega > n is wrong hence the formula is wrong. Please read yourself. In future I am not willing to explain everything > twice to you. > We discuss a diagonal with omega natural elements. Do you mean A n (n e omega -> n < omega) ? >> The lines have only finitely many elements. >> In a linear order of finitely many elements n exactly the following >> is implied: >> If for every n there exists an line L(n) such that L(n) contains >> all elements m <=n >> then there exists a line L such that for every n, L contains all >> elements m <=n . >> This is valid for the finitely many elements of every line. (How >> many lines there are is irrelevant.) >> Proof? It is a self evident truth for finitely many elements like the > elements of a line, because every finite set has a largest element. So you mean your implication is not universally valid? > There exists no counter example, and it is impossible to construct a > counter example. F. N. -- xyz === Subject: Re: Cantor Confusion >> 2. Since neither Virgil nor I did posit any claim which needs >> support, not even by naming one single element of the diagonal ..., >> we don't have to. Virgil claimed the diagonal be longer than any line. That needs > support. Of course. It is proven that for every single line l |l| < |diagonal|. Therefore he has to name an element which is not member of a > line. No Way. You reversed the quantifiers. WM does that a lot. When people catch him doing it, he denies it. -- David Marcus === Subject: Re: Cantor Confusion > >> [...] > And there are lines as long as the diagonal is. >> Name one. The elements of the diagonal are a subset of the line ends. Though Virgil posed this question: Name one single _line_ which is as > long as the diagonal ist. F. N. It is a question that WM dare not face directly, as facing it would show up his errors unmistakably. === Subject: Re: Cantor Confusion <455c7752$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455e0473$0$97237$892e7fe2@authen.yellow.readfreenews.net> <455f1091$0$97214$892e7fe2@authen.yellow.readfreenews.net> <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> The cardinality of omega is omega. >> The cardinality of omega is |omega| not omega. Kunen's Set Theory defines |A| to be the least ordinal that can be > bijected with A. So, with this definition, |omega| = omega. You are absolutely right. Well learned (after all)! Now try to understand the next step: If omega exists, then |omega| =/= omega & |omega| = omega. Then you will have reached a higher level of understanding math than most mathematicians. === Subject: Re: Cantor Confusion > >> The cardinality of omega is omega. >> The cardinality of omega is |omega| not omega. Kunen's Set Theory defines |A| to be the least ordinal that can be > bijected with A. So, with this definition, |omega| = omega. You are absolutely right. Well learned (after all)! Now try to understand the next step: If omega exists, then |omega| =/= omega & |omega| = omega. Then you will have reached a higher level of understanding math than > most mathematicians. What interpretation of |omega| =/= omega & |omega| = omega does WM suggest that is anything but false in any set version of set theory. What may be true in WM's notion of a set theory, need not be, and in the above example is not, true in any standard version of set theory. One of WM's assumed counterfactuals is that an infinite quantity (the length of the diagonal of his triangular list) cannot be greater than the length of every finite line in that infinite list. That assumption alone corrupts his theory fatally. === Subject: Re: Cantor Confusion > The cardinality of omega is omega. >> The cardinality of omega is |omega| not omega. >> Kunen's Set Theory defines |A| to be the least ordinal that can >> be bijected with A. So, with this definition, |omega| = omega. >> You are absolutely right. Well learned (after all)! Now try to understand the next step: If omega exists, then |omega| =/= omega & |omega| = omega. Sorry how did you arrive at |omega| =/= omega (*) ? F. N. -- xyz === Subject: Re: Cantor Confusion <455c7752$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455e0473$0$97237$892e7fe2@authen.yellow.readfreenews.net> <455f1091$0$97214$892e7fe2@authen.yellow.readfreenews.net> <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net>> The cardinality of omega is |omega| not omega. [1] >> Kunen's Set Theory defines |A| to be the least ordinal that can >> be bijected with A. So, with this definition, |omega| = omega. >> You are absolutely right. Well learned (after all)! Now try to understand the next step: If omega exists, then |omega| =/= omega & |omega| = omega. Sorry how did you arrive at |omega| =/= omega (*) Look a the first line [1], written by yourself. === Subject: Re: Cantor Confusion >> The cardinality of omega is |omega| not omega. [1] >> Kunen's Set Theory defines |A| to be the least ordinal that > can be bijected with A. So, with this definition, |omega| = > omega. >> You are absolutely right. >> Well learned (after all)! >> Now try to understand the next step: >> If omega exists, then |omega| =/= omega & |omega| = omega. >> Sorry how did you arrive at >> |omega| =/= omega (*) Look a the first line [1], written by yourself. <456068f6$0$97216$892e7fe2@authen.yellow.readfreenews.net> F. N. -- xyz === Subject: Re: Cantor Confusion <455c7752$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455e0473$0$97237$892e7fe2@authen.yellow.readfreenews.net> <455f1091$0$97214$892e7fe2@authen.yellow.readfreenews.net> <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net> <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net >> The cardinality of omega is |omega| not omega. [1] >> Kunen's Set Theory defines |A| to be the least ordinal that > can be bijected with A. So, with this definition, |omega| = > omega. >> You are absolutely right. >> Well learned (after all)! >> Now try to understand the next step: >> If omega exists, then |omega| =/= omega & |omega| = omega. >> Sorry how did you arrive at >> |omega| =/= omega (*) Look a the first line [1], written by yourself. <456068f6$0$97216$892e7fe2@authen.yellow.readfreenews.net> I cannot understand your explanation given there. If you say The cardinality of omega is |omega| not omega, so you must have had in mind |omega| =/= omega, which is wrong, (according to the late Cantor - you see it is useful to study him). Would it be meaningful to write Y is X but not X? === Subject: Re: Cantor Confusion The cardinality of omega is |omega| not omega. [1] >> Kunen's Set Theory defines |A| to be the least ordinal that >> can be bijected with A. So, with this definition, |omega| = >> omega. >> You are absolutely right. >> Well learned (after all)! >> Now try to understand the next step: >> If omega exists, then |omega| =/= omega & |omega| = omega. >> Sorry how did you arrive at >> |omega| =/= omega (*) >> Look a the first line [1], written by yourself. >> <456068f6$0$97216$892e7fe2@authen.yellow.readfreenews.net > I cannot understand your explanation given there. If you say The > cardinality of omega is |omega| not omega, so you must have had in > mind |omega| =/= omega, No Way! If you want to misapprehend me do so, but don't confuse your misapprehensions with theorems of set theory. > which is wrong, (according to the late Cantor - you see it is useful > to study him). Cantor is not generally normative on these issues as one of the proponents has already pointed out. Under a common definition of cardinal number omega is the cardinal number of omega. There is no doubt about it. Since you persistently refuse to give defintions wouldn't you call me presumptous if ever I uttered what you must have had in mind? > Would it be meaningful to write Y is X but not X? This is not a faithful representation of my sentence you have objected to because there is no character string which can be substituted for X to get my original sentence back. You seem to like removing vertical bars (and braces elsewhere) at will. F. N. -- xyz === Subject: Re: Cantor Confusion <455c7752$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455e0473$0$97237$892e7fe2@authen.yellow.readfreenews.net> <455f1091$0$97214$892e7fe2@authen.yellow.readfreenews.net> <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net> <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net> <4561d0e9$0$97261$892e7fe2@authen.yellow.readfreenews.net I cannot understand your explanation given there. If you say The > cardinality of omega is |omega| not omega, so you must have had in > mind |omega| =/= omega, No Way! If you want to misapprehend me do so, but don't confuse your > misapprehensions with theorems of set theory. Shall this sentence of yours express a difference between |omega| and omega or not? (Now I recognize why it is so difficult to convince the proponents sof set theory.) Would it be meaningful to write Y is X but not X? This is not a faithful representation of my sentence you have objected > to because there is no character string which can be substituted for X > to get my original sentence back. If you meant |omega| = omega, then we have X = |omega| = omega. === Subject: Re: Cantor Confusion I cannot understand your explanation given there. If you say The > cardinality of omega is |omega| not omega, so you must have had in > mind |omega| =/= omega, No Way! If you want to misapprehend me do so, but don't confuse your > misapprehensions with theorems of set theory. > Shall this sentence of yours express a difference between |omega| and > omega or not? (Now I recognize why it is so difficult to convince the > proponents sof set theory.) Franziska explained that what he meant was that the notation for the cardinality of omega is |omega|, not omega. It turns out (using a standard definition for cardinality) that |omega| = omega. -- David Marcus === Subject: Re: Cantor Confusion [...] Shall this sentence of yours express a difference between |omega| >> and omega or not? (Now I recognize why it is so difficult to convince >> the proponents sof set theory.) Franziska explained that what [s]he meant was that the notation for > the cardinality of omega is |omega|, not omega. It turns out > (using a standard definition for cardinality) that |omega| = omega. F. N. -- xyz === Subject: Re: Cantor Confusion <455e0473$0$97237$892e7fe2@authen.yellow.readfreenews.net> <455f1091$0$97214$892e7fe2@authen.yellow.readfreenews.net> <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net> <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net> <4561d0e9$0$97261$892e7fe2@authen.yellow.readfreenews.net> <45620094$0$97259$892e7fe2@authen.yellow.readfreenews.net > [...] >> Shall this sentence of yours express a difference between |omega| >> and omega or not? (Now I recognize why it is so difficult to convince >> the proponents sof set theory.) Franziska explained that what [s]he meant was that the notation for > the cardinality of omega is |omega|, not omega. It turns out > (using a standard definition for cardinality) that |omega| = omega. > A very sensible and understanding human being which not even can distinguish between female and male names. But the same gap of understanding becomes visible in his understanding of technical terms. It turns out that also the *notation* for the cardinality of omega is omega. === Subject: Re: Cantor Confusion > [...] >> Shall this sentence of yours express a difference between |omega| >> and omega or not? (Now I recognize why it is so difficult to convince >> the proponents sof set theory.) Franziska explained that what [s]he meant was that the notation for > the cardinality of omega is |omega|, not omega. It turns out > (using a standard definition for cardinality) that |omega| = omega. > A very sensible and understanding human being which not even can > distinguish between female and male names. In English, there are a large number of given names which are given to either gender, so that one cannot always be sure. For example, there someone who posts to this Ng as [Mr.} Lynn ..... because Lynn is one of those androgenous names. To someone not familiar with naming habits in languages foreign to them, it is an understandable error. Can WM guarantee to get the correct gender for given names of, say , Finnish, or Japanese, posters? But the same gap of understanding becomes visible in his understanding > of technical terms. It turns out that also the *notation* for the > cardinality of omega is omega. Not always. This would require that one use a specific definition of cardinality which is not universal. > === Subject: Re: Cantor Confusion [...] Shall this sentence of yours express a difference between |omega| >> and omega or not? (Now I recognize why it is so difficult to convince >> the proponents sof set theory.) Franziska explained that what [s]he Oops. Very sorry. > meant was that the notation for > the cardinality of omega is |omega|, not omega. It turns out > (using a standard definition for cardinality) that |omega| = omega. > You're welcome. -- David Marcus === Subject: Re: Cantor Confusion >> I cannot understand your explanation given there. If you say The >> cardinality of omega is |omega| not omega, so you must have had in >> mind |omega| =/= omega, >> No Way! If you want to misapprehend me do so, but don't confuse your >> misapprehensions with theorems of set theory. > And what I meant was: The cardinality of X is |X| not X. I have already clarified that my wording was misleading. > Shall this sentence of yours express a difference between |omega| and > omega or not? What exactly is so hard to understand? The cardinality of a set X is (written) |X| and -- in the case of omega -- equals under the common definition to omega. > (Now I recognize why it is so difficult to convince the proponents sof > set theory.) >> Would it be meaningful to write Y is X but not X? >> This is not a faithful representation of my sentence you have >> objected to because there is no character string which can be >> substituted for X to get my original sentence back. If you meant |omega| = omega, then we have X = |omega| = omega. Let Y = The cardinality of omega and case 1: let X = |omega| Y is X but not X transforms into The cardinality of omega is |omega| but not |omega|, case 2: let X = omega Y is X but not X transforms into The cardinality of omega is omega but not omega. I did neither write nor mean what case 1 nor case 2 state. F. N. -- xyz === Subject: Re: Cantor Confusion <455e0473$0$97237$892e7fe2@authen.yellow.readfreenews.net> <455f1091$0$97214$892e7fe2@authen.yellow.readfreenews.net> <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net> <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net> <4561d0e9$0$97261$892e7fe2@authen.yellow.readfreenews.net> <4561d945$0$97232$892e7fe2@authen.yellow.readfreenews.net >> I cannot understand your explanation given there. If you say The >> cardinality of omega is |omega| not omega, so you must have had in >> mind |omega| =/= omega, >> No Way! If you want to misapprehend me do so, but don't confuse your >> misapprehensions with theorems of set theory. > And what I meant was: The cardinality of X is |X| not X. I have > already clarified that my wording was misleading. You tried to say misleading, but your wording was wrong. The cardinality of omega is omega as well as |omega|. Shall this sentence of yours express a difference between |omega| and > omega or not? What exactly is so hard to understand? The cardinality of a set X is > (written) |X| and -- in the case of omega -- equals under the common > definition to omega. Therefore, the sentence The cardinality of omega is |omega| not omega is false? Or is it not false? (Now I recognize why it is so difficult to convince the proponents sof > set theory.) >> Would it be meaningful to write Y is X but not X? >> This is not a faithful representation of my sentence you have >> objected to because there is no character string which can be >> substituted for X to get my original sentence back. If you meant |omega| = omega, then we have X = |omega| = omega. Let Y = The cardinality of omega and case 1: let X = |omega| Y is X but not X transforms into > The cardinality of omega is |omega| but not |omega|, case 2: let X = omega Y is X but not X transforms into > The cardinality of omega is omega but not omega. I did neither write nor mean what case 1 nor case 2 state. So you have *not* yet learned that omega = |omega|? cardinality of omega is |omega| but not |omega|. (Now I recognize why it is so difficult to convince the proponents sof set theory.) === Subject: Re: Cantor Confusion I cannot understand your explanation given there. If you say The >> cardinality of omega is |omega| not omega, so you must have had in >> mind |omega| =/= omega, >> No Way! If you want to misapprehend me do so, but don't confuse your >> misapprehensions with theorems of set theory. > And what I meant was: The cardinality of X is |X| not X. I have > already clarified that my wording was misleading. You tried to say misleading, but your wording was wrong. The > cardinality of omega is omega as well as |omega|. Technically speaking, omega is an ordinal and |omega| = aleph_0 is a cardinal. They are distinct in the sense that here is nothing requiring a cardinal to to have any internal ordering in general, though by some definitions cardinals may be ordered sets. Any set of cardinals will be ordered, but nothing in the general properties of cardinality ( that two sets have the same cardinal if and only if there is a bijection between them) requires a single cardinal to be an ordered set. It is only when chooses something like the particular definition that a cardinal of a given set is the ordinally smallest ordinal that bijects with the given set, that any individual cardinal needs to be ordered at all, much less well-ordered. Shall this sentence of yours express a difference between |omega| and > omega or not? What exactly is so hard to understand? The cardinality of a set X is > (written) |X| and -- in the case of omega -- equals under the common > definition to omega. Therefore, the sentence The cardinality of omega is |omega| not omega > is false? Or is it not false? On the sense that the cardinality of omega does not need to be a well ordered set, it is true. So you have *not* yet learned that omega = |omega|? What is your definition of the cardinality of a set? the truth of omega = |omega| depends on that definition. === Subject: Re: Cantor Confusion > I cannot understand your explanation given there. If you say > The cardinality of omega is |omega| not omega, so you must > have had in mind |omega| =/= omega, [(****)] >> No Way! If you want to misapprehend me do so, but don't confuse > your misapprehensions with theorems of set theory. >> And what I meant was: The cardinality of X is |X| not X. I have >> already clarified that my wording was misleading. You tried to say misleading, but your wording was wrong. I will not rectify this once again. > The cardinality of omega is omega as well as |omega|. Under the common definition the cardinality of omega, also written as |omega|, is omega. Anyhow. I still see no support for your pretended interpretation (****), which I did not meant to write and did not write in that form. >> Shall this sentence of yours express a difference between |omega| >> and omega or not? >> What exactly is so hard to understand? The cardinality of a set X is >> (written) |X| and -- in the case of omega -- equals under the common >> definition to omega. Therefore, the sentence The cardinality of omega is |omega| not > omega is false? Or is it not false? Perhaps D. Marcus' understanding in may help you to cope with my sentence under discussion. >> (Now I recognize why it is so difficult to convince the proponents >> sof set theory.) >> Would it be meaningful to write Y is X but not X? >> This is not a faithful representation of my sentence you have > objected to because there is no character string which can be > substituted for X to get my original sentence back. >> If you meant |omega| = omega, then we have X = |omega| = omega. >> Let Y = The cardinality of omega and >> case 1: let X = |omega| Y is X but not X transforms into >> The cardinality of omega is |omega| but not |omega|, >> case 2: let X = omega Y is X but not X transforms into >> The cardinality of omega is omega but not omega. >> I did neither write nor mean what case 1 nor case 2 state. So you have *not* yet learned that omega = |omega|? My sentence under discussion is _not_ of the pretended _form_ (!) Y is X but not X Period. > cardinality of omega is |omega| but not |omega|. My sentence under discussion was not meant to be interpreted as an equation. F. N. -- xyz === Subject: Re: Cantor Confusion <455f1091$0$97214$892e7fe2@authen.yellow.readfreenews.net> <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net> <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net> <4561d0e9$0$97261$892e7fe2@authen.yellow.readfreenews.net> <4561d945$0$97232$892e7fe2@authen.yellow.readfreenews.net> <45622c2a$0$97236$892e7fe2@authen.yellow.readfreenews.net > I cannot understand your explanation given there. If you say > The cardinality of omega is |omega| not omega, so you must > have had in mind |omega| =/= omega, [(****)] >> No Way! If you want to misapprehend me do so, but don't confuse > your misapprehensions with theorems of set theory. >> And what I meant was: The cardinality of X is |X| not X. I have >> already clarified that my wording was misleading. You tried to say misleading, but your wording was wrong. I will not rectify this once again. The cardinality of omega is omega as well as |omega|. Under the common definition the cardinality of omega, also written as > |omega|, is omega. Anyhow. I still see no support for your pretended interpretation (****), > which I did not meant to write and did not write in that form. > Shall this sentence of yours express a difference between |omega| >> and omega or not? >> What exactly is so hard to understand? The cardinality of a set X is >> (written) |X| and -- in the case of omega -- equals under the common >> definition to omega. Therefore, the sentence The cardinality of omega is |omega| not > omega is false? Or is it not false? Perhaps D. Marcus' understanding in > 1) The cardinality of omega is |omega| not omega. And after learning from me that this is wrong, 2) The cardinality of omega, also written as |omega|, is omega. Now I am interested whether or not this is a contradiction in your eyes. If you say, no, it is slightly misleading but it is not a contradiction, then I am absolutely clear that ZFC will remain free of contradictions forever. But then I can tell the young students who not yet worship ZFC one more example of the logic of the worshepherds of transfinity. === Subject: Re: Cantor Confusion > Perhaps D. Marcus' understanding in > And after learning from me that this is wrong, > 2) The cardinality of omega, also written as |omega|, is omega. Now I am interested whether or not this is a contradiction in your > eyes. WM is guilty of much more serious trangressions of both logic and common sense that is involved with whether omega and |omega| are the same. Among others, WM's trangressions include repetitive instances of quantifier dyslexia. === Subject: Re: Cantor Confusion <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net> <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net> <4561d0e9$0$97261$892e7fe2@authen.yellow.readfreenews.net> <4561d945$0$97232$892e7fe2@authen.yellow.readfreenews.net> <45622c2a$0$97236$892e7fe2@authen.yellow.readfreenews.net> And after learning from me that this is wrong, > 2) The cardinality of omega, also written as |omega|, is omega. Now I am interested whether or not this is a contradiction in your > eyes. WM is guilty of much more serious trangressions of both logic and common > sense that is involved with whether omega and |omega| are the same. That is not the question any longer. The question is: Can a set theorist admit that she is in error? It seems impossible. They all are too well trained in defending ZFC. === Subject: Re: Cantor Confusion > > Perhaps D. Marcus' understanding in > And after learning from me that this is wrong, > 2) The cardinality of omega, also written as |omega|, is omega. Now I am interested whether or not this is a contradiction in your > eyes. WM is guilty of much more serious trangressions of both logic and common > sense than is involved with whether omega and |omega| are the same. That is not the question any longer. It is certainly beyond question that WM is guilty of much more serious trangressions of both logic and common sense than is involved with whether omega and |omega| are the same. > The question is: Can a set theorist admit that she is in error? WM has yet to find one which is in anywhere near as much error as WM. So let us see WM admit his own many fatal errors before bringing up the purported errors of others. > === Subject: Re: Cantor Confusion <455f74a9$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net> <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net> <4561d0e9$0$97261$892e7fe2@authen.yellow.readfreenews.net> <4561d945$0$97232$892e7fe2@authen.yellow.readfreenews.net> <45622c2a$0$97236$892e7fe2@authen.yellow.readfreenews.net> theorist admit that she is in error? It seems impossible. They all are > too well trained in defending ZFC. In error as to what? Set theorists admit mistakes. It is not uncommon for books to have errata sheets attached. MoeBlee === Subject: Re: Cantor Confusion <45605c6f$0$97253$892e7fe2@authen.yellow.readfreenews.net> <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net> <4561d0e9$0$97261$892e7fe2@authen.yellow.readfreenews.net> <4561d945$0$97232$892e7fe2@authen.yellow.readfreenews.net> <45622c2a$0$97236$892e7fe2@authen.yellow.readfreenews.net> theorist admit that she is in error? It seems impossible. They all are > too well trained in defending ZFC. In error as to what? Set theorists admit mistakes. It is not uncommon > for books to have errata sheets attached. Would you see a contradiction in these two statements? 1) The cardinality of omega is |omega| not omega. 2) The cardinality of omega, also written as |omega|, is omega. === Subject: Re: Cantor Confusion >> That is not the question any longer. The question is: Can a set >> theorist admit that she is in error? It seems impossible. They all >> are too well trained in defending ZFC. >> In error as to what? Set theorists admit mistakes. It is not uncommon >> for books to have errata sheets attached. Would you see a contradiction in these two statements? > 1) The cardinality of omega is |omega| not omega. > 2) The cardinality of omega, also written as |omega|, is omega. As 2) may be considered as an errata sheet correcting the wording and rectifiying 1) there is no contradiction. Not in what I intended to say and not in some set theory anyhow. F. N. -- xyz === Subject: Re: Cantor Confusion <456172f9$0$97254$892e7fe2@authen.yellow.readfreenews.net> <4561d0e9$0$97261$892e7fe2@authen.yellow.readfreenews.net> <4561d945$0$97232$892e7fe2@authen.yellow.readfreenews.net> <45622c2a$0$97236$892e7fe2@authen.yellow.readfreenews.net> <4565a0de$0$97221$892e7fe2@authen.yellow.readfreenews.net >> That is not the question any longer. The question is: Can a set >> theorist admit that she is in error? It seems impossible. They all >> are too well trained in defending ZFC. >> In error as to what? Set theorists admit mistakes. It is not uncommon >> for books to have errata sheets attached. Would you see a contradiction in these two statements? > 1) The cardinality of omega is |omega| not omega. > 2) The cardinality of omega, also written as |omega|, is omega. As 2) may be considered as an errata sheet correcting the wording and > rectifiying 1) there is no contradiction. Why did it take so long to switch from I don't need any advice from you and from slightly misleading to erratum? === Subject: Re: Cantor Confusion > Why did it take so long to switch from I don't need any advice from > you and from slightly misleading to erratum? It took so long because you made my wording an issue and because you prefered to ride this dead horse insistently claiming that it was not dead. F. N. -- xyz === Subject: Re: Cantor Confusion >> That is not the question any longer. The question is: Can a set >> theorist admit that she is in error? It seems impossible. They all are >> too well trained in defending ZFC. In error as to what? Set theorists admit mistakes. It is not uncommon >for books to have errata sheets attached. For what it's worth, Moe, I have yet to see any set theorists admit mistakes in the paradigm. ~v~~ === Subject: Re: Cantor Confusion [...] >> Therefore, the sentence The cardinality of omega is |omega| not >> omega is false? Or is it not false? >> Perhaps D. Marcus' understanding in >> may help you to cope with my sentence under discussion. > 1) The cardinality of omega is |omega| not omega. > And after learning from me that this is wrong, > 2) The cardinality of omega, also written as |omega|, is omega. Now I am interested whether or not this is a contradiction in your > eyes. If you say, no, it is slightly misleading but it is not a > contradiction, then I am absolutely clear that ZFC will remain free of > contradictions forever. But then I can tell the young students who not > yet worship ZFC one more example of the logic of the worshepherds of > transfinity. Nice try. F. N. -- xyz === Subject: Re: Cantor Confusion <455aeccb$0$97262$892e7fe2@authen.yellow.readfreenews.net> <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455db94f$0$97241$892e7fe2@authen.yellow.readfreenews.net> <455ee80f$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455f6f2c$0$97228$892e7fe2@authen.yellow.readfreenews.net Please look it up in any modern text book. You will find there that > omega is a cardinal number too. Anyway, omega is not a /natural/ number. And a cow is not a horse. Does that eliminate your error of claiming omega =/= |omega| in modern set theory? > You need not interpret n as a set (though you can do it). >> In contemporary set theory almost everything is a set. In ZFC everything is a set. Not in every set theory. I don't want to debate about the axioms of ZFC being sets. The point is > that you want to talk about columns which neither ZFC nor any other > contemporary set theory is about. I think that one can introduce new expressions, examples, and illustrations if they have been made sufficiently clear to a correspondent of average IQ. The reaction of William confirms that my ideas are understandable. So your reaction does not concern my writings but rather your means of reception. > A treatise in which variables (n) and number symbols (0, 1, >> ...) do not refer to sets is not a treatise _on_ set theory but >> a treatise of _application_ of set theory, if ever. Don't mistake set theory with ZF or ZFC. A theory of _sets_ is not a theory of _columns_. Experience has shown that practically all notions used in contemporary mathematics can be defined, and their mathematical properties derived, in ZFC. In this sense, the axiomatic set theory serves as a satisfactory foundation for al other branches of mathematics. It can describe every mathematical notion --- with the exception of what a column is? >> In contemporary set theory it is said that omega is _the_ _set_ of >> natural numbers [as Virgil pointed out the _ordered_ set]. The number >> (cardinality) of omega is named aleph_0. So it is said that the >> number (cardinality) of _the_ _set_ of natural numbers is aleph_0. You are completely in error. The number (Anzahl) of a set is its > ordinal number. Anzahl is your (or a historical person's) wording not mine. I can't > spot any error in my wording. That is due to your incompetence. Sometimes it is necessary to quote. In particular if you are > uninformed but nevertheless refuse to take advice from me. I don't need any advice from you. You don't know it. That' s why you would need to learn a lot. Look, you have learned from me meanwhile, even against your furious opposition, that omega = |omega| in modern set theory. Doesn't this case make you wonder whether there are other things which you do not yet know but which you could learn from me? > My opinion is A (n e omega & |{0, 1, 2, ..., n}| < |omega|) Seems an empty opinion. What is the symbol A refers to? > As the diagonal is defined to consist of the ends of terms >> of lines, this claim is easy to conradict. >> ends of terms of lines is your wording not mine. It is not your wording, because you prefer to veil your > inconsistencies, but it is your opinion. My opinion is |(d_nn) n e omega| = |omega| I know. You say there are less natural numbers than omega, but there are as many natural numbers as omega. Can you understand mathematical symbols? Can you understand d_nn <-- n? Writing down a bunch of symbols does not mean that you created a > mathematical notation. Rephrase what precisely There are more natural > numbers d_nn than natural numbers n. shall mean. It means that we have discovered an inconsistency in the assumption that there were infinitely many finite numbers. === Subject: Re: Cantor Confusion ADDENDUM > >> Please look it up in any modern text book. You will find there that >> omega is a cardinal number too. >> Anyway, omega is not a /natural/ number. And a cow is not a horse. Does that eliminate your error of claiming > omega =/= |omega| in modern set theory? ,----[ <455f1091$0$97214$892e7fe2@authen.yellow.readfreenews.net> ] | The cardinality of omega is |omega| not omega. `---- This may mistakenly be interpreted as the proposition |omega| =/= omega (*) which I did not want to posit. In general cardinality of set X is usually written card(X) or | X | and not X. In the case of the sets 0, 1, ..., omega | X | does equal X under the common cardinal assignment as David Marcus has already pointed out: ,----[ ] | Kunen's Set Theory defines |A| to be the least ordinal that can be | bijected with A. So, with this definition, |omega| = omega. `---- F. N. -- xyz === Subject: Re: Cantor Confusion [...] > You need not interpret n as a set (though you can do it). >> In contemporary set theory almost everything is a set. >> In ZFC everything is a set. Not in every set theory. >> I don't want to debate about the axioms of ZFC being sets. The point >> is that you want to talk about columns which neither ZFC nor any >> other contemporary set theory is about. I think that one can introduce new expressions, examples, and > illustrations if they have been made sufficiently clear to a > correspondent of average IQ. And obviously some increased IQ is necessary to distinguish propositions of set theory from those of columns. > The reaction of William confirms that my ideas are understandable. So > your reaction does not concern my writings but rather your means of > reception. You may even cite D. Marcus as your witness: ,----[ ] | I think we'll have more fun if we talk to him on his terms rather than | try to get him to be precise and formal. He doesn't know how to do the | latter. So, he just reverts to repeating himself. `---- > A treatise in which variables (n) and number symbols (0, 1, > ...) do not refer to sets is not a treatise _on_ set theory but > a treatise of _application_ of set theory, if ever. >> Don't mistake set theory with ZF or ZFC. >> A theory of _sets_ is not a theory of _columns_. Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in ZFC. In this sense, the axiomatic set theory serves as a > satisfactory foundation for al[l] other branches of mathematics. > It can describe every mathematical notion --- with the exception of > what a column is? I do not speculate about what is describable and what not. If you want to posit a definition do so! It is curious that you obviously don't like to give precise definitions of certain notions even when you have explicitly been asked for. [...] >> Sometimes it is necessary to quote. In particular if you are >> uninformed but nevertheless refuse to take advice from me. >> I don't need any advice from you. You don't know it. That' s why you would need to learn a lot. Look, > you have learned from me meanwhile, even against your furious > opposition, that omega = |omega| in modern set theory. You do no longer persue your plan to prove a contradiction in modern set theory? > Doesn't this case make you wonder whether there are other things which > you do not yet know but which you could learn from me? > My opinion is >> A (n e omega & |{0, 1, 2, ..., n}| < |omega|) Seems an empty opinion. What is the symbol A refers to? A n (n e omega & |{0, 1, 2, ..., n}| < |omega|) > As the diagonal is defined to consist of the ends of terms > of lines, this claim is easy to conradict. >> ends of terms of lines is your wording not mine. >> It is not your wording, because you prefer to veil your >> inconsistencies, but it is your opinion. >> My opinion is >> |(d_nn) n e omega| = |omega| I know. You say there are less natural numbers than omega, but there > are as many natural numbers as omega. What I say is: The cardinality of the sequence (d_nn) is the cardinality of omega. >> Can you understand mathematical symbols? Can you understand d_nn >> <--> n? >> Writing down a bunch of symbols does not mean that you created a >> mathematical notation. Rephrase what precisely There are more >> natural numbers d_nn than natural numbers n. shall mean. It means that we have discovered an inconsistency in the assumption > that there were infinitely many finite numbers. We -> You discovered -> invented F. N. -- xyz === Subject: Re: Cantor Confusion <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455db94f$0$97241$892e7fe2@authen.yellow.readfreenews.net> <455ee80f$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455f6f2c$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605b9a$0$97253$892e7fe2@authen.yellow.readfreenews.net> A theory of _sets_ is not a theory of _columns_. Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in ZFC. In this sense, the axiomatic set theory serves as a > satisfactory foundation for al[l] other branches of mathematics. > It can describe every mathematical notion --- with the exception of > what a column is? I do not speculate about what is describable and what not. If you want > to posit a definition do so! Practically all notions! If you again refuse to learn from me, look here: Experience has shown that practically all notions used in contemporary mathematics can be defined, and their mathematical properties derived, in this axiomatic system. In this sense, the axiomatic set theory serves as a satisfactory foundation for the other branches of mathematics. [Karel Hrbacek and Thomas Jech: Introduction to Set Theory Marcel Dekker Inc., New York, 1984, 2nd edition, p. 3] It is curious that you obviously don't like to give precise definitions > of certain notions even when you have explicitly been asked for. I defined the EIT. A column is a vertical row. The first column of it is the first vertical row. First is counted from the left hand side. Perhaps I should add that first means really first, not zeroth. [...] > Sometimes it is necessary to quote. In particular if you are >> uninformed but nevertheless refuse to take advice from me. >> I don't need any advice from you. You don't know it. That' s why you would need to learn a lot. Look, > you have learned from me meanwhile, even against your furious > opposition, that omega = |omega| in modern set theory. You do no longer persue your plan to prove a contradiction in modern set > theory? Perhaps I will find another proof, but I think those delivered are sufficient. Doesn't this case make you wonder whether there are other things which > you do not yet know but which you could learn from me? > My opinion is >> A (n e omega & |{0, 1, 2, ..., n}| < |omega|) Seems an empty opinion. What is the symbol A refers to? A n (n e omega & |{0, 1, 2, ..., n}| < |omega|) That is correct. All (initial) segments of natural numbers (which consist only of natural numbers!) have a finite number of members. What I say is: The cardinality of the sequence (d_nn) is the cardinality > of omega. And the ordinality of (d_nn) is omega too, like the ordinality of the first column. Therefore it is clear that all segments of natural numbers (which all are subsets of the numbers contained in the lines of the EIT) are finite. === Subject: Re: Cantor Confusion > You do no longer persue your plan to prove a contradiction in modern set > theory? Perhaps I will find another proof, but I think those delivered are > sufficient. Sufficient for what? Since all WM's arguments require assumptions outside of the axioms of ZF and which themselves contradict ZF, the contradictions are not within ZF, but between WM's system and the ZF system. Doesn't this case make you wonder whether there are other things which > you do not yet know but which you could learn from me? > My opinion is >> A (n e omega & |{0, 1, 2, ..., n}| < |omega|) Seems an empty opinion. What is the symbol A refers to? A n (n e omega & |{0, 1, 2, ..., n}| < |omega|) That is correct. All (initial) segments of natural numbers (which > consist only of natural numbers!) have a finite number of members. Only those initial segments which consist of all the naturals preceding some fixed natural are finite. Subsets of the naturals which cannot be contained within any such a bounded initial segment, and they exist, are not finite. What I say is: The cardinality of the sequence (d_nn) is the cardinality > of omega. And the ordinality of (d_nn) is omega too, like the ordinality of the > first column. Therefore it is clear that all segments of natural > numbers (which all are subsets of the numbers contained in the lines of > the EIT) are finite. All bounded initial segments, yes, but an unbounded segment exists. Supersedes: <4561c092$0$97239$892e7fe2@authen.yellow.readfreenews.net> === Subject: Re: Cantor Confusion > A theory of _sets_ is not a theory of _columns_. >> Experience has shown that practically all notions used in >> contemporary mathematics can be defined, and their mathematical >> properties derived, in ZFC. In this sense, the axiomatic set theory >> serves as a satisfactory foundation for al[l] other branches of >> mathematics. It can describe every mathematical notion --- with the >> exception of what a column is? >> I do not speculate about what is describable and what not. If you >> want to posit a definition do so! Practically all notions! If you again refuse to learn from me, look > here: Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. In this sense, the axiomatic set theory > serves as a satisfactory foundation for the other branches of > mathematics. [Karel Hrbacek and Thomas Jech: Introduction to Set > Theory Marcel Dekker Inc., New York, 1984, 2nd edition, p. 3] Again: Meet _your_ obligation and define column[s]. You have introduces this notion hence it is up to you to define it. >> It is curious that you obviously don't like to give precise >> definitions of certain notions even when you have explicitly been >> asked for. I defined the EIT. A column is a vertical row. What is a row in the language of (which?) set theory? > The first column of it is the first vertical row. First is counted > from the left hand side. Perhaps I should add that first means > really first, not zeroth. Would you please posit your claim in coherent sentences in the language of (which?) set theory now? >> [...] > Sometimes it is necessary to quote. In particular if you are > uninformed but nevertheless refuse to take advice from me. >> I don't need any advice from you. >> You don't know it. That' s why you would need to learn a lot. Look, >> you have learned from me meanwhile, even against your furious >> opposition, that omega = |omega| in modern set theory. >> You do no longer persue your plan to prove a contradiction in modern >> set theory? Perhaps I will find another proof, but I think those delivered are > sufficient. You have not yet found any. >> Doesn't this case make you wonder whether there are other things >> which you do not yet know but which you could learn from me? > My opinion is >> A (n e omega & |{0, 1, 2, ..., n}| < |omega|) >> Seems an empty opinion. What is the symbol A refers to? >> A n (n e omega & |{0, 1, 2, ..., n}| < |omega|) [(ISCARD)] That is correct. All (initial) segments of natural numbers (which > consist only of natural numbers!) have a finite number of > members. [(WMPHRASING)] I again refer to the definition of initial segment given in http://mathworld.wolfram.com/InitialSegment.html (ISDEF) 1. An initial segment is a special subset of a *set* and not of [...] numbers (plural). Your phrasing could lead to the (mis?)interpretation that segments of natural numbers means segments /consisting/ of natural numbers which is not the point to aim at. Perhaps you could clarify this. 2. Theorem: There is no set which is identical to one of its initial segments. Prove this as a homework and you will learn from yourself. You may discuss the three cases empty set, set with last element and set without last element separately. 3. A better translation of (ISCARD) would read Every single initial segment of _the_ _set_ of natural numbers has finite cardinality. {0, 1, 2, ..., n} is an initial segment of omega with respect to some element n + 1 e omega. Hence (ISCARD) states that every single initial segment of _the_ _set_ of natural numbers has a finite cardinality. >> What I say is: The cardinality of the sequence (d_nn) is the >> cardinality of omega. And the ordinality of (d_nn) is omega too, like the ordinality of the > first column. Whatever ordinality of the column formally is. > Therefore it is clear that all segments of natural numbers (which all > are subsets of the numbers contained in the lines of the EIT) are > finite. (ISCARD) is about initial segements not about plain-vanilla segments which still lack a definition of yours. F. N. -- xyz === Subject: Re: Cantor Confusion <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455db94f$0$97241$892e7fe2@authen.yellow.readfreenews.net> <455ee80f$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455f6f2c$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605b9a$0$97253$892e7fe2@authen.yellow.readfreenews.net> <4561c128$0$97239$892e7fe2@authen.yellow.readfreenews.net Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. In this sense, the axiomatic set theory > serves as a satisfactory foundation for the other branches of > mathematics. [Karel Hrbacek and Thomas Jech: Introduction to Set > Theory Marcel Dekker Inc., New York, 1984, 2nd edition, p. 3] Again: Meet _your_ obligation and define column[s]. You have > introduced this notion hence it is up to you to define it. No. The notion column is well known to anyone having studied the first semesters math. Further I defined it by means of the EIT as the union of all initial subsets of N. > It is curious that you obviously don't like to give precise >> definitions of certain notions even when you have explicitly been >> asked for. I defined the EIT. A column is a vertical row. What is a row in the language of (which?) set theory? I use the language of mathematics. > === Subject: Re: Cantor Confusion > Again: Meet _your_ obligation and define column[s]. You have > introduced this notion hence it is up to you to define it. No. Yes! >The notion column is well known to anyone having studied the > first semesters math. It is equally well known to those who study Greek architecture. Are yours Doric, Ionic or Corinthian? > Further I defined it by means of the EIT as the > union of all initial subsets of N. What is the EIT? The union of all initial subsets of N is just N itself, as every member of N is a member of at least one such initial subset. > It is curious that you obviously don't like to give precise >> definitions of certain notions even when you have explicitly been >> asked for. I defined the EIT. A column is a vertical row. What is a row in the language of (which?) set theory? I use the language of mathematics. WM's version of the language of mathematics is somehow never compatible with ZF to ZFC or NBG or NF or any standard foundation for mathematics. He must have learnt it in Babel. === Subject: Re: Cantor Confusion Again: Meet _your_ obligation and define column[s]. You have > introduced this notion hence it is up to you to define it. No. > > Yes! The notion column is well known to anyone having studied the > first semesters math. It is equally well known to those who study Greek architecture. > Are yours Doric, Ionic or Corinthian? Further I defined it by means of the EIT as the > union of all initial subsets of N. What is the EIT? That's WM's Equilateral Infinite Triangle, i.e., the list with line n having length n. Haven't you been paying attention? :) > The union of all initial subsets of N is just N itself, as every member > of N is a member of at least one such initial subset. > >> It is curious that you obviously don't like to give precise >> definitions of certain notions even when you have explicitly been >> asked for. I defined the EIT. A column is a vertical row. What is a row in the language of (which?) set theory? I use the language of mathematics. WM's version of the language of mathematics is somehow never > compatible with ZF to ZFC or NBG or NF or any standard foundation for > mathematics. He must have learnt it in Babel. Nor is it compatible with that of any historical mathematician, despite his frequent quotes from Cantor's papers. -- David Marcus === Subject: Re: Cantor Confusion > Further I defined it by means of the EIT as the > union of all initial subsets of N. What is the EIT? That's WM's Equilateral Infinite Triangle, i.e., the list with line n > having length n. Haven't you been paying attention? :) Then it must be an equilateral /right/ triangle if rows are horizontal and columns are vertical. So WM not only messes up set theory, he messes up geometry to boot. Unless it is spherical geometry, which does allow a triangle with 3 right angles. But then the edges would be finite. I use the language of mathematics. WM's version of the language of mathematics is somehow never > compatible with ZF to ZFC or NBG or NF or any standard foundation for > mathematics. He must have learnt it in Babel. Nor is it compatible with that of any historical mathematician, despite > his frequent quotes from Cantor's papers. === Subject: Re: Cantor Confusion >> Experience has shown that practically all notions used in >> contemporary mathematics can be defined, and their mathematical >> properties derived, in this axiomatic system. In this sense, the >> axiomatic set theory serves as a satisfactory foundation for the >> other branches of mathematics. [Karel Hrbacek and Thomas Jech: >> Introduction to Set Theory Marcel Dekker Inc., New York, 1984, >> 2nd edition, p. 3] >> Again: Meet _your_ obligation and define column[s]. You have >> introduced this notion hence it is up to you to define it. No. The notion column is well known to anyone having studied the > first semesters math. Why not stating it if it is so obvious? > Further I defined it by means of the EIT as the > union of all initial subsets of N. Does anybody keep track of your definitions? Where can I look them up? Do you mean initial subsets or initial segments according to the definition in http://mathworld.wolfram.com/InitialSegment.html ? > It is curious that you obviously don't like to give precise > definitions of certain notions even when you have explicitly been > asked for. >> I defined the EIT. A column is a vertical row. >> What is a row in the language of (which?) set theory? I use the language of mathematics. What is a possible formal definition of the set which represents a row? F. N. -- xyz === Subject: Re: Cantor Confusion Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. In this sense, the axiomatic set theory > serves as a satisfactory foundation for the other branches of > mathematics. [Karel Hrbacek and Thomas Jech: Introduction to Set > Theory Marcel Dekker Inc., New York, 1984, 2nd edition, p. 3] Again: Meet _your_ obligation and define column[s]. You have > introduced this notion hence it is up to you to define it. No. The notion column is well known to anyone having studied the > first semesters math. Further I defined it by means of the EIT as the > union of all initial subsets of N. By union of all initial subsets of N, do you mean N? >> It is curious that you obviously don't like to give precise >> definitions of certain notions even when you have explicitly been >> asked for. I defined the EIT. A column is a vertical row. What is a row in the language of (which?) set theory? I use the language of mathematics. Was that a joke? Because you are very funny. First, you claim that you are showing how set theory leads to an inconsistency. Then, when asked which version of set theory you are discussing, you switch and say you aren't really talking about set theory. Plus, you play this game of bait and switch repeatedly. Why don't you just admit that you are developing a new type of mathematics because you want to rather than insisting that you are doing it because the existing mathematics is inconsistent? You obviously are not familiar with existing mathematics, since you don't know any of the standard meanings of the many mathematical words you use. -- David Marcus === Subject: Re: Cantor Confusion <455b1275$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b8c15$0$97265$892e7fe2@authen.yellow.readfreenews.net> <455c4019$0$97239$892e7fe2@authen.yellow.readfreenews.net> <455db94f$0$97241$892e7fe2@authen.yellow.readfreenews.net> <455ee80f$0$97218$892e7fe2@authen.yellow.readfreenews.net> <455f6f2c$0$97228$892e7fe2@authen.yellow.readfreenews.net> <45605b9a$0$97253$892e7fe2@authen.yellow.readfreenews.net The reaction of William confirms that my ideas are understandable. So > your reaction does not concern my writings but rather your means of > reception. You may even cite D. Marcus as your witness: There is a proverb in Germany: Sage mir, mit wem du umgehst, und ich sage dir, wer du bist. Therefore, I wouldn't like to become too familiar with fools. === Subject: Re: Cantor Confusion The reaction of William confirms that my ideas are understandable. So > your reaction does not concern my writings but rather your means of > reception. You may even cite D. Marcus as your witness: There is a proverb in Germany: Sage mir, mit wem du umgehst, und ich > sage dir, wer du bist. Therefore, I wouldn't like to become too > familiar with fools. But royalty all have their fools like WM, and mathematics is the queen of sciences. === Subject: Re: Cantor Confusion > I do not speculate about what is describable and what not. If you want > to posit a definition do so! It is curious that you obviously don't like to give precise definitions > of certain notions even when you have explicitly been asked for. You've only been here a short time, so you probably missed my discussion with WM about what the word definition means. His definition of the word definition is very different from ours. To him, a definition is any sort of discussion/explanation. He denies that our sort of definition is possible or reasonable. -- David Marcus === Subject: Re: Cantor Confusion > >> I do not speculate about what is describable and what not. If you >> want to posit a definition do so! It is curious that you obviously don't like to give precise >> definitions of certain notions even when you have explicitly been >> asked for. You've only been here a short time, so you probably missed my > discussion with WM about what the word definition means. I had similar discussions with WM. > His definition of the word definition is very different from ours. > To him, a definition is any sort of discussion/explanation. He > denies that our sort of definition is possible or reasonable. Interestingly in WM's world there are wrong definitions: It seems that by some magic a truth value has been attached to a definition. This is in contrast to the meaning of wrong definition commonly refered to. F. N. -- xyz === Subject: Re: Cantor Confusion > You've only been here a short time, so you probably missed my > discussion with WM about what the word definition means. I had similar discussions with WM. His definition of the word definition is very different from ours. > To him, a definition is any sort of discussion/explanation. He > denies that our sort of definition is possible or reasonable. Interestingly in WM's world there are wrong definitions: It seems that > by some magic a truth value has been attached to a definition. Definition -> Explanation Wrong Definition -> Wrong Explanation > This is > in contrast to the meaning of wrong definition commonly refered to. I wonder what WM thinks the word wrong means. -- David Marcus === Subject: Re: Cantor Confusion idea in order to distinguish it from numbers which can be written in >lists and can be subject to a diagonal proof. I call only those >entities numbers which can be put in trichotomy with each other. I don't know what a diagonal proof and trichotomy may be. Excuse me. > 1) The diagonal proof by Cantor shows that any list of real numbers is > incomplete. For this purpose we use a list (= injective sequence) of > real numbers and exchange the n-th digit of the n-th number. The > changed digits put together yield a real number which differs from each > list entry at least at one position (it differs at position n from the > n-th list entry). This proof requires that all digits of numbers like > sqrt(2) do exist and can be exchanged. That assumption is wrong. WRONG AGAIN! Cantor's diagonal proof merely requires that the nth > number in the list have, in principle even if not in practice, a > determinable nth digit, which is quite a different issue. If not all digits of the diagonal number exist, then the diagonal number is not an example for a number which exists but is not in the list. For example, it is definitely the case that for any given position n, > the nth digit of sqrt(2) is, at least in principle, determinable, > however impractical it might be to carry out such a determination. No, it is clear for everyone who is not a fanatic that in principle and in praxis not every digit of sqrt(2) is determinable. === Subject: Re: Cantor Confusion > >Sqrt(2) does exist as the diagonal of the square. But I call that an >idea in order to distinguish it from numbers which can be written in >lists and can be subject to a diagonal proof. I call only those >entities numbers which can be put in trichotomy with each other. I don't know what a diagonal proof and trichotomy may be. Excuse me. > 1) The diagonal proof by Cantor shows that any list of real numbers is > incomplete. For this purpose we use a list (= injective sequence) of > real numbers and exchange the n-th digit of the n-th number. The > changed digits put together yield a real number which differs from each > list entry at least at one position (it differs at position n from the > n-th list entry). This proof requires that all digits of numbers like > sqrt(2) do exist and can be exchanged. That assumption is wrong. WRONG AGAIN! Cantor's diagonal proof merely requires that the nth > number in the list have, in principle even if not in practice, a > determinable nth digit, which is quite a different issue. If not all digits of the diagonal number exist, then the diagonal > number is not an example for a number which exists but is not in the > list. But there is nothing that prevents any digit from existing, provided only that the nth digit of the nth listed number is, at least in theory, determinable. For example, it is definitely the case that for any given position n, > the nth digit of sqrt(2) is, at least in principle, determinable, > however impractical it might be to carry out such a determination. No, it is clear for everyone who is not a fanatic that in principle and > in praxis not every digit of sqrt(2) is determinable. I did not say every, I said any. Does WM claim that there is ANY digit in the decimal expansion of sqrt(2) that is not, at least theoretically, determinable? If so he is a fool. === Subject: Re: Cantor Confusion number is not an example for a number which exists but is not in the > list. But there is nothing that prevents any digit from existing, Existing are such ideas which are accessible. No others are existing, because existence of ideas means existence in the mind of someone. For example, it is definitely the case that for any given position n, > the nth digit of sqrt(2) is, at least in principle, determinable, > however impractical it might be to carry out such a determination. No, it is clear for everyone who is not a fanatic that in principle and > in praxis not every digit of sqrt(2) is determinable. I did not say every, I said any. Does WM claim that there is ANY > digit in the decimal expansion of sqrt(2) that is not, at least > theoretically, determinable? I know that at most 10^100 digits of sqrt(2) can be determined, in principle. In praxis there are less digits possible. Therefore the decimal representation of this number can never be treated in a list. === Subject: Re: Cantor Confusion > If not all digits of the diagonal number exist, then the diagonal > number is not an example for a number which exists but is not in the > list. But there is nothing that prevents any digit from existing, Existing are such ideas which are accessible. No others are existing, > because existence of ideas means existence in the mind of someone. That something does not exist in WM's mind does not prevent it from existing in others' minds. For example, it is definitely the case that for any given position n, > the nth digit of sqrt(2) is, at least in principle, determinable, > however impractical it might be to carry out such a determination. No, it is clear for everyone who is not a fanatic that in principle and > in praxis not every digit of sqrt(2) is determinable. I did not say every, I said any. Does WM claim that there is ANY > digit in the decimal expansion of sqrt(2) that is not, at least > theoretically, determinable? I know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In praxis there are less digits possible. Therefore the > decimal representation of this number can never be treated in a list. Engineering is about practice. mathematics is about principles. While perhaps one cannot engineer a square root of two, one can imagine it, which is all that mathematics needs. === Subject: Re: Cantor Confusion number is not an example for a number which exists but is not in the > list. But there is nothing that prevents any digit from existing, Existing are such ideas which are accessible. No others are existing, > because existence of ideas means existence in the mind of someone. For example, it is definitely the case that for any given position n, > the nth digit of sqrt(2) is, at least in principle, determinable, > however impractical it might be to carry out such a determination. No, it is clear for everyone who is not a fanatic that in principle and > in praxis not every digit of sqrt(2) is determinable. I did not say every, I said any. Does WM claim that there is ANY > digit in the decimal expansion of sqrt(2) that is not, at least > theoretically, determinable? I know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In principle, if a is the sqrt(2) to 10^100 digits, then 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, or adding it to 1, or taking 0.5*(a + 2/a)? - Randy === Subject: Re: Cantor Confusion principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. If you don't believe me, simply try it. By about 330 calculations you should be able to produce the first 10^100 digits. If you have a slow computer you will need one second per calculation. So let it run for 5 minutes and you will know what prevents you from calculating 2/a. === Subject: Re: Cantor Confusion principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. That means it's hard in practice. But what prevents 2/a from existing in principle? > If you don't believe me, simply try it. By about 330 calculations you > should be able to produce the first 10^100 digits. If you have a slow > computer you will need one second per calculation. So let it run for 5 > minutes and you will know what prevents you from calculating 2/a. What difference does the speed of an actual computer make? We aren't talking about implementation, we're talking about in principle. How can a represent all bits available IN PRINCIPLE? In principle, there is no limit on the available bits. - Randy === Subject: Re: Cantor Confusion principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. That means it's hard in practice. But what prevents 2/a from existing in principle? It is not hard in practice, but it is impossible. There is no chance. That means in principle. If you don't believe me, simply try it. By about 330 calculations you > should be able to produce the first 10^100 digits. If you have a slow > computer you will need one second per calculation. So let it run for 5 > minutes and you will know what prevents you from calculating 2/a. What difference does the speed of an actual computer make? We > aren't talking about implementation, we're talking about in > principle. How can a represent all bits available IN PRINCIPLE? In principle, > there is no limit on the available bits. Do you think, in principle there are hidden variables, we only can't determine them? No, in principle we cannot circumvent the uncertainty relations and we cannot do what we know to be excluded in eternity. That is a meaningful understanding of in principle. If we accept your in principle, then in principle we can maintain the most ridiculous mess - and then we arrive at the finished infinity. === Subject: Re: Cantor Confusion > > I know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. That means it's hard in practice. But what prevents 2/a from existing in principle? It is not hard in practice, but it is impossible. There is no chance. > That means in principle. Then WM's principles are much less coherent that anyone else's. No, in principle we cannot circumvent the uncertainty relations and we > cannot do what we know to be excluded in eternity. That is a meaningful > understanding of in principle. If we accept your in principle, then > in principle we can maintain the most ridiculous mess - and then we > arrive at the finished infinity. Good! Let's do it and be done with it. I much prefer our own ridiculous mess to WM's even more ridiculous one. === Subject: Re: Cantor Confusion I know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. That means it's hard in practice. But what prevents 2/a from existing in principle? It is not hard in practice, but it is impossible. There is no chance. > That means in principle. No, that is not the meaning of in principle. - Randy === Subject: Re: Cantor Confusion I know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. That means it's hard in practice. But what prevents 2/a from existing in principle? It is not hard in practice, but it is impossible. There is no chance. > That means in principle. No, that is not the meaning of in principle. It is. What you mean is in the magical realm of transfinity-worshippers === Subject: Re: Cantor Confusion > It is not hard in practice, but it is impossible. There is no chance. > That means in principle. No, that is not the meaning of in principle. It is. What you mean is in the magical realm of > transfinity-worshippers > Only someone without principles could misinterpret what in principle means so badly. === Subject: Re: Cantor Confusion I know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. That means it's hard in practice. But what prevents 2/a from existing in principle? If you don't believe me, simply try it. By about 330 calculations you > should be able to produce the first 10^100 digits. If you have a slow > computer you will need one second per calculation. So let it run for 5 > minutes and you will know what prevents you from calculating 2/a. What difference does the speed of an actual computer make? We > aren't talking about implementation, we're talking about in > principle. How can a represent all bits available IN PRINCIPLE? In principle, > there is no limit on the available bits. Just a guess, but I think that for WM, the only things that are possible in principle are those that are possible in practice. It reminds me of the original Star Trek series: There was a barrier at the edge of the galaxy. So, if you got to the edge of the galaxy, you couldn't go any further (except in those episodes where they figured out how to get through the barrier). -- David Marcus === Subject: Re: Cantor Confusion principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. If you don't believe me, simply try it. By about 330 calculations you > should be able to produce the first 10^100 digits. If you have a slow > computer you will need one second per calculation. So let it run for 5 > minutes and you will know what prevents you from calculating 2/a. I think the bottom line here is that Mueckenheim doesn't have a clue what in principle means. (Amongst other things) Brian Chandler http://imaginatorium.org === Subject: Re: Cantor Confusion > I know that at most 10^100 digits of sqrt(2) can be determined, in >> principle. >> In principle, if a is the sqrt(2) to 10^100 digits, then >> 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. >> What in principle prevents me from calculating 2/a, >> or adding it to 1, or taking 0.5*(a + 2/a)? >> Lack of bits to represent 2/a if a already requires all bits available. >> If you don't believe me, simply try it. By about 330 calculations you >> should be able to produce the first 10^100 digits. If you have a slow >> computer you will need one second per calculation. So let it run for 5 >> minutes and you will know what prevents you from calculating 2/a. I think the bottom line here is that Mueckenheim doesn't have a clue >what in principle means. (Amongst other things) Whereas you do. ~v~~ === Subject: Re: Cantor Confusion I know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What in principle prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. If you don't believe me, simply try it. By about 330 calculations you > should be able to produce the first 10^100 digits. If you have a slow > computer you will need one second per calculation. So let it run for 5 > minutes and you will know what prevents you from calculating 2/a. You have a funny idea what the words in principle mean. Of course, you have a funny idea what almost all words mean. -- David Marcus === Subject: Re: Cantor Confusion > For example, it is definitely the case that for any given position n, > the nth digit of sqrt(2) is, at least in principle, determinable, > however impractical it might be to carry out such a determination. No, it is clear for everyone who is not a fanatic that in principle and > in praxis not every digit of sqrt(2) is determinable. I did not say every, I said any. Does WM claim that there is ANY > digit in the decimal expansion of sqrt(2) that is not, at least > theoretically, determinable? I'll bet he says that you can determine any digit, but not more than 10^ (10^100) of them at a time. > If so he is a fool. -- David Marcus === Subject: Re: Cantor Confusion For example, it is definitely the case that for any given position n, > the nth digit of sqrt(2) is, at least in principle, determinable, > however impractical it might be to carry out such a determination. No, it is clear for everyone who is not a fanatic that in principle and > in praxis not every digit of sqrt(2) is determinable. I did not say every, I said any. Does WM claim that there is ANY > digit in the decimal expansion of sqrt(2) that is not, at least > theoretically, determinable? I'll bet he says that you can determine any digit, but not more than 10^ > (10^100) of them at a time. > And for the diagonal, you only need one at a time. > If so he is a fool. === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net > I claim case iia: There is a potentially infinite sequence N = > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > treat all of its elements. In particular we can never complete this > set. We can never put it into a list OK. Knock yourself out. I read this several times from you. What does it mean? Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. But it does not say that it has an ordinal number and a cardinal > number. My case iia is in complete agreement with the axiom of > infinity. The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the real numbers can be well-ordered. We cannot construct or define or recognize a well ordering. The axiom of infinity does not say anything about ordinal or > cardinal numbers. However, given that the set N exists and > the defnition of ordinal and cardinal numbers, it is easy to > see that if N exists it must have both an ordinal and a cardinal > number. > No. You assume the possibility of a bijection of the set with itself. That is not proven from the mere existence of the set if we cannot recognize or treat all of its elements. And even if we could, the axiom of infinity does not prove that infinite ordinal and cardinal numbers are numbers, i.e., that they stand in trichotomy with each other. > Ordinal. Every natural number is an ordinal. > Every initial sequence of ordinals is an ordinal. > (an initial sequence is a set of ordinals, such that > if a is in the set every ordinal less than a is in the set). > The set of all natural numbers is an initial sequence of > ordinals. Therefore the set of natural numbers is an > ordinal. Cardinal. > Every natural number is a cardinal number too. > Cardinal numbers are equivalence classes of > sets under the equivalence relation bijection. Since any > set has a bijection to itself, Why? There are even models without a bijection to the natural numbers. > every set must be in an > equivalence class. So the set of natural numbers > has a cardinal number.. And even if it had. By means of the equilateral infinite triangle I proved that this cannot be the case. Therefore we have a contradiction. === Subject: Re: Cantor Confusion > The axiom of infinity does not say anything about ordinal or > cardinal numbers. However, given that the set N exists and > the defnition of ordinal and cardinal numbers, it is easy to > see that if N exists it must have both an ordinal and a cardinal > number. No. You assume the possibility of a bijection of the set with itself. > That is not proven from the mere existence of the set if we cannot > recognize or treat all of its elements. And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. Are you saying that the following is not a bijection of the natural numbers to themselves? 1 <-> 1 2 <-> 2 3 <-> 3 ... -- David Marcus === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net I claim case iia: There is a potentially infinite sequence N = > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > treat all of its elements. In particular we can never complete this > set. We can never put it into a list OK. Knock yourself out. I read this several times from you. What does it mean? Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. But it does not say that it has an ordinal number and a cardinal > number. My case iia is in complete agreement with the axiom of > infinity. The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the real > numbers can be well-ordered. We cannot construct or define or > recognize a well ordering. The fact that we cannot construct a real ordering does not stop us from proving such an ordering exists and using the existence of such an ordering. The axiom of infinity does not say anything about ordinal or > cardinal numbers. However, given that the set N exists and > the defnition of ordinal and cardinal numbers, it is easy to > see that if N exists it must have both an ordinal and a cardinal > number. No. You assume the possibility of a bijection of the set with itself. > That is not proven from the mere existence of the set if we cannot > recognize or treat all of its elements. Piffle >And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. Check the definitions of ordinal and cardinal below. Look for the term trichotomy. Fail to find the term trichotomy. Draw the obvious conclusion. Ordinal. Every natural number is an ordinal. > Every initial sequence of ordinals is an ordinal. > (an initial sequence is a set of ordinals, such that > if a is in the set every ordinal less than a is in the set). > The set of all natural numbers is an initial sequence of > ordinals. Therefore the set of natural numbers is an > ordinal. Cardinal. > Every natural number is a cardinal number too. Cardinal numbers are equivalence classes of > sets under the equivalence relation bijection. Since any > set has a bijection to itself, Why? There are even models without a bijection to the natural numbers. Piffle. If the natural numbers exist, then the identity map is a bijection. - William Hughes === Subject: Re: Cantor Confusion <455b11ff$0$97238$892e7fe2@authen.yellow.readfreenews.net> <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the real > numbers can be well-ordered. We cannot construct or define or > recognize a well ordering. The fact that we cannot construct a real ordering does not > stop us from proving such an ordering exists LOL. I know. > and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility or definability of a real ordering. The existence of a set does not guarantee the existence or definability of a bijection or an identity mapping. And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. Check the definitions of ordinal and cardinal below. Look for > the term trichotomy. Fail to find the term trichotomy. Draw > the obvious conclusion. My conclusion is: You don't know this term. Nevertheless, you will agree if you learn what trichotomy means: a < b or a = b or a > b for any two numbers a and b. > There are even models without a bijection to the natural numbers. Piffle. If the natural numbers exist, then the identity map is > a bijection. If it exists, of course. === Subject: Re: Cantor Confusion <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the real > numbers can be well-ordered. We cannot construct or define or > recognize a well ordering. The fact that we cannot construct a real ordering does not > stop us from proving such an ordering exists LOL. I know. and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility > or definability of a real ordering. > The existence of a set does not guarantee the existence or definability > of a bijection or an identity mapping. No. To say that a set N exists is to say that all elements n of N exist. Thus the mapping n ->n for all elements n of N exists. >And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. Check the definitions of ordinal and cardinal below. Look for > the term trichotomy. Fail to find the term trichotomy. Draw > the obvious conclusion. My conclusion is: You don't know this term. No, this is not the obvious conclusion (it is also wrong). The obvious conclusion is that the fact that the term tricotomy is not used when defining either the ordinals or the cardinals, means that we do not need to show trichotomy to show that something is an ordinal or a cardinal. > Nevertheless, you will > agree if you learn what trichotomy means: a < b or a = b or a > b for > any two numbers a and b. > Trichotomy is a property of ordered sets, not of sets. We need to put an ordering on the ordinals or the cardinals before we can say whether trichotomy holds. If we put the natural ordering on the ordinals, then trichotomy holds for all ordinals (finite and infinite). If we put a partial ordering on the cardinals by a<=b if there exists an injection from a to b, trichotomy will hold for all sets if we assume AC. If we do not assume AC, trichotomy may not hold for all sets, but it does hold for all sets of natural numbers (including the set of all natural numbers). > There are even models without a bijection to the natural numbers. Piffle. If the natural numbers exist, then the identity map is > a bijection. If it exists, of course. > Piffle. If the set of natural numbers exists then the identiy map on the natural numbers exists. - William Hughes === Subject: Re: Cantor Confusion <455b22ef$0$97227$892e7fe2@authen.yellow.readfreenews.net> <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the real > numbers can be well-ordered. We cannot construct or define or > recognize a well ordering. The fact that we cannot construct a real ordering does not > stop us from proving such an ordering exists LOL. I know. and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility > or definability of a real ordering. > The existence of a set does not guarantee the existence or definability > of a bijection or an identity mapping. No. To say that a set N exists is to say that all elements n of N exist. > Thus the mapping n ->n for all elements n of N exists. That is exaggerated. There are models of ZFC (including countably many elements a part of which could be interpreted as the set of natural numbers) where no mapping on N exists. Why should it fail if you were right? And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. Check the definitions of ordinal and cardinal below. Look for > the term trichotomy. Fail to find the term trichotomy. Draw > the obvious conclusion. My conclusion is: You don't know this term. No, this is not the obvious conclusion (it is also wrong). > The obvious conclusion is that the > fact that the term tricotomy is not used when defining > either the ordinals or the cardinals, means that we > do not need to show trichotomy to show that > something is an ordinal or a cardinal. It may be called by another name but trichotomy is implied. In fact all ordinals are asserted to stand in trichotomy with each other. Nevertheless, you will > agree if you learn what trichotomy means: a < b or a = b or a > b for > any two numbers a and b. > Trichotomy is a property of ordered sets, not of sets. Only ordered sets have ordinal numbers / are ordinal numbers. > We need to put > an ordering on the ordinals or the cardinals before we can say > whether trichotomy holds. Ordinals are ordered sets by definition. We need not put anything. > If we put the natural ordering on the > ordinals, then trichotomy holds for all ordinals (finite and infinite). Yes, why then do you dislike the name? > Piffle. If the natural numbers exist, then the identity map is > a bijection. If it exists, of course. > Piffle. If the set of natural numbers exists then the identiy map > on the natural numbers exists. > You are wrong. But this carelessness is typical for set theory. === Subject: Re: Cantor Confusion The existence of a set does not guarantee the existence or definability > of a bijection or an identity mapping. No. To say that a set N exists is to say that all elements n of N exist. > Thus the mapping n ->n for all elements n of N exists. That is exaggerated. There are models of ZFC (including countably many > elements a part of which could be interpreted as the set of natural > numbers) where no mapping on N exists. In what model of ZF does no identity mapping from N to itself exist? > Why should it fail if you were > right? What proof does WM have that there is ANY model of ZF for which no identity mapping from N to N is possible? > Check the definitions of ordinal and cardinal below. Look for > the term trichotomy. Fail to find the term trichotomy. Draw > the obvious conclusion. My conclusion is: You don't know this term. No, this is not the obvious conclusion (it is also wrong). > The obvious conclusion is that the > fact that the term tricotomy is not used when defining > either the ordinals or the cardinals, means that we > do not need to show trichotomy to show that > something is an ordinal or a cardinal. It may be called by another name but trichotomy is implied. In fact all > ordinals are asserted to stand in trichotomy with each other. Trichotomy for ordinals and cardinals is a consequence of other properties, and is not an ur-property. It is provable for cardinals in ZFC and for ordinals in ZF, but is a mere consequence of other properties of cardinality/ordinality. === Subject: Re: Cantor Confusion and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility > or definability of a real ordering. > The existence of a set does not guarantee the existence or definability > of a bijection or an identity mapping. No. To say that a set N exists is to say that all elements n of N exist. > Thus the mapping n ->n for all elements n of N exists. That is exaggerated. There are models of ZFC (including countably many > elements a part of which could be interpreted as the set of natural > numbers) where no mapping on N exists. Can you give an example of such a model? > Why should it fail if you were right? -- David Marcus === Subject: Re: Cantor Confusion <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net > The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the real > numbers can be well-ordered. We cannot construct or define or > recognize a well ordering. The fact that we cannot construct a real ordering does not > stop us from proving such an ordering exists LOL. I know. and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility > or definability of a real ordering. > The existence of a set does not guarantee the existence or definability > of a bijection or an identity mapping. No. To say that a set N exists is to say that all elements n of N exist. > Thus the mapping n ->n for all elements n of N exists. That is exaggerated. There are models of ZFC (including countably many > elements a part of which could be interpreted as the set of natural > numbers) where no mapping on N exists. Why should it fail if you were > right? Piffle. Any defintion of mapping under which we can have an existing set of elements, but no existing identity map goes under the technical term of stupid. And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. Check the definitions of ordinal and cardinal below. Look for > the term trichotomy. Fail to find the term trichotomy. Draw > the obvious conclusion. My conclusion is: You don't know this term. No, this is not the obvious conclusion (it is also wrong). > The obvious conclusion is that the > fact that the term tricotomy is not used when defining > either the ordinals or the cardinals, means that we > do not need to show trichotomy to show that > something is an ordinal or a cardinal. It may be called by another name but trichotomy is implied. In fact all > ordinals are asserted to stand in trichotomy with each other. If you want to include the natural ordering as part of the ordering of the ordinals, yes. ALL ordinals, including infinite ordinals, stand in trichotomy with each other. In particluar, N, the set of all natural numbers, stands in trichotomy with all other ordinals. Nevertheless, you will > agree if you learn what trichotomy means: a < b or a = b or a > b for > any two numbers a and b. > Trichotomy is a property of ordered sets, not of sets. Only ordered sets have ordinal numbers / are ordinal numbers. > Yes. But not every set that has an ordering/satisfies trichotomy is the ordinals (e.g. real numbers with the usual ordering). > We need to put > an ordering on the ordinals or the cardinals before we can say > whether trichotomy holds. Ordinals are ordered sets by definition. We need not put anything. > If you insist. I would define the set of ordinals first, then put an order on it, but, you can do both simultaneously if you want. Note, however, cardinals are not ordered by definition. > If we put the natural ordering on the > ordinals, then trichotomy holds for all ordinals (finite and infinite). Yes, why then do you dislike the name? I don't. Since trichotomy holds for both omega (for all ordinals) and for aleph_0 (at least within sets of natural numbers and within all sets if we assume AC ) there is no problem. I merely point out that it is not necessary to decide whether or not trichotomy holds in order to show that omega is an ordinal and aleph_0 is a cardinal. Neither the definition for ordinal, nor the definition for cardinal contains the term trichotomy or any term that means the same thing. - William Hughes === Subject: Re: Cantor Confusion <455b90b2$0$97249$892e7fe2@authen.yellow.readfreenews.net > The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the real > numbers can be well-ordered. We cannot construct or define or > recognize a well ordering. The fact that we cannot construct a real ordering does not > stop us from proving such an ordering exists LOL. I know. and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility > or definability of a real ordering. > The existence of a set does not guarantee the existence or definability > of a bijection or an identity mapping. No. To say that a set N exists is to say that all elements n of N exist. > Thus the mapping n ->n for all elements n of N exists. That is exaggerated. There are models of ZFC (including countably many > elements a part of which could be interpreted as the set of natural > numbers) where no mapping on N exists. Why should it fail if you were > right? Piffle. Any defintion of mapping under which we can have > an existing set of elements, but no existing identity map goes > under the technical term of stupid. That is a strong argument. You never heard of countable uncountable models? What about all contructible numbers or all words? They cannot be mapped on N though they are a countable set. >And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. Check the definitions of ordinal and cardinal below. Look for > the term trichotomy. Fail to find the term trichotomy. Draw > the obvious conclusion. My conclusion is: You don't know this term. No, this is not the obvious conclusion (it is also wrong). > The obvious conclusion is that the > fact that the term tricotomy is not used when defining > either the ordinals or the cardinals, means that we > do not need to show trichotomy to show that > something is an ordinal or a cardinal. It may be called by another name but trichotomy is implied. In fact all > ordinals are asserted to stand in trichotomy with each other. If you want to include the natural ordering as part of the ordering > of the ordinals, yes. ALL ordinals, including infinite ordinals, stand > in trichotomy with each other. Correct. that's what I said. > Nevertheless, you will > agree if you learn what trichotomy means: a < b or a = b or a > b for > any two numbers a and b. > Trichotomy is a property of ordered sets, not of sets. Only ordered sets have ordinal numbers / are ordinal numbers. > Yes. But not every set that has an ordering/satisfies trichotomy > is the ordinals (e.g. real numbers with the usual ordering). These examples are called order types but not ordinals. We need to put > an ordering on the ordinals or the cardinals before we can say > whether trichotomy holds. Ordinals are ordered sets by definition. We need not put anything. If you insist. Even if I wouldn't insist. It is the case by definition. > I would define the set of ordinals first, then put an > order on it, but, you can do both simultaneously if you want. Note, however, cardinals are not ordered by definition. Note however that in order to attach a definite cardinal number to every set, well-ordering must be possible for every set. That is why Cantor insisted on well-ordering of all sets. > If we put the natural ordering on the > ordinals, then trichotomy holds for all ordinals (finite and infinite). Yes, why then do you dislike the name? I don't. Since trichotomy holds for both omega (for all ordinals) > and for aleph_0 (at least within sets of natural numbers > and within all sets if we assume AC ) > there is no problem. I merely point out that it is not necessary > to decide whether or not trichotomy holds in order to show > that omega is an ordinal and aleph_0 is a cardinal. > Neither the definition for ordinal, nor the definition for cardinal > contains the term trichotomy or any term that means > the same thing. Nevertheless we can recognize ordinals by trichotomy: If two numbers both are ordinals, then they show trichotomy. If two numbers don't show trichotomy, then they are not both ordinals. === Subject: Re: Cantor Confusion > The axiom of infinity says that the set N exists. Your iia > says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the > real > numbers can be well-ordered. We cannot construct or define or > recognize a well ordering. The fact that we cannot construct a real ordering does not > stop us from proving such an ordering exists LOL. I know. and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility > or definability of a real ordering. > The existence of a set does not guarantee the existence or > definability > of a bijection or an identity mapping. No. To say that a set N exists is to say that all elements n of N exist. > Thus the mapping n ->n for all elements n of N exists. That is exaggerated. There are models of ZFC (including countably many > elements a part of which could be interpreted as the set of natural > numbers) where no mapping on N exists. Why should it fail if you were > right? Piffle. Any defintion of mapping under which we can have > an existing set of elements, but no existing identity map goes > under the technical term of stupid. That is a strong argument. You never heard of countable uncountable models? What about all contructible numbers or all words? They cannot be mapped > on N though they are a countable set. We are talking about one set mapping to ITSELF under the identity function x |--> x, not about one set being mapped to another. Such an identity mapping exists for any set because it is a set, at least in ZF or NBG. Does WM know what WM is talking about? > Trichotomy is a property of ordered sets, not of sets. Only ordered sets have ordinal numbers / are ordinal numbers. > Yes. But not every set that has an ordering/satisfies trichotomy > is the ordinals (e.g. real numbers with the usual ordering). These examples are called order types but not ordinals. The examples are *ordered sets* . Different ordered sets can exhibit different order types, but are not themselves order types. An order type applies to a set of ordered sets in which the order relations have in common some property over and above merely being an order relation. Note, however, cardinals are not ordered by definition. Note however that in order to attach a definite cardinal number to > every set, well-ordering must be possible for every set. That is why > Cantor insisted on well-ordering of all sets. That depends on one's definition of cardinal. If one is allowed to take an arbitrary representative of each cardinality , rather than the least ordinal of a given cardinality, then well-ordering of all sets is not necessary, though in that case, one does not have a provable trichotomy on cardinals without an axiom of choice. If we put the natural ordering on the > ordinals, then trichotomy holds for all ordinals (finite and infinite). Yes, why then do you dislike the name? I don't. Since trichotomy holds for both omega (for all ordinals) > and for aleph_0 (at least within sets of natural numbers > and within all sets if we assume AC ) > there is no problem. I merely point out that it is not necessary > to decide whether or not trichotomy holds in order to show > that omega is an ordinal and aleph_0 is a cardinal. > Neither the definition for ordinal, nor the definition for cardinal > contains the term trichotomy or any term that means > the same thing. Nevertheless we can recognize ordinals by trichotomy: If two numbers > both are ordinals, then they show trichotomy. If two numbers don't show > trichotomy, then they are not both ordinals. Numbers is ambiguous. What of the cases where numbers show trichotomy but are not ordinals, or are not even numbers in any usual sense? If numbers includes cardinals and our definition of cardinals does not identify them with ordinals and we do not have an axiom of choice, then it is quite possible to have two distinct cardinals neither of which is smaller than the other. === Subject: Re: Cantor Confusion > The axiom of infinity says that the set N exists. Your iia says we cannot recognize or treat all of its elements This is no contradiction to the axiom. Compare the proof that the real > numbers can be well-ordered. We cannot construct or define or > recognize a well ordering. The fact that we cannot construct a real ordering does not > stop us from proving such an ordering exists LOL. I know. and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility > or definability of a real ordering. > The existence of a set does not guarantee the existence or definability > of a bijection or an identity mapping. No. To say that a set N exists is to say that all elements n of N exist. > Thus the mapping n ->n for all elements n of N exists. That is exaggerated. There are models of ZFC (including countably many > elements a part of which could be interpreted as the set of natural > numbers) where no mapping on N exists. Why should it fail if you were > right? Piffle. Any defintion of mapping under which we can have > an existing set of elements, but no existing identity map goes > under the technical term of stupid. That is a strong argument. You never heard of countable uncountable models? I've never heard of something which is both countable (can be bijected to N) and uncountable (can not be bijected to N). That would seem to require both P and (not P) to be true for some proposition P. > What about all contructible numbers or all words? They cannot be mapped > on N though they are a countable set. The set of all finite words, the set of polynomials, or any countable set can be put in bijection with N. Where do you get your view that it can't be mapped to N? - Randy === Subject: Re: Cantor Confusion You never heard of countable uncountable models? I've never heard of something which is both countable > (can be bijected to N) and uncountable (can not be bijected > to N). That would seem to require both P and (not P) to > be true for some proposition P. Indeed, that is my impression too. But ZFC cannot tolerate P and notP. Therefore people have dispelled it. If ZFC is correct, then, according to a theorem of Skolem, it must have a countable model. But the most important theorem of ZFC proves the existence of uncountable sets. Therefore the worshippers of transfinity proclaim the existence of a set which is countable (as Skolem requires) from outside but does not contain the bijection with N internally. Therefore inside the model, countability cannot be proved - and all is fine. What about all contructible numbers or all words? They cannot be mapped > on N though they are a countable set. The set of all finite words, the set of polynomials, or any > countable set can be put in bijection with N. Where do you > get your view that it can't be mapped to N? The list of all finite words cannot be constructed. The list of all constructible numbers cannot be constructed. That means, these bijections cannot be constructed. === Subject: Re: Cantor Confusion If ZFC is correct, then, according > to a theorem of Skolem, it must have a countable model. If ZFC is consistent, then it has a countable model. If one finds that to be an unappealing feature of ZFC, then fine, one can choose not to use ZFC or to seek some other theory. But ZFC is not inconsistent simply for Skolem's paradox nor is ZFC even rendered prohibitively counterintutitive by Skolem's paradox. There exists a bijection from the universe of the countable model onto N; but that bijection is not itself a member of the universe of the countable model. And why should we assume that the universe of such a countable model must have such a function as a member? MoeBlee === Subject: Re: Cantor Confusion > If ZFC is correct, then, according > to a theorem of Skolem, it must have a countable model. If ZFC is consistent, then it has a countable model. If one finds that to be an unappealing feature of ZFC, then fine, one > can choose not to use ZFC or to seek some other theory. If some other theory is consistent, then it has a countable model. > But ZFC is not > inconsistent simply for Skolem's paradox nor is ZFC even rendered > prohibitively counterintutitive by Skolem's paradox. That's why Skolem liked ZFC soo much? > There exists a > bijection from the universe of the countable model onto N; but that > bijection is not itself a member of the universe of the countable > model. And why should we assume that the universe of such a countable > model must have such a function as a member? Because the bijection from the model (N') onto N *is* a countable set. And if the model contains countable sets (N'), why then does it not contain this set, which bijects N' and N? Why do you think that our arithmetic (which is not a model of ZFC) does contain such a bijection? === Subject: Re: Cantor Confusion If ZFC is correct, then, according > to a theorem of Skolem, it must have a countable model. If ZFC is consistent, then it has a countable model. If one finds that to be an unappealing feature of ZFC, then fine, one > can choose not to use ZFC or to seek some other theory. If some other theory is consistent, then it has a countable model. If a FIRST ORDER theory of predicate logic HAS AN INFINITE model, then it has a countable model. > But ZFC is not > inconsistent simply for Skolem's paradox nor is ZFC even rendered > prohibitively counterintutitive by Skolem's paradox. That's why Skolem liked ZFC soo much? Yes, Skolem was critical of set theory. > There exists a > bijection from the universe of the countable model onto N; but that > bijection is not itself a member of the universe of the countable > model. And why should we assume that the universe of such a countable > model must have such a function as a member? Because the bijection from the model (N') onto N *is* a countable set. > And if the model contains countable sets (N'), why then does it not > contain this set, which bijects N' and N? That's ridiculous reasoning. Just because a set has as members certain countable sets doesn't demand that the set have as members all countable sets. It would help if you knew just what a structure for a language is. > Why do you think that our > arithmetic (which is not a model of ZFC) does contain such a bijection? I don't know what bijection you are referring to. Exactly what bijection in exactly what set do you claim existence? MoeBlee === Subject: Re: Cantor Confusion > If some other theory is consistent, then it has a countable model. If a FIRST ORDER theory of predicate logic HAS AN INFINITE model, then > it has a countable model. Of course, I implied these properties. One cannot repeat always everything. But HAS AN INFINITE is superfluous, at least the capitals, because if the theory has not an infinite model, then it has to have a countable model too. (Every finite model is per definition countable. The meaning of countable set covers finite set as well as denumerable infinite set. Didn't you study set theory and its definitions?) But ZFC is not > inconsistent simply for Skolem's paradox nor is ZFC even rendered > prohibitively counterintutitive by Skolem's paradox. That's why Skolem liked ZFC soo much? Yes, Skolem was critical of set theory. What do you think, why? Was he too stupid to recognize this big advantage of mathematics? > language is. Why do you think that our > arithmetic (which is not a model of ZFC) does contain such a bijection? I don't know what bijection you are referring to. Exactly what > bijection in exactly what set do you claim existence? No bijection in any infinite set. But why do you think that there is a bijection N <--> Q? === Subject: Re: Cantor Confusion My previous post formatted the quote incorrectly. The whole quote: If ZFC is correct, then, according to a theorem of Skolem, it must have a countable model. is WM's. === Subject: Re: Cantor Confusion You never heard of countable uncountable models? I've never heard of something which is both countable > (can be bijected to N) and uncountable (can not be bijected > to N). That would seem to require both P and (not P) to > be true for some proposition P. Indeed, that is my impression too. But ZFC cannot tolerate P and notP. > Therefore people have dispelled it. If ZFC is correct, then, according > to a theorem of Skolem, it must have a countable model. But the most > important theorem of ZFC proves the existence of uncountable sets. > Therefore the worshippers of transfinity proclaim the existence of a > set which is countable (as Skolem requires) from outside but does not > contain the bijection with N internally. Therefore inside the model, > countability cannot be proved - and all is fine. More piffle. Skolem's paradox does not concern us. Yes there is a countable model of ZFC. Yes, within this model there is a set of natural numbers N', and a set of real numbers R' such that within the model there is no bijection between N' and R'. But what you claimed was that there was no bijection between N' and N'. - William Hughes === Subject: Re: Cantor Confusion You never heard of countable uncountable models? I've never heard of something which is both countable > (can be bijected to N) and uncountable (can not be bijected > to N). That would seem to require both P and (not P) to > be true for some proposition P. Indeed, that is my impression too. But ZFC cannot tolerate P and notP. > Therefore people have dispelled it. If ZFC is correct, then, according > to a theorem of Skolem, it must have a countable model. But the most > important theorem of ZFC proves the existence of uncountable sets. > Therefore the worshippers of transfinity proclaim the existence of a > set which is countable (as Skolem requires) from outside but does not > contain the bijection with N internally. Therefore inside the model, > countability cannot be proved - and all is fine. More piffle. Skolem's paradox does not concern us. It does. It is not only a paradox but an antinomy. > Yes there > is a countable model of ZFC. Yes, within this model there is a > set of natural numbers N', and a set of real numbers R' such that > within the model there is no bijection between N' and R'. But what > you claimed > was that there was no bijection between N' and N'. If someone would have claimed, before 1920, that between a countable set N' and a countable set R' no bijection can exist, you would have answered piffle or so. Now you are in the same situation. === Subject: Re: Cantor Confusion > You never heard of countable uncountable models? I've never heard of something which is both countable > (can be bijected to N) and uncountable (can not be bijected > to N). That would seem to require both P and (not P) to > be true for some proposition P. Indeed, that is my impression too. But ZFC cannot tolerate P and notP. > Therefore people have dispelled it. If ZFC is correct, then, according > to a theorem of Skolem, it must have a countable model. But the most > important theorem of ZFC proves the existence of uncountable sets. > Therefore the worshippers of transfinity proclaim the existence of a > set which is countable (as Skolem requires) from outside but does not > contain the bijection with N internally. Therefore inside the model, > countability cannot be proved - and all is fine. More piffle. Skolem's paradox does not concern us. It does. It is not only a paradox but an antinomy. Yes there > is a countable model of ZFC. Yes, within this model there is a > set of natural numbers N', and a set of real numbers R' such that > within the model there is no bijection between N' and R'. But what > you claimed > was that there was no bijection between N' and N'. If someone would have claimed, before 1920, that between a countable > set N' and a countable set R' no bijection can exist, you would have > answered piffle or so. Now you are in the same situation. Yes I would have replied piffle. I would have been right. The only way this can be true is if you reinterpret countable. Do not confuse the meaning of countable within a nonstandard model with the meaning of countable in the standard interpretation. - William Hughes === Subject: Re: Cantor Confusion Yes there > is a countable model of ZFC. Yes, within this model there is a > set of natural numbers N', and a set of real numbers R' such that > within the model there is no bijection between N' and R'. But what > you claimed > was that there was no bijection between N' and N'. If someone would have claimed, before 1920, that between a countable > set N' and a countable set R' no bijection can exist, you would have > answered piffle or so. Now you are in the same situation. Yes I would have replied piffle. I would have been right. > The only way this can be true is if you reinterpret > countable. Do not confuse the meaning of countable > within a nonstandard model There is no standard model of ZFC, so there is no non-standard model. > with the meaning of countable in the > standard interpretation. Countable means: There exists a bijection with N. After having recognized that this simple definition leads to a contradiction, people defined countable in and outside differently. And if they will recognize that there is really a contradiction in ZFC, then they will define contradiction differently. I am sure, they will not fail. === Subject: Re: Cantor Confusion > Countable means: There exists a bijection with N. After having > recognized that this simple definition leads to a contradiction, We've been over this and over this and over this, and you still persist incorrectly. You've not shown a statement P in the language of a Z set theory such that both P and ~P are theorems of the theory. > people > defined countable in and outside differently. I don't know who does that. I don't. > And if they will > recognize that there is really a contradiction in ZFC, then they will > define contradiction differently. I am sure, they will not fail. If there is shown a contradiction in ZFC, then I won't change any definitions; I'll just recognize that ZFC is inconsistent. But you've not shown a contradiction in ZFC. MoeBlee === Subject: Re: Cantor Confusion > Countable means: There exists a bijection with N. After having > recognized that this simple definition leads to a contradiction, We've been over this and over this and over this, and you still persist > incorrectly. You've not shown a statement P in the language of a Z set > theory such that both P and ~P are theorems of the theory. people > defined countable in and outside differently. I don't know who does that. I don't. It is necessary to avoid Skolems antinomy. If you have the same definition of countability inside and outside the countable model, then you will have the same result. But that would be a desaster. And if they will > recognize that there is really a contradiction in ZFC, then they will > define contradiction differently. I am sure, they will not fail. If there is shown a contradiction in ZFC, then I won't change any > definitions; I'll just recognize that ZFC is inconsistent. But you've > not shown a contradiction in ZFC. Yes, the same track should have been followed by set theorists after Russell had shown his antinomy. But I am sure (and in fact I know) that you would not agree that a contradiction is a contradiction. And you are right. That's the simplest way to avoid learning that many of your efforts have been wasted in vain. === Subject: Re: Cantor Confusion What about all contructible numbers or all words? They cannot be mapped > on N though they are a countable set. The set of all finite words, the set of polynomials, or any > countable set can be put in bijection with N. Where do you > get your view that it can't be mapped to N? The list of all finite words cannot be constructed. It can be constructed inductively, which means that in ZF or NBG, it can be constructed. > The list of all > constructible numbers cannot be constructed. The list of all naturals can be constructed inductively, which means that in ZF or NBG, it can be constructed. > That means, these > bijections cannot be constructed. In ZF they can! > === Subject: Re: Cantor Confusion > What about all contructible numbers or all words? They cannot be mapped > on N though they are a countable set. The set of all finite words, the set of polynomials, or any > countable set can be put in bijection with N. Where do you > get your view that it can't be mapped to N? The list of all finite words cannot be constructed. It can be constructed inductively, which means that in ZF or NBG, it can > be constructed. The natural numbers cannot be constructed inductively. Induction proofs