mm-1209 === Subject: Extension [CapitalThorn]elds how can I prove the following: Let E_1, E_2 be sub[CapitalThorn]elds of K/F, and [E_i:F] [CapitalThorn]nite. Then [E:F]<=[E_1:F][E_2:F], where E ist the sub[CapitalThorn]eld generated by E_1, E_2. The statement must be provable with elementary knowledge about extension [CapitalThorn]elds. === Subject: Re: Extension [CapitalThorn]elds > how can I prove the following: > Let E_1, E_2 be sub[CapitalThorn]elds of K/F, and [E_i:F] [CapitalThorn]nite. Then > [E:F]<=[E_1:F][E_2:F], where E ist the sub[CapitalThorn]eld generated by E_1, E_2. > The statement must be provable with elementary knowledge about extension > [CapitalThorn]elds. > Who can help me? > Joachim Say a_1, ... a_n are a basis for E_1 over F and b_1, ... , b_m are a basis for E_2 over F. You can assume that one of each set is the element 1 (why?) and therefore it is clear that the elements a_i b_j for 1 <= i <= n, 1 <= j <= m span E (why? Use the de[CapitalThorn]nition that E is the smallest extension of F containing E_1 and E_2, and exploit the fact that we included 1 explicitly in each basis). There are [E_1 : F][E_2 : F] many elements of this form, and a genuine basis for E therefore has no more than this. -- Ryan Reich ryanr@uchicago.edu === Subject: Re: Extension [CapitalThorn]elds > how can I prove the following: > Let E_1, E_2 be sub[CapitalThorn]elds of K/F, and [E_i:F] [CapitalThorn]nite. Then > [E:F]<=[E_1:F][E_2:F], where E ist the sub[CapitalThorn]eld generated by E_1, E_2. > The statement must be provable with elementary knowledge about extension > [CapitalThorn]elds. Use |E:F| = |E:E1| |E_1:F|. You then need some way of relating |E:E_1| to |E_2:F|. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Extension [CapitalThorn]elds >> how can I prove the following: >> Let E_1, E_2 be sub[CapitalThorn]elds of K/F, and [E_i:F] [CapitalThorn]nite. Then >> [E:F]<=[E_1:F][E_2:F], where E ist the sub[CapitalThorn]eld generated by E_1, E_2. >> The statement must be provable with elementary knowledge about extension >> [CapitalThorn]elds. > Use |E:F| = |E:E1| |E_1:F|. > You then need some way of relating |E:E_1| to |E_2:F|. and i think that |E:E_1|<=|E_2:F|. But i dont know how to prove it, because of my lack of linear-algebra-knowledge. If {v_1,...,v_n} is a base of E_2/F then it is clear that over E_1 is a subspace of E/E_1. It contains surely E_1 and E_2. But i dont know whether =E/E_1. Joachim === Subject: Re: Extension [CapitalThorn]elds > > how can I prove the following: > Let E_1, E_2 be sub[CapitalThorn]elds of K/F, and [E_i:F] [CapitalThorn]nite. Then > [E:F]<=[E_1:F][E_2:F], where E ist the sub[CapitalThorn]eld generated by E_1, E_2. > > The statement must be provable with elementary knowledge about extension > [CapitalThorn]elds. >> Use |E:F| = |E:E1| |E_1:F|. >> You then need some way of relating |E:E_1| to |E_2:F|. >and i think that |E:E_1|<=|E_2:F|. But i dont know how to prove it, because >of my lack of linear-algebra-knowledge. >If {v_1,...,v_n} is a base of E_2/F then it is clear that over >E_1 is a subspace of E/E_1. It contains surely E_1 and E_2. But i dont know >whether =E/E_1. This is not good notation. What you want to show is that {v_1,...,v_n} generate E over E_1. Remember what it means to be a base: since {v_1,...,v_n} is a base for E_2/F, that means that every element of E_2 can be written as an F-linear combination of the v_i: f_1*v_1 + ... + f_n*v_n with f_i in F. What you want to show is that every element of E can be written as an E_1-linear combination of v1,...,vn. You also know that E is the sub[CapitalThorn]eld generated by E_1 and E_2, which means that every element of E can be written as a ([CapitalThorn]nite) sum of products of elements of E_1 and E_2. So let e in E be an element of E. Then we can write it as a1*b1 + ... + ar*br with a_i in E_1 and bi in E_2. Now, we can also write every b_i as an F-linear combination of {v1,...,vn}. Do so, and substitute into the equation above; then reorganize the sum, and remember that F is contained in E_1. Can you conclude that e may be written as an E_1-linear combination of {v1,...,vn}? -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: phi-issad In ninety three, I had a physical system to analyze. I thus sat down at the edge of water and I observed. What I saw and measured was astonishing, and in particular, which relates to time. I saw that the matter attracts the matter, and if this one has a direction, the attracted matter will take the same direction, to some extent it w a s the same principle which wants to think of happy acts attracts happiness. I continued the observation and I saw the swirls were formed with the feet of the sluice valves, I saw that a phenomenon, (with the particular laws) could involve another phenomenon much more powerful (governs by new laws) a little as in a water pan, Si made to you turn the object by accompanying it, therefore while imposing, this one will go at the speed of your handle, but if you give some small impulses, it will create for itself a swirl a speed tending towards the in[CapitalThorn]nite one. It is still the same principle used by Hom.8fre to teach the matter and Saint Exup.8ery, the spirit. They do not dictate, it propose ideas which accelerate the spirit, this stimulated spirit of small implulsions, creates many other ideas much more powerful, particular to each one. You know in summer, one likes yourselves at the edge of water, also I continued the observation, and I found the key which enabled me to analyze this channel, thing that nobody had stated to know to make. What I observed, it is that: each thing is analyzed only compared to itself, thus the lines of the ßuids have their own lives, in fact these principles physical were analysable only compared to themselves in an exponential way. I could only lean me on my perception; over the time of the event, we were still in the [CapitalThorn]rst days of my training course, however the [CapitalThorn]rst contacts were taken, I had posed to my marks, thus I recognized the time of the event, the time which makes that each new experiment tends towards the in[CapitalThorn]nite one. The concept was not long in developing, and revealed an individual time which was particular for me, in connection with my age, id for these young secretaries or this benevolent manager. Social time, this common time which marks the appointment was present like ogre, but the physical systems erased it while smiling, making the stop watch ubuesque from its operation. I had carried out the objective, then I continued to observe, and I saw that the some either way traversed by water, it always arose by the same sluice valves, like the light during the lesson on the interference, or like x-rays, thus the some or way traversed, the thoughts of the large philosophers are found at the same point, the difference is the manner, the borrowed way. This history of time seems to me extraordinary of share the many applications, then I still offer myself a little observations, and I re[CapitalThorn]ne the second principle in a new principle separating the laws from operation, the laws of globality being completely different and independent of the laws of the characteristics which one cr.8e.8eent. The [CapitalThorn]fth law is the part placed between bracket in the second law. Andre pierre jocelyn http://www.letime.net y=k(1-exp(-t/jo)) === Subject: Book recommendations in these areas. Id like book recommendations in the following areas and why you believe the book to be good: - Number Theory - Real Analysis - Abstract Algebra - Single and Multi variable calculus - Linear Algebra - Proofs Brett === Subject: Re: Book recommendations in these areas. > Id like book recommendations in the following areas and why you believe the > book to be good: Good for what? for reading on the bus? for learning on your own? for lecturing from? for setting as a text? > - Number Theory For enjoyable reading Id recommend Beiler, Recreations in the Theory of Numbers; Guy, Unsolved Problems in Number Theory; van der Poorten, Notes on Fermats Last Theorem (disclaimer: the author was my boss); Sierpinski, A Selection of Problems in the Theory of Numbers; Conway, The Sensual Quadratic Form; Havil, Gamma; Klee & Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory. As texts of one sort or another, Niven, Zuckerman and Montgomery, An Introduction to the Theory of Numbers; Leveque, Fundamentals of Number Theory; Roberts, Elementary Number Theory; Ireland & Rosen, A Classical Introduction to Modern Number Theory; Hardy & Wright, The Theory of Numbers; Shapiro, Introduction to the Theory of Numbers; Rosen, Elementary Number Theory and its Applications; Landau, Elementary Number Theory. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Book recommendations in these areas. > Id like book recommendations in the following areas and why you believe the > book to be good: > - Number Theory > - Real Analysis > - Abstract Algebra > - Single and Multi variable calculus > - Linear Algebra > - Proofs Vellemans book is a good introduction to mathematical structures/proofs. > Brett === Subject: Re: Book recommendations in these areas. > Id like book recommendations in the following areas and why you believe the > book to be good: > - Number Theory Hardy & Wright The Theory of Numbers > - Real Analysis Burkill A First Course in Mathematical Analysis > - Abstract Algebra Birkhoff & Mac Lane A Survey of Modern Algebra > - Single and Multi variable calculus Courant & John Introduction to Calculus and Analysis > - Linear Algebra Birkhoff & Mac Lane again > - Proofs ?? The above have proofs in them. > Brett === Subject: Good intro to Lie Groups & Algebras I am looking for a good book and/or online literatures that provides a good introduction to Lie groups and algebras for someone who has not gone through a course in differential geometry. I have a background in advanced calculus, point set topology, real and functional analysis, and basic complex analysis. Unfortunately, I have not taken a course in diff. geom. nor understand the concept of manifolds. If someone knows of a book that is a good intro to diff. geom. that covers Lie groups/algebras and manifolds, that would be great. --john PS: Please post responses to newsgroup. Do not email to me directly. === Subject: Re: Good intro to Lie Groups & Algebras > I am looking for a good book and/or online literatures that provides a > good introduction to Lie groups and algebras for someone who has not > gone through a course in differential geometry. > I have a background in advanced calculus, point set topology, real and > functional analysis, and basic complex analysis. Unfortunately, I have > not taken a course in diff. geom. nor understand the concept of manifolds. > If someone knows of a book that is a good intro to diff. geom. that > covers Lie groups/algebras and manifolds, that would be great. In my view, you lose very little if you restrict yourself to linear groups - which include all semisimple Lie groups. I think Adams Lectures on Lie groups is a very good introduction along these lines. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Good intro to Lie Groups & Algebras > I am looking for a good book and/or online literatures that provides a > good introduction to Lie groups and algebras for someone who has not > gone through a course in differential geometry. > I have a background in advanced calculus, point set topology, real and > functional analysis, and basic complex analysis. Unfortunately, I have > not taken a course in diff. geom. nor understand the concept of manifolds. > If someone knows of a book that is a good intro to diff. geom. that > covers Lie groups/algebras and manifolds, that would be great. > --john > PS: Please post responses to newsgroup. Do not email to me directly. Helgason: _Differential Geometry, Lie Groups, and Symmetric Spaces_. -- Chris Henrich God just doesnt [CapitalThorn]t inside a single religion. === Subject: Re: Good intro to Lie Groups & Algebras > If someone knows of a book that is a good intro to diff. geom. that > covers Lie groups/algebras and manifolds, that would be great. Warner, Foundations of Differentiable Manifolds and Lie Groups (Springer) LD === Subject: Has This Been Noted? The statement that the last Fermat prime is 65537 is equivalent to certain statements about Bernoulli numbers: i. If n>=16, then the Bernoulli number B_(2^n) has denominator 8589934590. ii. If n>=16, then B_(2^(n+1)) - B_(2^n) is an integer. iii. If n>=16, then ceiling(B_(2^n))-B_(2^n)=1/30 + 1/17 + 1/257 + 1/65537. This equivalence is because the Bernoulli number B_2n is I-sum(1/p), where I is an integer and the sum is taken over primes p such that p-1|2n. It would be my guess that this has been noticed before and that no one has pursued this line of attack because we dont know how else to handle the denominator of the 2^n-th Bernoulli number. Does anyone know anything about this? Any help would be appreciated. ---- David To send me email, move the r from the beginning to the end of the part before the @ and insert alum. at the beginning of the part after the @. === Subject: Problem from Atiyah-McDonald I have a problem with Exercise 11 from Chapter 2. It states a question: (Let A be a commutative ring with 1, f - A-module homomorphism (what is important - not necessarily a ring homomorphism)) If f: A^m -> A^n is injective is it always the case that m <= n ? Form of the question suggests that it is not always the case. However, I couldnt [CapitalThorn]nd any example - I would greatly appreciate if someone gave me one (or appropriate proof). What is sure, A cannot be [CapitalThorn]nite (because of set-theoretic argument), [CapitalThorn]eld (obvious) or PID (since f(A^m) is a submodule of A^n, it is isomorphic with A^k, k<=n and hence has a base consisting of k vectors. Every base has a same naumber of elements, thus m <= n. All these refer to facts valid in PIDs) What was at [CapitalThorn]rst strange for me - it can happen that A^2 is isomorphic with A as a ring, but not as an A-module (take A = Z x Z x Z x ...) sirix. === Subject: Re: Problem from Atiyah-McDonald > I have a problem with Exercise 11 from Chapter 2. It states a question: > (Let A be a commutative ring with 1, f - A-module homomorphism (what > is important - not necessarily a ring homomorphism)) > If f: A^m -> A^n is injective is it always the case that m <= n ? Yes. Suppose that m > n. We can assume that m = n + 1. We have a map f: A^m -> A^n. We have to show it has a non-trivial kernel. It is represented by a matrix of size n+1 by n. Call this matrix M. Then we can write down a kernel element. Let M_j be M with row j deleted. Then (det M_1, -det M_2, det M_3, ....) is in the kernel. Of course this might be zero. But then all the M_j have zero determinant. So we reduce the problem to showing that a map g: A^n -> A^n represented by a zero-determinant matrix N has a non-trivial kernel. Unless N is zero (nothing to prove then), then N has a largest submatrix with nonzero determinant. We can suppose that it is the top left k by k matrix. Let P be the top left k+1 by k+1 submatrix of N. Consider the [CapitalThorn]rst row of the adjugate of P. If we add n - k - 1 zeros to the end of this we get a nonzero element in the kernel of g. Im sure all this can be more succintly expressed using exterior algebra. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Problem from Atiyah-McDonald > Yes. What a surprise. I wonder who of them, Atiyah whether McDonald, has put it this way - it should be forbidden :-) However, thank You very much (You were using notation very strange to me (row instead of column, [CapitalThorn]rst instead of last, etc. :-) but it turned to be in my favour all in all - I had to [CapitalThorn]gure out some details by myself). sirix. === Subject: General solutions to (dr/dt)^2=k The problem Im working on is the general solution to: (dr/dt)^2 = k, 0The problem Im working on is the general solution to: >(dr/dt)^2 = k, 0r is the position vector. >This is equivalent to gravity free motion with conservation of energy >and k is double the per unit mass kinetic energy T of a body with mass >m (2T/m) >A solution is planar uniform circular motion of constant radius a, >r(t) = a and omega = constant. >As some of you have pointed out already, there are many parametric >solutions to this equation in the most general case, like in 3-D >motion. >The question is whether there is a general classi[CapitalThorn]cation of these >types of motions. Any references will be appreciated. Vectors dont have squares, so the equation doesnt make any sense square of the magnitude of the velocity? If so the motion can be along _any_ curve in space, as long as (Or mathematically, the solution can be any differentiable curve, parametrized by arclength.) >Mike ************************ David C. Ullrich === Subject: higher reduced K-groups Following the notation of Atiyah: In topological K-theory, the reduced group K~(X) for a pointed space is de[CapitalThorn]ned as the kernel of the canonical group morphism induced by collapsing to the basepoint. Is there a similar relation between the higher reduced and unreduced K-groups? I am trying to a corresponding splitting based on Atiyahs de[CapitalThorn]nitions, but I am always going in suspensions (eh, circles ;-) TIA, Tobias -- everyone who casts a shadow seems to stand in the sun reverse my forename for mail! - saibot === Subject: Re: higher reduced K-groups > Following the notation of Atiyah: > In topological K-theory, the reduced group K~(X) for a pointed space is > de[CapitalThorn]ned as the kernel of the canonical group morphism induced by > collapsing to the basepoint. Ups, I forgot to say: this induces a splitting K(x) = K~(X) + K(pt) where K(pt) = Z. > Is there a similar relation between the > higher reduced and unreduced K-groups? I am trying to a corresponding > splitting based on Atiyahs de[CapitalThorn]nitions, but I am always going in > suspensions (eh, circles ;-) > TIA, > Tobias -- everyone who casts a shadow seems to stand in the sun reverse my forename for mail! - saibot === Subject: Re: higher reduced K-groups >>Following the notation of Atiyah: >>In topological K-theory, the reduced group K~(X) for a pointed space is >>de[CapitalThorn]ned as the kernel of the canonical group morphism induced by >>collapsing to the basepoint. > Ups, I forgot to say: this induces a splitting K(x) = K~(X) + K(pt) where > K(pt) = Z. >>Is there a similar relation between the >>higher reduced and unreduced K-groups? I am trying to a corresponding >>splitting based on Atiyahs de[CapitalThorn]nitions, but I am always going in >>suspensions (eh, circles ;-) By the way, how are you liking Atiyahs book? Its amazing in many ways, but at least the copy I had (which was borrowed, so I dont know how old it may be) was a tad disorganized and I think its not necessarily the best introduction to the subject. I dont know what is; I spent a quarter last year suffering from lack of such an introduction. In particular, when you hit his proof of Botts periodicity theorem, you may know what I mean. The beginning sections arent bad, but his proof of that theorem, while possibly ideal, is rather harsh on the reader. -- Ryan Reich ryanr@uchicago.edu === Subject: Re: higher reduced K-groups >>Following the notation of Atiyah: >>In topological K-theory, the reduced group K~(X) for a pointed space is >>de[CapitalThorn]ned as the kernel of the canonical group morphism induced by >>collapsing to the basepoint. > Ups, I forgot to say: this induces a splitting K(x) = K~(X) + K(pt) where > K(pt) = Z. >>Is there a similar relation between the >>higher reduced and unreduced K-groups? I am trying to a corresponding >>splitting based on Atiyahs de[CapitalThorn]nitions, but I am always going in >>suspensions (eh, circles ;-) I imagine, since the higher K-groups are (as you say) just the K^0 groups of suspensions of X, that the same relation holds. Another way to construct the reduced K-groups is to take them as the kernel of the dimension map on K(X) (each vector bundle gets sent to its dimension; since anything in K(X) is given by a formal difference of vector bundles you can get zero from this even for Also, could you perhaps mean for your de[CapitalThorn]nition that K~(X) is the kernel of the morphism induced by the _inclusion_ of a basepoint? After all, the K functor is contravariant...in this case, it is not hard to show that my de[CapitalThorn]nition and yours are the same, since the induced map preserves dimension of vector bundles, and therefore _is_ just the dimension map (since theres only one bundle of each dimension on a point). -- Ryan Reich ryanr@uchicago.edu === Subject: Re: higher reduced K-groups >Following the notation of Atiyah: >In topological K-theory, the reduced group K~(X) for a pointed space is >de[CapitalThorn]ned as the kernel of the canonical group morphism induced by >collapsing to the basepoint. >> Ups, I forgot to say: this induces a splitting K(x) = K~(X) + K(pt) where >> K(pt) = Z. >Is there a similar relation between the >higher reduced and unreduced K-groups? I am trying to a corresponding >splitting based on Atiyahs de[CapitalThorn]nitions, but I am always going in >suspensions (eh, circles ;-) > I imagine, since the higher K-groups are (as you say) just the K^0 groups > of > suspensions of X, that the same relation holds. No. Let X be a pointed space. Applying the K-functor on the basepoint inclusion of S^n(X) gives a homomorphism K(S^n(X)) --> K(pt), collapsing to pt gives the morphism in the other way so that K(S^n(X)) splits as K(pt) times K~(S^n(X)) = K~^(-n)(X). But now what is the relation between K(S^n(X)) and K^(-n)(X)? > Another way to construct > the reduced K-groups is to take them as the kernel of the dimension map on > K(X) (each vector bundle gets sent to its dimension; since anything in > K(X) is given by a formal difference of vector bundles you can get zero > from this even for > Also, could you perhaps mean for your de[CapitalThorn]nition that K~(X) is the kernel > of the > morphism induced by the _inclusion_ of a basepoint? After all, the K > functor > is contravariant...in this case, it is not hard to show that my de[CapitalThorn]nition > and yours are the same, since the induced map preserves dimension of > vector bundles, > and therefore _is_ just the dimension map (since theres only one bundle > of each dimension on a point). time now on this, a break would be helpful... Currently I am attending an introductory seminar (given by a student of Hirzebruch, btw!) where we use Atiyahs book in a version from 1988. I do not like it very much because constructions are poorly motivated, many details are omitted and in this version, even some things are in the wrong order. I didnt look at the Bott part yet because my talk is about the de[CapitalThorn]nition and homotopy-theoretic description, but I heard that it is very hard! -- everyone who casts a shadow seems to stand in the sun reverse my forename for mail! - saibot === Subject: Re: higher reduced K-groups > >> Following the notation of Atiyah: >> >> In topological K-theory, the reduced group K~(X) for a pointed space is >> de[CapitalThorn]ned as the kernel of the canonical group morphism induced by >> collapsing to the basepoint. > > > Ups, I forgot to say: this induces a splitting K(x) = K~(X) + K(pt) where > K(pt) = Z. > > > >> Is there a similar relation between the higher reduced and unreduced >> K-groups? I am trying to a corresponding splitting based on Atiyahs >> de[CapitalThorn]nitions, but I am always going in suspensions (eh, circles ;-) >> I imagine, since the higher K-groups are (as you say) just the K^0 groups >> of suspensions of X, that the same relation holds. > No. Let X be a pointed space. Applying the K-functor on the basepoint > inclusion of S^n(X) gives a homomorphism K(S^n(X)) --> K(pt), collapsing to > pt gives the morphism in the other way so that K(S^n(X)) splits as K(pt) > times K~(S^n(X)) = K~^(-n)(X). But now what is the relation between K(S^n(X)) > and K^(-n)(X)? I am probably forgetting something, then, but how is K^{-n}(X) de[CapitalThorn]ned if not as K(S^n(X))? Looking at the notes I have from last year, I have it as K^{-n}(X) = K(X / S^n) = K(S^n(X)) in those exact words. Granted, I was using books aside from Atiyah, but I dont know how you would do it otherwise. Oh, I think I remember the dif[CapitalThorn]culty now; does Atiyah neglect to cover the de[CapitalThorn]nitions of ALL the possible combinations of exponent and reduction for the K-groups? That is, does he de[CapitalThorn]ne K~^{-n}(X) but not K^{-n}(X)? >> Another way to construct the reduced K-groups is to take them as the kernel >> of the dimension map on K(X) (each vector bundle gets sent to its >> dimension; since anything in K(X) is given by a formal difference of vector >> the decomposition is sort of obvious. >> Also, could you perhaps mean for your de[CapitalThorn]nition that K~(X) is the kernel >> of the morphism induced by the _inclusion_ of a basepoint? After all, the >> K functor is contravariant...in this case, it is not hard to show that my >> de[CapitalThorn]nition and yours are the same, since the induced map preserves >> dimension of vector bundles, and therefore _is_ just the dimension map >> (since theres only one bundle of each dimension on a point). > time now on this, a break would be helpful... > Currently I am attending an introductory seminar (given by a student of > Hirzebruch, btw!) where we use Atiyahs book in a version from 1988. I do not > like it very much because constructions are poorly motivated, many details > are omitted and in this version, even some things are in the wrong order. I > didnt look at the Bott part yet because my talk is about the de[CapitalThorn]nition and > homotopy-theoretic description, but I heard that it is very hard! Ah, yes, the wrong order. I remember doing a search of the [CapitalThorn]rst several sections to [CapitalThorn]nd one lemma which he cites but, as far as I can tell, does not include. Like I said, I had great dif[CapitalThorn]culties [CapitalThorn]nding a decent introduction to the subject. I also used a book by Husemoller called Fibre Bundles, which is vastly more complete than Atiyah but which I am hesitant to suggest that you take it too literally since there are other problems with it. However, I would use it whenever Atiyah was being too terse. And if you need a reference on the homotopy-theoretic description, theres a book by Aguilar, Gitler, and Prieto called (I think) Algebraic Topology from a Homotopical Viewpoint or something close to that which my partner in that course used, and which likewise is more expansive than Atiyah. Sounds like a cool class, though. Where are you at school? -- Ryan Reich ryanr@uchicago.edu === Subject: Re: higher reduced K-groups > I am probably forgetting something, then, but how is K^{-n}(X) de[CapitalThorn]ned if > not as > K(S^n(X))? Looking at the notes I have from last year, I have it as > K^{-n}(X) = > K(X / S^n) = K(S^n(X)) in those exact words. Granted, I was using > books > aside from Atiyah, but I dont know how you would do it otherwise. Oh, I > think I remember the dif[CapitalThorn]culty now; does Atiyah neglect to cover the > de[CapitalThorn]nitions of > ALL the possible combinations of exponent and reduction for the K-groups? > That is, does he de[CapitalThorn]ne K~^{-n}(X) but not K^{-n}(X)? For some space X without basepoint, K^(-n)(X) is de[CapitalThorn]ned as K~(S^n(X+)) where X+ is the basepointed-space obtained by attaching a separate basepoint giving a new connected component. IMO, the problem is that S^n(X) makes only sense for pointed spaces (S is the reduced suspension). But I am too tired now to think clearly (as you have already seen)... > Sounds like a cool class, though. Where are you at school? I am at Heidelberg University, Germany, the seminar is taught by Matthias Kreck. -- everyone who casts a shadow seems to stand in the sun reverse my forename for mail! - saibot === Subject: Re: higher reduced K-groups >>I am probably forgetting something, then, but how is K^{-n}(X) de[CapitalThorn]ned if >>not as >>K(S^n(X))? Looking at the notes I have from last year, I have it as >>K^{-n}(X) = >>K(X / S^n) = K(S^n(X)) in those exact words. Granted, I was using >>books >>aside from Atiyah, but I dont know how you would do it otherwise. Oh, I >>think I remember the dif[CapitalThorn]culty now; does Atiyah neglect to cover the >>de[CapitalThorn]nitions of >>ALL the possible combinations of exponent and reduction for the K-groups? >>That is, does he de[CapitalThorn]ne K~^{-n}(X) but not K^{-n}(X)? > For some space X without basepoint, K^(-n)(X) is de[CapitalThorn]ned as K~(S^n(X+)) > where X+ is the basepointed-space obtained by attaching a separate > basepoint giving a new connected component. > IMO, the problem is that S^n(X) makes only sense for pointed spaces (S is > the reduced suspension). But I am too tired now to think clearly (as you > have already seen)... Well, Ive tried to think about this for some hours now and I admit Im at a loss for what to say. The answer to your original question appears to be no, in that higher K-groups are reduced by default and therefore equal to their own reductions. This should make some sort of sense by analogy with homology or cohomology, where H~^n(X) and H~_n(X) are both equal to H^n(X) and H_n(X) for any n > 0. I also know that the Bott periodicity theorem is not, in the form K^{n + 2}(X) = K^n(X), actually true for unreduced K-groups, whereas it is for reduced ones (instead, for unreduced groups you get K(X x S^2) = K(X)). The homotopy interpretation of K-theory is likewise more elegant for reduced K-groups. However, although I can throw out nuggets like this I am not sure how to proceed with the actual discussion, and I am not sure what motivates this de[CapitalThorn]nition of the higher K-groups in reality. >>Sounds like a cool class, though. Where are you at school? > I am at Heidelberg University, Germany, the seminar is taught by Matthias > Kreck. Im at the University of Chicago (the e-mail gives it away) and I did some kind of independent study in K-theory last year with Jesper Grodal (hes quite young, Im not expecting you to know of him). -- Ryan Reich ryanr@uchicago.edu === Subject: solid angle Hello. I read somewhere the following equation for the solid angle covered by a PET-camera: solid angle = 4 * pi * sin(arctg(A/D)) A PET camera is a device used to detect and image positron emitters. Nowadays, it consists of a ring detector, into which the patient is brought. A in the equation above refers to the depth of the ring detector, ie. the dimension parallel to the ring axis. D stands for the diameter of the ring. I cannot see how to get to the equation. Using a plane through the axis, I would believe that in such a plane the angle covered by the detectors would equal 4 * arctg(A/D)). But where does the sinus come from? So I looked up the de[CapitalThorn]nition of solid angle in Mathworld and tried to make a calculation. I used as the origin the point on the ring axis halfway between its borders. The x-axis was placed on the ring axis. Then, using the de[CapitalThorn]nition of solid angle and the fact that for all points on the detector y^2 + z^2 = R^2 because of the ring geometry (R being the radius of the ring detector), I arrived at the following integral: Int (-R -- +R) Int (-A/2 -- +A/2) (Sqrt(R^2-y^2) / (x^2 + R^2)^3/2) dx dy I was unable to compute this integral (even with the help of Mathematica), and cannot see how it could lead to an equation as the one cited above. So I probably made several mistakes. I would appreciate any help to sort out this mess. Frank === Subject: Re: solid angle >Hello. >I read somewhere the following equation for the solid angle covered by >a PET-camera: >solid angle = 4 * pi * sin(arctg(A/D)) >A PET camera is a device used to detect and image positron emitters. >Nowadays, it consists of a ring detector, into which the patient is >brought. >A in the equation above refers to the depth of the ring detector, ie. >the dimension parallel to the ring axis. >D stands for the diameter of the ring. >I cannot see how to get to the equation. Using a plane through the >axis, I would believe that in such a plane the angle covered by the >detectors would equal 4 * arctg(A/D)). But where does the sinus come >from? A picture would surely help. And are you sure D is the diameter and not the radius? I tried to look up ring detector on Google and was overwhelmed with hits about doorbell rings and network software etc. Im trying to picture a ring on the surface of a sphere and [CapitalThorn]gure out the solid angle subtended by it. Apparently that isnt the right picture because it doesnt give their answer either. Can you post a picture somewhere? --Lynn === Subject: Re: solid angle >I read somewhere the following equation for the solid angle covered by >a PET-camera: >solid angle = 4 * pi * sin(arctg(A/D)) >A PET camera is a device used to detect and image positron emitters. >Nowadays, it consists of a ring detector, into which the patient is >brought. >A in the equation above refers to the depth of the ring detector, ie. >the dimension parallel to the ring axis. >D stands for the diameter of the ring. >I cannot see how to get to the equation. Using a plane through the >axis, I would believe that in such a plane the angle covered by the >detectors would equal 4 * arctg(A/D)). But where does the sinus come >from? I agree that the equation you have written down is incorrect. TIn addition, A and D are not clearly de[CapitalThorn]ned. This looks like a language translation problem. I think that a PET detector is a gamma-camera arrangement with a Soller collimator in front of a scintillator crystal, often of sodium iodide. The Soller collimator is a slab, typically of lead, densely perforated with high-aspect-ratio cylindrical holes. These holes may be parallel, but can also be drilled so that their axes all point towards a common point, where you put the object to be imaged. The collimator and crystal are typically both circular. By ring do they mean the collimator and/or scintillator? In this case A is roughly the distance of the collimator from object to be imaged, and D is the diameter of the collimator. The solid angle omega subtended by the Soller collimator at the object depends on the half-angle theta. theta is given by an arctangent all right, but arctg(D/A/2). Since the area of a circle goes as the square of its radius, for small theta omega should vary as theta^2. The formula you have read gives totally wrong behavior. I make it: omega = 4*pi*sin{arctg(D/A/2)/2}^2 Zigoteau. === Subject: roots vs zeros I have found several cases where an author has presented a method for [CapitalThorn]nding the zeros of arbitrary functions and has called those zeros Ôroots of the function. But it seems to me that the term Ôroots for zeros has special meaning for polynomials, as they can be constructed from the zeros. Is there a consensus among mathematicians on whether the terms Ôzeros and Ôroots are interchangeable? -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: roots vs zeros >I have found several cases where an author has presented a >method for [CapitalThorn]nding the zeros of arbitrary functions and has >called those zeros Ôroots of the function. But it seems to >me that the term Ôroots for zeros has special meaning for >polynomials, as they can be constructed from the zeros. >Is there a consensus among mathematicians on whether the >terms Ôzeros and Ôroots are interchangeable? I was taught to say that a function has zeroes and an equation has roots. The zeroes of a function f(x) are the roots of the equation f(x) = 0. As I was taught, all of that is true whether the function is a polynomial or not. But I understand that many people do use the terms root and zero interchangeably. Id be curious to see how people on this group feel. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: roots vs zeros >I was taught to say that a function has zeroes and an equation has >roots. The zeroes of a function f(x) are the roots of the equation >f(x) = 0. As I was taught, all of that is true whether the function >is a polynomial or not. One does say roots for a polynomial, as well as a polynomial equation. But a polynomial is not the same as a polynomial function (the difference being most notable over a [CapitalThorn]nite [CapitalThorn]eld). My preference is to reserve roots for polynomials and polynomial equations; in general functions have zeros and equations have solutions. But thats just a preference, and I wouldnt be surprised to [CapitalThorn]nd that Ive occasionally used roots for non-polynomials. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: roots vs zeros >I have found several cases where an author has presented a > method for [CapitalThorn]nding the zeros of arbitrary functions and has > called those zeros Ôroots of the function. But it seems to > me that the term Ôroots for zeros has special meaning for > polynomials, as they can be constructed from the zeros. > Is there a consensus among mathematicians on whether the > terms Ôzeros and Ôroots are interchangeable? I know that the use of the term roots as the zeros of a polynomial is historically based on the fundamental theorem of algebra. Whether this is applicable to other functions I dont know. Since I teach my students the relationship between factoring and roots, I would avoid using the term outside of polynomials simply to avoid confusing the students. === Subject: roots vs. zeros In my experience, the terms are interchangeable, at least when we are concerned with [CapitalThorn]nding x where f(x)=0. f(x) may be a polynomial or any other function. *-----------------------* www.GroupSrv.com *-----------------------* === Subject: Re: roots vs zeros >Is there a consensus among mathematicians on whether the >terms Ôzeros and Ôroots are interchangeable? I dont know if there is a consensus, but my vote would be that the terms are equivalent for polynomials, and that using the term Ôroots in other contexts is least technically incorrect. Of course, just like people do in real elections, I am voting without researching the question. :-) --Lynn === Subject: Fermat 420 1564.140 in reply to 1564.139 Fermats Last Theorem Ben Ito 11-21-04 I will show that Fermats n=4 and Wiles proofs are invalid then prove Fermats last theorem using the law of cosine. l. Introduction Fermats last theorem states that X^n + Y^n = Z^n, (equ 1) when n>2 does not form integer solutions of X, Y and Z. 2. Fermats n=4 Proof Fermats n=4 proof is described. Fermat uses the integer solution equations of n=2, X = 2uv, Y = u^2 - v^2, and Z = u^2 + v^2 (equ 2a,b,c), to derive, X^2 = 2uv, Y^2 = u^2 - v^2, and Z = u^2 + v^2 (equ 3a,b,c), (Shanks, p.141). Equations 3a,b,c are used to prove that n=4 does not form integer solutions. Fermats proof is only proving equations 3a,b,c do not form integer solutions. Proving n=4 using only equations 3a,b,c violates logic. There are an in[CapitalThorn]nite number of integer combinations of X and Y that are not proven in Fermats n=4 proof; Fermat is only proving that the equations 3a,b,c do not form solutions; therefore, Fermats n=4 proof is incomplete and therefore, invalid. 3. Wiles Proof Wiles proof of Fermats last theorem uses Fermats elliptic curve. The elliptic curve equation is derived using the integer solution equations of n=2 (Osserman, p.21), X = 2uv, Y = u^2 - v^2, and Z = u^2 + v^2 (equ 4), Therefore, the elliptic curve is only valid for n=2. Wiles proof of Fermat last theorem is using Fermats elliptic curve to prove n>2; therefore, Wiles proof using elliptic curves is invalid since the elliptic curve is only valid for n=2. 4. Proof I will prove Fermats last theorem. The variables X, Y and Z represent the sides of a triangle. When n>2, the following equation describes the length of Z, Z = (X^n + Y^n)^(1/n). (equ 5) The length of Z is represented with the law of cosine. Z^2 = X^2 + Y^2 - 2XYcos(A). (equ 6) therefore, Z = [X^2 + Y^2 - 2XYcos(A)]^(1/2) (equ 7) Equating equations 5 and 7, Z = (X^n + Y^n)^(1/n) = [X^2 + Y^2 - 2XYcos(A)]^(1/2) (equ 8) Since X=Y=Z =/ 0, the minimum and maximum angle of A is, 0 < A< 60 degrees (equ 9) I will prove that X^2 + Y^2 - 2XYcos(A) (equ 10) of equation 8 always forms a non-integer by showing that 2XYcos(A) always forms a non-integer using the cosine expansion, cos(A) = 1 - (A^2)/2! + (A^4)/4! - (A^6)/6! + ............. . (equ 11). Using equation 11 in equation 10, the value of 2XYcos(A) is always a non-integer; therefore, equation 10 forms a non-integer which proves that equation 8 can never form a integer value of Z which complete the proof of Fermats Last Theorem. 5. Conclusion Fermats n=4 proof is a deception that implies that equations 3a,b,c represent all integers; however, 3a,b,c does not include all integers of X and Y; therefore, Fermats n=4 proof is incomplete and therefore invalid. Fermats elliptic curves are derived using the integer solution equations of n=2; therefore, an elliptic curve can not be used to prove Fermats last theorem when n>2 since the elliptic curve is only valid when n=2. I will prove Fermats last theorem by showing that the law of cosine never forms a integer value of Z when n>2. 6. References Robert Osserman. Fermats Last Theorem (a supplement to the video). MSRI. 1994 Daniel Shanks. Solved and Unsolved Problems in Number Theory. Chelsea Pub. 1985. *-----------------------* www.GroupSrv.com *-----------------------* === Subject: question for math teachers A quick question for anyone whos taught at the high school level, esp. the AP math classes. Having 2 teenagers in high school (9th and 10th grades) both in the Ôhigh math classes for their grades, I have been wondering this for a while. Why is the order of classes set up like: 8th grade, Algebra 1 9th grade, geometry 10th grade, Algebra 2 11 th grade, trig 12th grade, calculus instead of algebra 1, 2 then geometry, trig? and their friends, these kids forget a lot of algebra between 8th and 10th grade! The [CapitalThorn]rst month is then spent on review, whereas if it was just a summer away, it would be far less review. Is there a teaching-philosophy reason for this? k wallace === Subject: Re: question for math teachers > A quick question for anyone whos taught at the high school level, esp. > the AP math classes. > Having 2 teenagers in high school (9th and 10th grades) both in the > Ôhigh math classes for their grades, I have been wondering this for a > while. > Why is the order of classes set up like: > 8th grade, Algebra 1 > 9th grade, geometry > 10th grade, Algebra 2 > 11 th grade, trig > 12th grade, calculus > instead of algebra 1, 2 then geometry, trig? > and their friends, these kids forget a lot of algebra between 8th and > 10th grade! The [CapitalThorn]rst month is then spent on review, whereas if it was > just a summer away, it would be far less review. > Is there a teaching-philosophy reason for this? > k wallace The way geometry is taught, you need to know some algebra. The way algebra 2 is taught, you need to know some geometry. Etc. You can teach geometry without algebra, or upper algebra without geometry, but the result will be something like the physics without calculus taught in business schools. === Subject: Re: question for math teachers rphenry@home.com comments on the sequence of jr.hi and hischool math courses: >The way geometry is taught, you need to know some algebra. >The way algebra 2 is taught, you need to know some geometry. >Etc. Actually, Geometry in high school requires at least introductory algebra knowledge. Also, the proof-based Euclidean Geometry in high school using has returned, and very strongly. The usual sequence of course, sometimes switching the order of Algebra-Intermediate and Geometry, is necessary because the subject matter build as one studies each successive course. Strong algebra skill is used in Trigonometry; strong algebra skill and some details of Trigonometry are used in Calculus. The Geometry is a foundation course that may help in understanding of trigonometry, but most students will forget most of so many different theorems, and such forgetting seems not to be a serious problem in learning trigonometry, since the numeric sense of algebra secures the students progress. Most students will [CapitalThorn]nd trigonometry to be easier and more fun to study than Euclidean geometry. G C === Subject: Re: question for math teachers >A quick question for anyone whos taught at the high school level, >esp. the AP math classes. > Having 2 teenagers in high school (9th and 10th grades) both in the > Ôhigh math classes for their grades, I have been wondering this for a > while. > Why is the order of classes set up like: > 8th grade, Algebra 1 > 9th grade, geometry > 10th grade, Algebra 2 > 11 th grade, trig > 12th grade, calculus > instead of algebra 1, 2 then geometry, trig? > and their friends, these kids forget a lot of algebra between 8th and > 10th grade! The [CapitalThorn]rst month is then spent on review, whereas if it > was just a summer away, it would be far less review. > Is there a teaching-philosophy reason for this? > k wallace Forgetting mathematics you once really understood is a straw man proposed only by those who never really got it in the [CapitalThorn]rst place. Im not saying the order you propose is a poor one, just to watch out for those who propose various excuses to make up for basic de[CapitalThorn]cits. John Lowry Flight Physics === Subject: Re: question for math teachers posting-account=OT2XzgwAAACfOom0T1e3nydGQHkWwi3r of algebra were taught during the freshman and sophomore years of highschool. However, at the time most schools required only 2 years of mathematics to graduate, and it was thought that everyone should see axiomatic mathematics if they were to graduate from high school. At the time a geometry class meant a [CapitalThorn]rm grounding in the theory and application of logic; the class would help students not only in mathematics but whenever they needed to make logical assesments about the world around them. As things stand now, rigorous geometry classes based on Euclid have become a thing of the past, and college-prep-for-everyone curricula have upped the math requirement, so the old reasons dont apply so much any more. On the other hand, if your daughters have a whole lot of trouble with algebra after one year away, rearranging the schedule so that they have two years away and then jump right into their calculus classes would probably be even more painful, because introductory calculus depends much more on algebra than on synthetic geometry. (Most schools actually have a class called analytic geometry, which is an introduction to Cartesian geometry and serves both a refresher in algebra, geometry, and trigonometry and as a demonstration of how they can be integrated. This class is generally taken immediately before the [CapitalThorn]rst calculus class an in fact is generally a more advanced class than the calculus series.) === Subject: Re: question for math teachers > A quick question for anyone whos taught at the high school level, esp. > the AP math classes. > Having 2 teenagers in high school (9th and 10th grades) both in the > Ôhigh math classes for their grades, I have been wondering this for a > while. > Why is the order of classes set up like: > 8th grade, Algebra 1 > 9th grade, geometry > 10th grade, Algebra 2 > 11 th grade, trig > 12th grade, calculus > instead of algebra 1, 2 then geometry, trig? > and their friends, these kids forget a lot of algebra between 8th and > 10th grade! The [CapitalThorn]rst month is then spent on review, whereas if it was > just a summer away, it would be far less review. > Is there a teaching-philosophy reason for this? As I remember from high school, Algebra 2 is concerned with analytic geometry -- the application of alebraic techniques to geometric problems. I remember how cool it it seemed to me that given a geometric problem, instead of trying to guess at trying to [CapitalThorn]nd a clever approach involving in drawing in auxiliary lines and such, you could just put the origin of a coordinate system anyplace it seemed most convenient; write down a couple of equations; and grind out the answer by straightforward calculation. That was a great advance in mathematics, generally attributed to Descartes; and thats why this is the natural way of teaching analytic geometry, AFTER a year of plane geometry. My two cents, Im not a math teacher, just a former student. === Subject: Solid geometry calculation Where can I [CapitalThorn]nd Solid Geometry calculation library? My program needs some functions such as calculation of distance from point to surface. I searched in Matlab, but seemed to [CapitalThorn]nd nothing related there. Ting === Subject: Dif[CapitalThorn]cult Running Time posting-account=7TU2Dw0AAAC7Wj_U6RdHrF_8rRW4SXhl I am writing a paper about the unrestricted partition function p(n). One method of determining the parity of p(n) is to use the following recurrence: 2), 2), 2), where a_i = i(8i +/- 1), b_i = i(8i +/- 3), c_i = i(8i +/- 5), d_i = i(8i +/- 7). Now p(N)=0 for all N<0, so all we need to compute the parity is a few base cases: . . . Im having a little dif[CapitalThorn]culty determining the running time of this algorithm. Thinking of the recursion as a tree, the branching factor is initially O( sqrt(n) ), and the depth of the tree is O( log_4(n) ), but. . . I get stuck from here. One approach might be to make a convenient simpli[CapitalThorn]cation (for the sake of determining an upper bound on running time) and assume that p(4n) = p(n) + p(n) + ... + p(n) for O( sqrt(n) ) copies of p(n). This way, at the mth level on the tree, each node has (1/4^m)^(1/2) children. Or something like that. Any thoughts? === Subject: Parametrized Curves If I have a vector function r(t)=(t^2-7t)i + (8 ln t)j + (sin 4t)k, how can === Subject: Re: Parametrized Curves >If I have a vector function r(t)=(t^2-7t)i + (8 ln t)j + (sin 4t)k, how can correctly often helps lead to a method of solution. --Lynn === Subject: Re: Parametrized Curves >>If I have a vector function r(t)=(t^2-7t)i + (8 ln t)j + (sin 4t)k, how >>can > correctly often helps lead to a method of solution. > --Lynn === Subject: Re: Parametrized Curves >If I have a vector function r(t)=(t^2-7t)i + (8 ln t)j + (sin 4t)k, how can Hint: when is the tangent vector orthogonal to j? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Parametrized Curves r(t)=(t^2-7t)i + (sin 4t)k My understanding puts this vector in the xz plane orthogonal to j. Am I way off here? >>If I have a vector function r(t)=(t^2-7t)i + (8 ln t)j + (sin 4t)k, how >>can > Hint: when is the tangent vector orthogonal to j? > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada === Subject: Re: Parametrized Curves > r(t)=(t^2-7t)i + (sin 4t)k > My understanding puts this vector in the xz plane orthogonal to j. Am I way > off here? >>If I have a vector function r(t)=(t^2-7t)i + (8 ln t)j + (sin 4t)k, how >>can orthogonal to i and to j also. As far as I can tell, tangent lines to your curve are never orthogonal to j. Perhaps you want to have the position vector given by r(t)=(t^2-7t)i + (8 ln t)j + (sin 4t)k to be orthogonal to the vector j? If this is what you want, it will happen only when 8*ln(t) = 0, === Subject: Re: Parametrized Curves >r(t)=(t^2-7t)i + (sin 4t)k >My understanding puts this vector in the xz plane orthogonal to j. Am I way >off here? Yes. Look at Roberts hint again. Do you know how to compute the tangent vector to a trajectory? --Lynn === Subject: Re: Parametrized Curves r(t) is the equation of the tangent vector.....but I am getting lost in terminology. This is a self-taught course so I am trying to understand, but the text is not very good. I can [CapitalThorn]nd r(t) to get the tangent vector, but Roberts hint led me to believe I needed to do something with only the j-component of the vector function. >>r(t)=(t^2-7t)i + (sin 4t)k >>My understanding puts this vector in the xz plane orthogonal to j. Am I >>way >>off here? > Yes. Look at Roberts hint again. Do you know how to compute the > tangent vector to a trajectory? > --Lynn === Subject: Re: Parametrized Curves > r(t) is the equation of the tangent vector.....but I am getting lost in > terminology. This is a self-taught course so I am trying to understand, but > the text is not very good. > I can [CapitalThorn]nd r(t) to get the tangent vector, but Roberts hint led me to > believe I needed to do something with only the j-component of the vector > function. >> >>r(t)=(t^2-7t)i + (sin 4t)k >> >>My understanding puts this vector in the xz plane orthogonal to j. Am I >>way >>off here? > Yes. Look at Roberts hint again. Do you know how to compute the > tangent vector to a trajectory? > --Lynn > The problem cannot be asking about the tangent vector to the curve, because that tangent vector is never perpendicular to j. I suspect that the problem is asking to have the position vector of the point perpendicular to j, which is quite easy, and involves only the j component of the vector. === Subject: Re: Parametrized Curves >The problem cannot be asking about the tangent vector to the curve, >because that tangent vector is never perpendicular to j. >I suspect that the problem is asking to have the position vector of the >point perpendicular to j, which is quite easy, and involves only the j >component of the vector. On the contrary, I suspect that the problem is asking about the tangent parallel to the xz plane which makes no sense, it should say when the the answer is, never. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: max-min problem of a set of points in space? Hi all, Suppose I have a a N-dimension space and I want to place a set of M points into this space. I want to maximize the minimum distance between each pair of the M points among all points. Where shall I place those M points? If the possible locations for the M points are con[CapitalThorn]ned to within a certain region, (I dont know how to specify this region in N-dimensional space... maybe I should say the con[CapitalThorn]ning region is de[CapitalThorn]ned by a function? or I should say the region is de[CapitalThorn]ned by transforming a unit hyper-sphere by some linear or non-linear transformation? ) Anyway, con[CapitalThorn]ned into within this region, how can I place those M-points in order to maximize the minimum distance between each pair of the M points among all points? Any thoughts? === Subject: Re: max-min problem of a set of points in space? >Suppose I have a a N-dimension space and I want to place a set of M points >into this space. >I want to maximize the minimum distance between each pair of the M points >among all points. >Where shall I place those M points? >If the possible locations for the M points are con[CapitalThorn]ned to within a certain >region, (I dont know how to specify this region in N-dimensional space... >maybe I should say the con[CapitalThorn]ning region is de[CapitalThorn]ned by a function? or I >should say the region is de[CapitalThorn]ned by transforming a unit hyper-sphere by some >linear or non-linear transformation? ) There are lots of ways to specify it, but you must specify it to have a well-de[CapitalThorn]ned problem. >Anyway, con[CapitalThorn]ned into within this region, how can I place those M-points in >order to maximize the minimum distance between each pair of the M points >among all points? This is essentially a sphere-packing problem: you want the largest r so that M non-overlapping spheres of radius r can be placed with their centres in your region. It is a very dif[CapitalThorn]cult problem. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: max-min problem of a set of points in space? >>Suppose I have a a N-dimension space and I want to place a set of M points >>into this space. >>I want to maximize the minimum distance between each pair of the M points >>among all points. >>Where shall I place those M points? >>If the possible locations for the M points are con[CapitalThorn]ned to within a >>certain >>region, (I dont know how to specify this region in N-dimensional space... >>maybe I should say the con[CapitalThorn]ning region is de[CapitalThorn]ned by a function? or I >>should say the region is de[CapitalThorn]ned by transforming a unit hyper-sphere by >>some >>linear or non-linear transformation? ) > There are lots of ways to specify it, but you must specify it to have > a well-de[CapitalThorn]ned problem. >>Anyway, con[CapitalThorn]ned into within this region, how can I place those M-points >>in >>order to maximize the minimum distance between each pair of the M points >>among all points? > This is essentially a sphere-packing problem: you want the largest r so > that M non-overlapping spheres of radius r can be placed with their > centres in your region. It is a very dif[CapitalThorn]cult problem. > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada Hi Robert, thank you for your information. Maybe I did not complete understand you, I did not see the relationship between my question and the sphere-packing problem. I did some googling following your pointer, the sphere-packing problem wants to [CapitalThorn]nd way of packing as many as spheres into some crate. But for my problem, the points has no size. You can put a point in a very small area, you dont need to be able to put in a complete sphere. All that is required is to maximize the minimum distance among all pair of points. Suppose the con[CapitalThorn]ning region is a unit hyper-sphere, then you want to put all points on the sphere, and their grativity center to be 0. Am I right? Do you have a proof for this? Suppose the con[CapitalThorn]ning region is a unit hyper-sphere lineary transformed by a matrix T, into an eplitical-sphere, with long axis and short axis. Where should you place those points? If the con[CapitalThorn]ning region is a concave shape, where should you place those points? === Subject: Integrals In Spherical Coordinates This is homework so I dont want the answer....just some hints. Let W be the solid bounded above by x^2+y^2+z^2=9 and below by phi=pi/3. Calculate the mass of W if the density at each point is directly proportional to the distance above the xy plane. As I have it drawn, it is an upward cone about the origin with a slanted plane of z = -x - y +3 for a top. I am having a little dif[CapitalThorn]culty setting up the triple integral as well as the density function. Here is what I have so far: 2 pi pi/6 ??? / / / | | | ??? p^2 sin phi d-rho d-phi d-theta / / / 0 0 0 Anyone got any pointers on this one? === Subject: Re: Integrals In Spherical Coordinates >This is homework so I dont want the answer....just some hints. >Let W be the solid bounded above by x^2+y^2+z^2=9 and below by phi=pi/3. >Calculate the mass of W if the density at each point is directly >proportional to the distance above the xy plane. >As I have it drawn, it is an upward cone about the origin with a slanted >plane of z = -x - y +3 for a top. I am having a little dif[CapitalThorn]culty setting >up the triple integral as well as the density function. Here is what I have >so far: >2 pi pi/6 ??? > / / / > | | | ??? p^2 sin phi d-rho d-phi d-theta >/ / / >0 0 0 >Anyone got any pointers on this one? You have the rectangular coordinate equation for the top plane; change it to spherical coordinates and solve for rho on the outer surface to get: rho-on-the-top = function of phi and theta, for your inside upper limit. Your density ??? = delta = z, change to spherical coordinates too. Also think about your middle upper limit again. --Lynn === Subject: Re: Integrals In Spherical Coordinates guessing it is supposed to be pi/3. >>This is homework so I dont want the answer....just some hints. >>Let W be the solid bounded above by x^2+y^2+z^2=9 and below by phi=pi/3. >>Calculate the mass of W if the density at each point is directly >>proportional to the distance above the xy plane. >>As I have it drawn, it is an upward cone about the origin with a slanted >>plane of z = -x - y +3 for a top. I am having a little dif[CapitalThorn]culty setting >>up the triple integral as well as the density function. Here is what I >>have >>so far: >>2 pi pi/6 ??? >> / / / >> | | | ??? p^2 sin phi d-rho d-phi d-theta >>/ / / >>0 0 0 >>Anyone got any pointers on this one? > You have the rectangular coordinate equation for the top plane; change > it to spherical coordinates and solve for rho on the outer surface > to get: > rho-on-the-top = function of phi and theta, for your inside upper > limit. > Your density ??? = delta = z, change to spherical coordinates too. > Also think about your middle upper limit again. > --Lynn === Subject: Q on inner product in reali[CapitalThorn]ed complex space Hi Id appreciate if anyone can explain and elaborate on the following. The n-dimensional complex vector space V has an inner product de[CapitalThorn]ned on it. Through Ôreali[CapitalThorn]cation or Ôdecomplexi[CapitalThorn]cation a 2n-dimensional real vector space V_R is created with complex structure j, ie (V_R,j) through the usual correspondence iv <-> j(v). Does the inner product on V induce an inner product on (V_R,j) - if so how? And if so then in particular if is symmetric, skew-symmetric or Hermitian what is the corresponding inner product in (V_R,j)? Ron Jones === Subject: Re: Q on inner product in reali[CapitalThorn]ed complex space >Id appreciate if anyone can explain and elaborate on the following. >The n-dimensional complex vector space V has an inner product de[CapitalThorn]ned >on it. Through Ôreali[CapitalThorn]cation or Ôdecomplexi[CapitalThorn]cation a 2n-dimensional real >vector space V_R is created with complex structure j, ie (V_R,j) through the >usual correspondence iv <-> j(v). Does the inner product on V induce an >inner product on (V_R,j) - if so how? And if so then in particular if >is symmetric, skew-symmetric or Hermitian what is the corresponding >inner product in (V_R,j)? De[CapitalThorn]ne (u|v) = Re . If is a Hermitian inner product, then (u|v) is a (symmetric) real inner product. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Cantors diagonal proof wrong? >But, is there some way to use language where we cross over from >very-unlikely, to ßat out impossible? I think there might be. Im trying >to understand if that point exists and how to describe it. >Im trying to understand if some [CapitalThorn]elds of math might have wondered off >into the ßat out impossible land. n spatial dimensions can be described in mathematics, yet theres reason to believe only 3 spatial dimensions may be possible. There we seem to have math (over 3D space) with no analog in nature ßowing smoothly from math with a natural analog (the 3Ds of space). >And if they have, what it means for >those [CapitalThorn]elds of reason and how they relate to the [CapitalThorn]elds of reason which >have not left the land of the possible. Seamlessly. If there was something distinct about the subset of mathematics with analogs in the natural world, then you could [CapitalThorn]nd and know about nature a priori by investigating mathematics, but thats not the case. The natural world is only known by empirical means, ie, by observation. We can only know if some subset of math has an analog in nature a posteriori. num num === Subject: Re: Cantors diagonal proof wrong? Discussion, linux) > I do not know all the other proofs and it will take me some time to > learn their language, and come up with counter examples as I have > done here to show their weakness. And to be honest, I dont have > enough interest in math to spend the time it would take to clean up > the mess that I believe has been created in the past 100 or so > years. The world sighs at the missed opportunity. -- Thats all the legacy I ever wanted, to have people remember me like a shooting star streaking across their Life sky, illuminating, for just one moment, unparalleled beauty unique to itself. -- Weblogs are a particularly humble medium, unique to themselves. === Subject: Re: Cantors diagonal proof wrong? posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs I didnt wade through this whole thread, but I saw several times that you claimed this proof had not been addressed. > Lets create a table of integers like this: > ...000000 > ...000001 > ...000002 > ...000010 > ...000123 > Its just a normal list of integers, but instead of following the normal > convention of leaving off the leading zeros (which we all know are implied > even if we dont write them) I include them in that table. The positive integers are de[CapitalThorn]ned by the Peano axioms. This precludes the possibility of any integers with in[CapitalThorn]nitely many digits. > So lets use Cantors logic on this table and see if we can construct a > number which is not in the table. We take the numbers from the diagonal, > and construct the number ...111111 just like we did above. Well, we can construct the string. And what youve done is prove that the number of strings with countably many characters is uncountable. But we already knew that. What you havent done is construct a new integer. Thats not an integer. It does not arise from the Peano axioms. It is not [CapitalThorn]nite. It is not a member of N. I do know that people have been trying to tell you this, but you havent seemed to hear them. - Randy === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > I didnt wade through this whole thread, but I saw several times > that you claimed this proof had not been addressed. > I do know that people have been trying to tell you this, but > you havent seemed to hear them. > - Randy Yeah, I heard it. It just had to be repeated a few times before I understand the point of what they were trying to say. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~( 5eZ41to5f%E@ELIi $t^ VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > I didnt wade through this whole thread, but I saw several times > that you claimed this proof had not been addressed. >> Lets create a table of integers like this: >> ...000000 >> ...000001 >> ...000002 >> ...000010 >> ...000123 >> Its just a normal list of integers, but instead of following the >> normal convention of leaving off the leading zeros (which we all >> know are implied even if we dont write them) I include them in >> that table. > The positive integers are de[CapitalThorn]ned by the Peano axioms. Not really. They Peano axioms just de[CapitalThorn]ne the natural numbers, and the natural numbers not even have addition de[CapitalThorn]ned. > This precludes the possibility of any integers with in[CapitalThorn]nitely many > digits. This is patent nonsense. The number of digits is a pure artifact of some chosen representation, and most certainly you can choose to place an in[CapitalThorn]nitude of leading zeros in the common representation without changing the value. It is just that something which does _not_ ultimately (after a [CapitalThorn]nite number of digits) has only 0 remaining, does not describe a natural. So he is free to place leading zeros. >> So lets use Cantors logic on this table and see if we can >> construct a number which is not in the table. We take the numbers >> from the diagonal, and construct the number ...111111 just like we >> did above. > Well, we can construct the string. And what youve done is prove > that the number of strings with countably many characters is > uncountable. But we already knew that. More or less. > What you havent done is construct a new integer. Thats not an > integer. It does not arise from the Peano axioms. It is not > [CapitalThorn]nite. It is not a member of N. Quite so. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cantors diagonal proof wrong? >> I didnt wade through this whole thread, but I saw several times >> that you claimed this proof had not been addressed. > Lets create a table of integers like this: > ...000000 > ...000001 > ...000002 > ...000010 > ...000123 > Its just a normal list of integers, but instead of following the > normal convention of leaving off the leading zeros (which we all > know are implied even if we dont write them) I include them in > that table. >> The positive integers are de[CapitalThorn]ned by the Peano axioms. >Not really. They Peano axioms just de[CapitalThorn]ne the natural numbers, and >the natural numbers not even have addition de[CapitalThorn]ned. >> This precludes the possibility of any integers with in[CapitalThorn]nitely many >> digits. >This is patent nonsense. The number of digits is a pure artifact of >some chosen representation, and most certainly you can choose to place >an in[CapitalThorn]nitude of leading zeros in the common representation without >changing the value. It is just that something which does _not_ >ultimately (after a [CapitalThorn]nite number of digits) has only 0 remaining, >does not describe a natural. >So he is free to place leading zeros. > So lets use Cantors logic on this table and see if we can > construct a number which is not in the table. We take the numbers > from the diagonal, and construct the number ...111111 just like we > did above. >> Well, we can construct the string. And what youve done is prove >> that the number of strings with countably many characters is >> uncountable. But we already knew that. >More or less. >> What you havent done is construct a new integer. Thats not an >> integer. It does not arise from the Peano axioms. It is not >> [CapitalThorn]nite. It is not a member of N. >Quite so. Please tell me: Where do you take the time to answer all these questions? Im trying to read this thread, meaning that I lok at every,say, 100th post. How do you do keep up with all this stuff and still are able to answer? Matthias K.8ay -- www.kcc.ch === Subject: Re: Cantors diagonal proof wrong? > Has any one else put forth this same argument (or others) that Cantors > proof is invalid? Hi Curt, at [CapitalThorn]rst let me apologize me for my bad English. I try my best. Several months ago I had found a similar method like yours to map the integers to the rational numbers. Than I try to [CapitalThorn]x the ßaw in Cantors diagonal proof and start the thread: Hat Cantor doch geirrt? in de.sci.mathematik in German language. I had not publish the method of mirroring the numbers at decimal point (how I called the method) so as far as I know, you are the [CapitalThorn]rst one who published it. At [CapitalThorn]rst, when I found the method, I had thought it is a counter-proof to Cantor like you do. In the meantime I think this method supports Cantor. If we accept the existence of irrational numbers with in[CapitalThorn]nit decimal places and we also accept that integers had only [CapitalThorn]nite [CapitalThorn]gures, the mirroring denumerables only the subset of rational numbers of the set of real numbers. But at this point I see a important ßaw in the set theorie. We talk of the set of real numbers. Is this right? I think not. The de[CapitalThorn]nition of sets needs to have objects as elements of the set. Objects must be individually, distinguishable, discrete. But the reals build the continuum. They seems to us to be discrete, like sqrt(2). But can we distinguish sqrt(2) from the in[CapitalThorn]nite real numbers which lies in an epsilon-surrounding of sqrt(2)? No, we cant. So in my opinion there exists exact two kinds of in[CapitalThorn]nity: The in[CapitalThorn]nity of objects which is denumerable and the in[CapitalThorn]nity of the continuum which is corresponding to the real numbers. And thats it. Beside this the diagonal proof of Cantor has very much points which are critical. Albrecht Storz === Subject: Re: Cantors diagonal proof wrong? at 02:09 PM, albstorz@gmx.de (albrecht) said: >Objects must be individually, distinguishable, discrete. Discrete is not the same as individually distiguishable. >They seems to us to be discrete, like sqrt(2). Discrete and continuous are properties of sets, not of individual numbers. >But can we distinguish sqrt(2) from the in[CapitalThorn]nite real numbers which >lies in an epsilon-surrounding of sqrt(2)? No, we cant. Of course we can. Only one of them is sqrt(2). -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantors diagonal proof wrong? > at 02:09 PM, albstorz@gmx.de (albrecht) said: >>But can we distinguish sqrt(2) from the in[CapitalThorn]nite real numbers which >>lies in an epsilon-surrounding of sqrt(2)? No, we cant. > Of course we can. Only one of them is sqrt(2). Distinguish and distinguish makes two. You can distinguish sqrt(2) in a mathematical sense, but you cant do the same when you are measuring a certain edge of a certain rectangular triangle with a non-ideal ruler. Albrecht is referring to the latter, obviously. Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? >> at 02:09 PM, albstorz@gmx.de (albrecht) said: >But can we distinguish sqrt(2) from the in[CapitalThorn]nite real numbers which >lies in an epsilon-surrounding of sqrt(2)? No, we cant. >> Of course we can. Only one of them is sqrt(2). > Distinguish and distinguish makes two. You can distinguish sqrt(2) in > a mathematical sense, but you cant do the same when you are measuring > a certain edge of a certain rectangular triangle with a non-ideal ruler. Every one of those non-sqrt(2) real numbers can be distinguished from the actual sqrt(2) with a non-ideal ruler. Some of them require better rulers than others. But none requires a perfect ruler. John Briggs === Subject: Re: Cantors diagonal proof wrong? >>Distinguish and distinguish makes two. You can distinguish sqrt(2) in >>a mathematical sense, but you cant do the same when you are measuring >>a certain edge of a certain rectangular triangle with a non-ideal ruler. > Every one of those non-sqrt(2) real numbers can be distinguished from > the actual sqrt(2) with a non-ideal ruler. > Some of them require better rulers than others. But none requires a > perfect ruler. Physics forbids you to make a ruler which can measure distances smaller than the Planck length h/(m_p.c) where m_p = sqrt(h.c/G) . Here h = Planks constant, c = speed of light, G = gravitation constant. Therefore an _absolute lower bound_ for the accuracy of your ruler is: 1.6 x 10^(-36) m , approximately. And nothing can be done about it. Absolute rigour is a phantom. Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? >>Distinguish and distinguish makes two. You can distinguish sqrt(2) in >>a mathematical sense, but you cant do the same when you are measuring >>a certain edge of a certain rectangular triangle with a non-ideal ruler. > Every one of those non-sqrt(2) real numbers can be distinguished from > the actual sqrt(2) with a non-ideal ruler. > Some of them require better rulers than others. But none requires a > perfect ruler. > Physics forbids you to make a ruler which can measure distances smaller > than the Planck length h/(m_p.c) where m_p = sqrt(h.c/G) . > Here h = Planks constant, c = speed of light, G = gravitation constant. > Therefore an _absolute lower bound_ for the accuracy of your ruler is: > 1.6 x 10^(-36) m , approximately. And nothing can be done about it. > Absolute rigour is a phantom. So continuity is an illusion, and everything is really discrete mathematics. ;-) Paul -- Hanging on in quiet desperation is the English way. The time is gone, the song is over. Thought Id something more to say. === Subject: Re: Cantors diagonal proof wrong? > > >>Distinguish and distinguish makes two. You can distinguish sqrt(2) in >>a mathematical sense, but you cant do the same when you are measuring >>a certain edge of a certain rectangular triangle with a non-ideal ruler. > Every one of those non-sqrt(2) real numbers can be distinguished from > the actual sqrt(2) with a non-ideal ruler. > Some of them require better rulers than others. But none requires a > perfect ruler. > > Physics forbids you to make a ruler which can measure distances smaller > than the Planck length h/(m_p.c) where m_p = sqrt(h.c/G) . > Here h = Planks constant, c = speed of light, G = gravitation constant. > > Therefore an _absolute lower bound_ for the accuracy of your ruler is: > 1.6 x 10^(-36) m , approximately. And nothing can be done about it. > > Absolute rigour is a phantom. > So continuity is an illusion, and everything is really discrete > mathematics. ;-) And does it mean that for suitably small circles, Pi would really be 3 after all! Phil -- ... one Marine noticed one of the prisoners was still breathing. A Marine can be heard saying on the pool footage provided to Reuters Television: Hes ing faking hes dead. He faking hes ing dead. The Marine then raises his riße and [CapitalThorn]res into the mans head. The pictures are too graphic for us to broadcast, Sites said. === Subject: Re: Cantors diagonal proof wrong? > So continuity is an illusion, and everything is really discrete > mathematics. ;-) Thats not far besides the truth. Continuity is looking at the discrete with your eyes half closed :-) Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > So continuity is an illusion, and everything is really discrete > mathematics. ;-) Actually most of the math used in physics is about contiuous locally compact structures. If we used just [CapitalThorn]nite math it would soon become intractable. Can you say differential equation? Sure you can. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > Absolute rigour is a phantom. > So continuity is an illusion, and everything is really discrete > mathematics. ;-) Id say so - or at least, all continuum below a certain level is noise and not useful information. So we are forced to work in discrete systems above the noise ßoor. The brain works that way. Neurons [CapitalThorn]re or they dont. Its a discrete representation of data. Its not a continuous analog format. The high resolution continuum for the brain is in the time domain, but because its all noise below a given precision, its effectively a discrete system which limits the ability to other neurons to detect time variations below the resolution of the system. However, as is well known, that doesnt stop us from thinking about an ideal world where things are continuous, and that doesnt stop us from [CapitalThorn]nding applications in this world for those continuous models. High resolution discrete systems are suf[CapitalThorn]ciently close to continuous to make the models very useful. And in many cases, the continuous models are far simpler than the high resolution discrete models. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? ... > However, as is well known, that doesnt stop us from thinking about an > ideal world where things are continuous, and that doesnt stop us from > [CapitalThorn]nding applications in this world for those continuous models. High > resolution discrete systems are suf[CapitalThorn]ciently close to continuous to make > the models very useful. And in many cases, the continuous models are far > simpler than the high resolution discrete models. Discrete models of the reals only work because you use discretisation on a continuous model. There is a whole branch of mathematics devoted to this subject (numerical mathematics). In continuous models you can prove quite a bit because in that model the reals form a metric space. This is not the case in discrete models. Numerical methematics checks in what ways algorithms (that would work perfectly in the continuous model) can be applied in a discrete model, and what problems we can encounter. For instance, how do you prove convergence of an algorithm directly in a discrete model, when that is not even a metric space? -- 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: Cantors diagonal proof wrong? > So in my opinion there exists exact two kinds of in[CapitalThorn]nity: > The in[CapitalThorn]nity of objects which is denumerable > and the in[CapitalThorn]nity of the continuum which is corresponding to the real > numbers. > And thats it. Yep. And it has a model which can be idealized from the real world. Imagine an ideal computer with N -> oo (read 32) bits for the integers and M+N -> oo (read 64) bits for the ßoating point numbers. Assume that N bits are reserved for the mantissa and M bits for the exponent of the latter. Then it is obvious that the cardinality of the continuum (= R = ßoating point) is 2^M times the cardinality of the discrete (= Q = integers). Now do M,N -> oo and a proof of Cantors famous Continuum Hypothesis has been found herewith: there is no cardinality between that of the naturals and the continuum (for the simple reason that only integers and ßoating point numbers exist in our machine). Moreover, the cardinality of the continuum is between Aleph_0 and 2^Aleph_0, if we only assume that the number of bits for the exponent is less than the number of bits for the mantissa, for ever and ever. Have fun :-) Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > Imagine an ideal computer with N -> oo (read 32) bits for the > integers and M+N -> oo (read 64) bits for the ßoating point > numbers. Assume that N bits are reserved for the mantissa and > M bits for the exponent of the latter. Then it is obvious that > the cardinality of the continuum (= R = ßoating point) is 2^M > times the cardinality of the discrete (= Q = integers). You have now a model where the triangular inequality does not hold. That is, in your [CapitalThorn]nite model it is always possible to [CapitalThorn]nd a, b and c such that |c - b| + |b - a| < |c - a|, regardless the rounding method used. But on the other hand, that is a basic property used in the proofs of many theorems. So in your [CapitalThorn]nite model those theorems are false, or unproven. The triangular inequality only holds in the idealised world with all reals. -- 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: Cantors diagonal proof wrong? > You have now a model where the triangular inequality does not hold. Uhm, allright, something had to go wrong here ;-) > The triangular inequality only holds in the idealised world with all > reals. Hmm, thats an important thing to keep in mind. Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? >> You have now a model where the triangular inequality does not hold. > Uhm, allright, something had to go wrong here ;-) >> The triangular inequality only holds in the idealised world with all >> reals. > Hmm, thats an important thing to keep in mind. And then I added Iff this is true and deleted it again. Wrong! So let me [CapitalThorn]nish this post now in the proper way. The triangle inequality IS valid in an idealized world with the whole numbers alone. And it IS valid in the rationals without the irrationals. So your statement that all reals are needed is false. Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > >> You have now a model where the triangular inequality does not hold. > > Uhm, allright, something had to go wrong here ;-) > >> The triangular inequality only holds in the idealised world with all >> reals. > > Hmm, thats an important thing to keep in mind. > And then I added Iff this is true and deleted it again. Wrong! So let > me [CapitalThorn]nish this post now in the proper way. > The triangle inequality IS valid in an idealized world with the whole > numbers alone. And it IS valid in the rationals without the irrationals. > So your statement that all reals are needed is false. inequality does not hold in a ßoating-point model of the real 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: Cantors diagonal proof wrong? >> So in my opinion there exists exact two kinds of in[CapitalThorn]nity: >> The in[CapitalThorn]nity of objects which is denumerable >> and the in[CapitalThorn]nity of the continuum which is corresponding to the real >> numbers. >> And thats it. > Yep. > And it has a model which can be idealized from the real world. > Imagine an ideal computer with N -> oo (read 32) bits for the > integers and M+N -> oo (read 64) bits for the ßoating point > numbers. Assume that N bits are reserved for the mantissa and > M bits for the exponent of the latter. Then it is obvious that > the cardinality of the continuum (= R = ßoating point) is 2^M > times the cardinality of the discrete (= Q = integers). Or 2^(M-1) if you forget to normalize your mantissas and utilize the hidden bit. > Now do > M,N -> oo More detail needed here. If you take the in[CapitalThorn]nite union of all positive integers expressible in N bits as N takes on all values from 0 (inclusive) to oo (exclusive), you get the ordinary natural numbers. If you take the in[CapitalThorn]nite union of all ßoating point values expressible with an N bit mantissa and an M bit exponent as N and M range from 0 (inclusive) to oo (exclusive), you get a subset of the rational numbers -- the rational numbers that can be expressed with a denominator that is a power of two. Those two sets have identical cardinality. If you close the latter set by including those values that are approached with arbitrary accuracy by an unlimited precision/range binary ßoating point value then you wind up with the [positive] reals and a useful set of arithmetic operations. If you close the former set by including integers with in[CapitalThorn]nite binary expansions then you run into some arithmetic issues and wind up with the 2-adics. Again, these two sets have identical cardinality. > and a proof of Cantors famous Continuum Hypothesis > has been found herewith: there is no cardinality between that > of the naturals and the continuum (for the simple reason that > only integers and ßoating point numbers exist in our machine). And subsets of integers and subsets of ßoating point numbers. Some of which might plausibly have cardinality different from the set of all integers and from the set of all ßoats. But thats assuming that your machine does in[CapitalThorn]nite precision rather than unlimited precision. If youre on unlimited precision hardware then youre restricted to countable sets. John Briggs === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > Has any one else put forth this same argument (or others) that Cantors > proof is invalid? > Hi Curt, > at [CapitalThorn]rst let me apologize me for my bad English. Let me apologize for not knowing any German. :) > I try my best. > Several months ago I had found a similar method like yours to map the > integers to the rational numbers. Than I try to [CapitalThorn]x the ßaw in > Cantors diagonal proof and start the thread: Hat Cantor doch > geirrt? in > de.sci.mathematik in German language. > I had not publish the method of mirroring the numbers at decimal > point (how I called the method) so as far as I know, you are the > [CapitalThorn]rst one who published it. > At [CapitalThorn]rst, when I found the method, I had thought it is a counter-proof > to Cantor like you do. In the meantime I think this method supports > Cantor. If we accept the existence of irrational numbers with in[CapitalThorn]nit > decimal places and we also accept that integers had only [CapitalThorn]nite > [CapitalThorn]gures, the mirroring denumerables only the subset of rational > numbers of the set of real numbers. I dont actually accept the existence of in[CapitalThorn]nite length anything including irrational numbers. I only accept that the descriptions for the generation of in[CapitalThorn]nite length irrational numbers exist. That is, we have a description for what the square root of two is. We dont actually have, the full, square root of two, written down anywhere. Likewise, we dont have the full list of natural numbers written down anywhere. The mirroring the digits at the decimal point algrorithm, is a description for generating real values, that will generate the same numbers, which any of the descriptions anyone can provide for any irrational (converted to decimal), will also generate. So to me, my description will, through obvious inspection, produce a one to one mapping to any decimal description of an irrational numbers that I know about. But, what I do accept, is this is not how mathematicians choose to de[CapitalThorn]ne or think about these things. So, before I can understand what they have done, and why they think its valid, I have to learn their language. Theres a huge body of work formalizing how they do think, which I have not yet mastered. What remains a mystery to me is that I still dont see how any formal system could be both consistent, and produce the results which Cantors proof produces. If you completely drop the idea of natrual numebers and reals, and simply work with sets, I dont see how it can be justi[CapitalThorn]ed that all in[CapitalThorn]nite sets are not one-to-one mapable. So there is still, what looks to me, to be some real odd stuff that has been done in the [CapitalThorn]eld of mathematics. I will [CapitalThorn]nd the answer to what they have done and how they did it. > But at this point I see a important ßaw in the set theorie. We talk > of the set of real numbers. Is this right? I think not. The de[CapitalThorn]nition > of sets needs to have objects as elements of the set. Objects must be > individually, distinguishable, discrete. But the reals build the > continuum. They seems to us to be discrete, like sqrt(2). But can we > distinguish sqrt(2) from the in[CapitalThorn]nite real numbers which lies in an > epsilon-surrounding of sqrt(2)? No, we cant. > So in my opinion there exists exact two kinds of in[CapitalThorn]nity: > The in[CapitalThorn]nity of objects which is denumerable > and the in[CapitalThorn]nity of the continuum which is corresponding to the real > numbers. > And thats it. Well, as an intuitive idea, that sounds valid. As a formal proof, that seems to go nowhere. In a formal system, if there is a way to de[CapitalThorn]ne two type of in[CapitalThorn]nite sets, there should likewise be a way to describe an in[CapitalThorn]nite number of different types of in[CapitalThorn]nite sets. The confusing part is the simple fact that we are taught informally to think of numbers in many different ways. But once you decide you want to get formal with it, you must drop all that intuition and pick the ones you want to base your formal system on. And thats clearly been done in mathematics - and Cantors work is an outgrowth of that (as far as I know). -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? >I dont actually accept the existence of in[CapitalThorn]nite length anything >including irrational numbers. Then why are you writing about in[CapitalThorn]nite strings of digits? >I dont actually accept the existence of in[CapitalThorn]nite length anything >including irrational numbers. I only accept that the descriptions >for the generation of in[CapitalThorn]nite length irrational numbers exist. That >is, we have a description for what the square root of two is. We >dont actually have, the full, square root of two, written down >anywhere. Its existence is not dependent on having it written down. >The mirroring the digits at the decimal point algrorithm, is a >description for generating real values, No. Its a description for generating in[CapitalThorn]nite strings of digits that dont represent numbers. >So to me, my description will, through obvious inspection, produce a >one to one mapping to any decimal description of an irrational >numbers that I know about. Thats because youre ignoring the obvious inconsistency in what you >Well, as an intuitive idea, that sounds valid. As a formal proof, >that seems to go nowhere. In a formal system, if there is a way to >de[CapitalThorn]ne two type of in[CapitalThorn]nite sets, there should likewise be a way to >describe an in[CapitalThorn]nite number of different types of in[CapitalThorn]nite sets. Correct; there is a hierarchy of cardinalities. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantors diagonal proof wrong? > I dont actually accept the existence of in[CapitalThorn]nite length anything including > irrational numbers. I only accept that the descriptions for the generation > of in[CapitalThorn]nite length irrational numbers exist. That is, we have a > description for what the square root of two is. We dont actually have, > the full, square root of two, written down anywhere. The length corresponding to the square root of two is easility constructble and laid out before your eyes. Construct a unit square and draw a line segement between a pair of diagonally opposed corners of the squre. There it is, the square root of two. Right in front of your eyes. There is also a [CapitalThorn]nite algorithm which will yield the n-th digit of the decimal expansion for the square root of two. > Likewise, we dont have the full list of natural numbers written down > anywhere. Not enough paper or disk space. But we can generate as many as we need. There is no limit. > The mirroring the digits at the decimal point algrorithm, is a > description for generating real values, that will generate the same > numbers, which any of the descriptions anyone can provide for any > irrational (converted to decimal), will also generate. > So to me, my description will, through obvious inspection, produce a one to > one mapping to any decimal description of an irrational numbers that I know > about. > But, what I do accept, is this is not how mathematicians choose to de[CapitalThorn]ne > or think about these things. So, before I can understand what they have > done, and why they think its valid, I have to learn their language. > Theres a huge body of work formalizing how they do think, which I have not > yet mastered. And you probably never will. Yoda says: Hold not your breath young Curt, else purple turn you will. > What remains a mystery to me is that I still dont see how any formal > system could be both consistent, and produce the results which Cantors > proof produces. That is admirably correct. You dont see. Why dont you try a line of thought where you CAN see (i.e. understans). Bob Kolker === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > I dont actually accept the existence of in[CapitalThorn]nite length anything > including irrational numbers. I only accept that the descriptions for > the generation of in[CapitalThorn]nite length irrational numbers exist. That is, > we have a description for what the square root of two is. We dont > actually have, the full, square root of two, written down anywhere. > The length corresponding to the square root of two is easility > constructble and laid out before your eyes. Construct a unit square and > draw a line segement between a pair of diagonally opposed corners of the > squre. There it is, the square root of two. Right in front of your eyes. But any physical drawing you do will only, once again, be a representation of the concept. The length of the line you draw will never be exactly the square root of two. It is really no different than saying 1.414 is the square root of two because most lines you draw will have less precision than that. > There is also a [CapitalThorn]nite algorithm which will yield the n-th digit of the > decimal expansion for the square root of two. Cool. How much space does it take to run for large values of N? Does the space increase with the size of N at some rate more than log2(N)? > That is admirably correct. You dont see. > Why dont you try a line of thought where you CAN see (i.e. understans). For the same reason you wont [CapitalThorn]nd me spending much time playing tic-tac-toe. :) > Bob Kolker -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? >> But at this point I see a important ßaw in the set theorie. We talk >> of the set of real numbers. Is this right? I think not. The de[CapitalThorn]nition >> of sets needs to have objects as elements of the set. Objects must be >> individually, distinguishable, discrete. But the reals build the >> continuum. They seems to us to be discrete, like sqrt(2). But can we >> distinguish sqrt(2) from the in[CapitalThorn]nite real numbers which lies in an >> epsilon-surrounding of sqrt(2)? No, we cant. >> So in my opinion there exists exact two kinds of in[CapitalThorn]nity: >> The in[CapitalThorn]nity of objects which is denumerable >> and the in[CapitalThorn]nity of the continuum which is corresponding to the real >> numbers. >> And thats it. What about the power set of the reals? > Well, as an intuitive idea, that sounds valid. As a formal proof, that > seems to go nowhere. In a formal system, if there is a way to de[CapitalThorn]ne two > type of in[CapitalThorn]nite sets, there should likewise be a way to describe an > in[CapitalThorn]nite number of different types of in[CapitalThorn]nite sets. Correct. Writing N for the naturals, P(X) for the power set of X, and |.| for cardinality of, we have |N| < |P(N)| = |R| < |P(P(N))| < |P(P(P(N)))| < ... which is an in[CapitalThorn]nite hierarchy of in[CapitalThorn]nities. -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > Correct. Writing N for the naturals, P(X) for the power set of X, and > |.| for cardinality of, we have > |N| < |P(N)| = |R| < |P(P(N))| < |P(P(P(N)))| < ... > which is an in[CapitalThorn]nite hierarchy of in[CapitalThorn]nities. Speeking of which, the notion that |P(N)| = |R| has been put forth many times in this thead and in the links Ive recently read, but I dont know where or how that relation was proven. Can someone point me to that or explain it if its short? -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? >> Correct. Writing N for the naturals, P(X) for the power set of X, and >> |.| for cardinality of, we have >> |N| < |P(N)| = |R| < |P(P(N))| < |P(P(P(N)))| < ... >> which is an in[CapitalThorn]nite hierarchy of in[CapitalThorn]nities. > Speeking of which, the notion that |P(N)| = |R| has been put forth many > times in this thead and in the links Ive recently read, but I dont know > where or how that relation was proven. Can someone point me to that or > explain it if its short? Its pretty trivial. First, lets establish a bijection between the set of all subsets of the naturals -- that is P(N) -- and the set of all binary sequences. So, take an element of P(N) and call it SubSet We need to come up with a corresponding binary sequence and call it Seq Now, for each natural number n: If n is in SubSet then bit n in Seq is a 1 If n is not in SubSet then bit n in Seq is a 0 Clearly, this is a bijection. For each subset, there is exactly one corresponding binary sequence. And for each binary sequence there is exactly one corresponding subset. Use that binary sequence as a binary expansion with an implied decimal point to the left of the [CapitalThorn]rst digit. If we ignore (as it is legitimate to do) the countable set of duplicate representations, the cardinality of the set of binary expansions is the same as the cardinality of the set of real numbers in the range from 0 to 1. And that cardinality is, in turn, the same as the cardinality of the set of all real numbers. So |(P(N))| = |binary sequences| = |reals in [0,1]| = |R| John Briggs === Subject: Re: Cantors diagonal proof wrong? >Speeking of which, the notion that |P(N)| = |R| has been put forth many >times in this thead and in the links Ive recently read, but I dont know >where or how that relation was proven. Can someone point me to that or >explain it if its short? Let f: P(N) -> [0,1] be de[CapitalThorn]ned as: f(A) = Sum_k=0^oo[ 2^(-k-1) * X_k(A) ] where { 1, if k in A X_k(A) = { { 0, if k not in A Then f is a bijection. Let g: [0,1]->R s.t. g is a bijection, for example g(x) = tan(Pi(x-1/2)). Then g(f(A)) is a bijection P(N) -> R => |P(N)| = |R|. -- Im not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: Re: Cantors diagonal proof wrong? >>Speeking of which, the notion that |P(N)| = |R| has been put forth many >>times in this thead and in the links Ive recently read, but I dont know >>where or how that relation was proven. Can someone point me to that or >>explain it if its short? > Let f: P(N) -> [0,1] be de[CapitalThorn]ned as: > f(A) = Sum_k=0^oo[ 2^(-k-1) * X_k(A) ] > where > { 1, if k in A > X_k(A) = { > { 0, if k not in A > Then f is a bijection. Let g: [0,1]->R s.t. g is a bijection, for > example g(x) = tan(Pi(x-1/2)). Then g(f(A)) is a bijection P(N) -> R >=> |P(N)| = |R|. As David Ullrich points out, its not a bijection. The easiest way to show that a bijection exists is to use the Cantor-Bernstein theorem. If you can exhibit an injection f: P(N) -> [0,1] and an injection g: [0,1] -> P(N), then the theorem guarantees that a bijection exists. For an injection f: P(N) -> [0,1], a slight modi[CapitalThorn]cation of the scheme above will do. Let f(A) = Sum_{k in A} 2/3^(k-1). This is a bijection between P(N) and the Cantor set, which is a subset of [0,1]. For the reverse mapping, let x be given and suppose we have a binary representation x = sum_{k=1}^oo 2^(-k+1), where for de[CapitalThorn]niteness we choose the representation that does not end with in[CapitalThorn]nitely many 0s. Then de[CapitalThorn]ne g(x) = { k : the kth binary digit of x is a 1 }. Then g: [0,1] -> P(N) is an injection, as required. -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Cantors diagonal proof wrong? >>Speeking of which, the notion that |P(N)| = |R| has been put forth many >>times in this thead and in the links Ive recently read, but I dont know >>where or how that relation was proven. Can someone point me to that or >>explain it if its short? >Let f: P(N) -> [0,1] be de[CapitalThorn]ned as: >f(A) = Sum_k=0^oo[ 2^(-k-1) * X_k(A) ] >where > { 1, if k in A >X_k(A) = { > { 0, if k not in A >Then f is a bijection. Alas its not _quite_ a bijection, for example f({1}) = f({2,3,4,5,...}). >Let g: [0,1]->R s.t. g is a bijection, for >example g(x) = tan(Pi(x-1/2)). Then g(f(A)) is a bijection P(N) -> R >=> |P(N)| = |R|. ************************ David C. Ullrich === Subject: Re: Cantors diagonal proof wrong? > > Has any one else put forth this same argument (or others) that Cantors > proof is invalid? > Hi Curt, > at [CapitalThorn]rst let me apologize me for my bad English. > Let me apologize for not knowing any German. :) > I try my best. > Several months ago I had found a similar method like yours to map the > integers to the rational numbers. Than I try to [CapitalThorn]x the ßaw in > Cantors diagonal proof and start the thread: Hat Cantor doch > geirrt? in > de.sci.mathematik in German language. > I had not publish the method of mirroring the numbers at decimal > point (how I called the method) so as far as I know, you are the > [CapitalThorn]rst one who published it. > At [CapitalThorn]rst, when I found the method, I had thought it is a counter-proof > to Cantor like you do. In the meantime I think this method supports > Cantor. If we accept the existence of irrational numbers with in[CapitalThorn]nit > decimal places and we also accept that integers had only [CapitalThorn]nite > [CapitalThorn]gures, the mirroring denumerables only the subset of rational > numbers of the set of real numbers. > I dont actually accept the existence of in[CapitalThorn]nite length anything including > irrational numbers. I only accept that the descriptions for the generation > of in[CapitalThorn]nite length irrational numbers exist. That is, we have a > description for what the square root of two is. We dont actually have, > the full, square root of two, written down anywhere. > Likewise, we dont have the full list of natural numbers written down > anywhere. That assumes, wrongly, that the only legitimate way to have a number is in decimal, or possibly other base, notation. In fact, rational numbers are better expressed as fractions, and square roots of positive non-square integers as radical expressions. > The mirroring the digits at the decimal point algrorithm, is a > description for generating real values, that will generate the same > numbers, which any of the descriptions anyone can provide for any > irrational (converted to decimal), will also generate. > So to me, my description will, through obvious inspection, produce a one to > one mapping to any decimal description of an irrational numbers that I know > about. > But, what I do accept, is this is not how mathematicians choose to de[CapitalThorn]ne > or think about these things. So, before I can understand what they have > done, and why they think its valid, I have to learn their language. > Theres a huge body of work formalizing how they do think, which I have not > yet mastered. > What remains a mystery to me is that I still dont see how any formal > system could be both consistent, and produce the results which Cantors > proof produces. If you completely drop the idea of natrual numebers and > reals, and simply work with sets, I dont see how it can be justi[CapitalThorn]ed that > all in[CapitalThorn]nite sets are not one-to-one mapable. So there is still, what > looks to me, to be some real odd stuff that has been done in the [CapitalThorn]eld of > mathematics. > I will [CapitalThorn]nd the answer to what they have done and how they did it. > But at this point I see a important ßaw in the set theorie. We talk > of the set of real numbers. Is this right? I think not. The de[CapitalThorn]nition > of sets needs to have objects as elements of the set. Objects must be > individually, distinguishable, discrete. But the reals build the > continuum. They seems to us to be discrete, like sqrt(2). But can we > distinguish sqrt(2) from the in[CapitalThorn]nite real numbers which lies in an > epsilon-surrounding of sqrt(2)? No, we cant. > So in my opinion there exists exact two kinds of in[CapitalThorn]nity: > The in[CapitalThorn]nity of objects which is denumerable > and the in[CapitalThorn]nity of the continuum which is corresponding to the real > numbers. > And thats it. > Well, as an intuitive idea, that sounds valid. As a formal proof, that > seems to go nowhere. In a formal system, if there is a way to de[CapitalThorn]ne two > type of in[CapitalThorn]nite sets, there should likewise be a way to describe an > in[CapitalThorn]nite number of different types of in[CapitalThorn]nite sets. > The confusing part is the simple fact that we are taught informally to > think of numbers in many different ways. But once you decide you want to > get formal with it, you must drop all that intuition and pick the ones you > want to base your formal system on. And thats clearly been done in > mathematics - and Cantors work is an outgrowth of that (as far as I know). === Subject: Re: Cantors diagonal proof wrong? >> Likewise, we dont have the full list of natural numbers written down >> anywhere. > That assumes, wrongly, that the only legitimate way to have a number > is in decimal, or possibly other base, notation. In fact, rational > numbers are better expressed as fractions, and square roots of positive > non-square integers as radical expressions. Indeed. We see that Welch is yet another crass formalist, confusing a representation of a number with the number itslef, and like so many of that tribe, fetishizing one particular system of representation, decimal notation. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Cantors diagonal proof wrong? > Indeed. We see that Welch is yet another crass formalist, confusing > a representation of a number with the number itslef, and like > so many of that tribe, fetishizing one particular system of representation, > decimal notation. He goes further than that. He denies in[CapitalThorn]nite anything. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? > I dont actually accept the existence of in[CapitalThorn]nite length anything including > irrational numbers. I only accept that the descriptions for the generation > of in[CapitalThorn]nite length irrational numbers exist. That is, we have a > description for what the square root of two is. We dont actually have, > the full, square root of two, written down anywhere. Does that mean you dont believe in in[CapitalThorn]nite series? > Likewise, we dont have the full list of natural numbers written down > anywhere. Writing things down is not a mathematical concept. Even if you only accept [CapitalThorn]nite numbers such as 1, 2, 3, 4, ... there is some number large enough that you could never write it down in the age of the universe. Does that number exist or not? If not, then whats the largest natural number that exists? Call it E. Does E+1 exist or not? > But, what I do accept, is this is not how mathematicians choose to de[CapitalThorn]ne > or think about these things. So, before I can understand what they have > done, and why they think its valid, I have to learn their language. > Theres a huge body of work formalizing how they do think, which I have not > yet mastered. The information you need is very simple. You need to understand what is meant by the sum of an in[CapitalThorn]nite series such as 1/2 + 1/4 + 1/8 + . . . You need to then understand what it means, mathematically, for such a series to be said to converge to 1. Once youve got that -- maybe an hour or twos work reviewing a freshman calculus text -- then you de[CapitalThorn]ne a real number such as sqrt(2) as 1 + 4/10 + 1/100 + 4/1000 + ... People have been trying to explain this to you for days. You would rather keep saying how huge is this body of knowledge to be mastered, than just sit down and work through the basics of in[CapitalThorn]nite series in about two hours. > What remains a mystery to me is that I still dont see how any formal > system could be both consistent, and produce the results which Cantors > proof produces. If you completely drop the idea of natrual numebers and > reals, and simply work with sets, I dont see how it can be justi[CapitalThorn]ed that > all in[CapitalThorn]nite sets are not one-to-one mapable. The proof that any set X is strictly smaller than its power set P(X) has been demonstrated to you in this thread several times already. > I will [CapitalThorn]nd the answer to what they have done and how they did it. You are working very hard not to. > But at this point I see a important ßaw in the set theorie. Are you a crank or a serious student of this subject? Get to work, do your homework, come back in a couple of days knowing what it means for an in[CapitalThorn]nite series to convergs. === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > Does that mean you dont believe in in[CapitalThorn]nite series? You know, I just wasted an hour writing a long reply to you. By then you said: > Get to work, do > your homework, come back in a couple of days knowing what it means for > an in[CapitalThorn]nite series to convergs. And I decided it was pointless waste of my time so I deleted it. I mastered the concept of in[CapitalThorn]nite series over 30 years ago when I took calculus in high school. That was before 5 odd years of higher math clases getting a CS degree. I am not here asking questions so I can become a mathematican. Im here searching for answers about things that have nothing to do with math. My search just happened to wander off into some ideas about math, so I can here looking for some expert advice - and Ive gotten a lot of it and Ive been given many good pointers. Suggesting I learn stuff that was trival for me to understand 30 years ago is not some of the most expert advice Ive received here. Suggesting I stop wasting my time was good advice. :) -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > Does that mean you dont believe in in[CapitalThorn]nite series? That is what he says. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? > Curt, whats the point, man? > Ive addressed that to some extend in a previous post now. My interst > comes from my exploring the ideas of AI. Im not trying to [CapitalThorn]x math, Im > trying to understand what has happened. > My interest is in understanding the relationship between what we can do > with language, and what exists in the physical world. My interest is to > try and understand if there is a clear point where we violate some > important principle and end up describing something with language, that can > never exist in our universe. > I can make up a story about a blue book on my desk. There is no blue book > on my desk, so Ive just used language to describe something that does not > exist. However, just because it does not happen to exist does not mean it > is impossible for it to exist. I can describe a blue book on my desk which > is talking to me. Thats something we see in the cartoons all the time. > And as far as we all know, such a thing does not exist in real life > anywhere in the universe. But it could exist for all we know. > But, is there some way to use language where we cross over from > very-unlikely, to ßat out impossible? I think there might be. Im trying > to understand if that point exists and how to describe it. > Im trying to understand if some [CapitalThorn]elds of math might have wondered off > into the ßat out impossible land. And if they have, what it means for > those [CapitalThorn]elds of reason and how they relate to the [CapitalThorn]elds of reason which > have not left the land of the possible. > Anyways, Curt, some people are very attached to their notions ... > Thats key. People use language to justify what they believe. They seldom > if ever, really understand why they believe what they believe, yet, if they > can construct elaborate language to justify it, it makes them feel good, so > they do it. We all work this way. And thats part of the danger. The > language we use to justify everything exists simply because it makes us > feel good. Separating truth, from good feelings is much harder to do > than most people understand because in the end, none of cares as much about > truth as we do about feeling good. (but thats another argument for another > group). > You used a lot of language about math in your post which I do not > understand. I have a lot of work to do before I could discuss those issues > with you. But I did [CapitalThorn]nd your post interesting. Yeah, Skolem is useful for those who dont care to deny that two in[CapitalThorn]nite sets are each in[CapitalThorn]nite, if only as an example to give to others, besides peer-reviewed papers of dubious validity on ArXiv, for example, repository of much of cutting edge modern mathematical physics. While that is so, Cantors results, as mathematical results, demand evaluation. In terms of the naturals and reals, any mapping of the naturals to and from the reals would have to have characteristics thus that Cantors, and the few other quite similar results, about the naturals and reals, do not hold, as each of the integers and reals are in[CapitalThorn]nite. In[CapitalThorn]nite: in-[CapitalThorn]nite, not [CapitalThorn]nite, neverending, without cessation, theres always one more. Casual examination leads to reevaluation of the de[CapitalThorn]nitions of the reals, because standard de[CapitalThorn]nitions of the real numbers are inadequate to express them. That leads to formal expressions. Besides the natural and real numbers, there is the consideration of the set, and its powerset. One reason set theory is deemed suitable for the expression of what is considered the foundations of mathematics is that it is so simple. Where in[CapitalThorn]nite sets are equivalent, and the naive interpretation of the in[CapitalThorn]nite powerset leads to what would be a paradox, or contradiction, which is not allowed, then that leads to basically dual representation, an enlightened interpretation. You might have heard of proper classes, where someone says in ZF, the set of all sets does not exist, there is only the proper class of all sets. Under some naive and correct de[CapitalThorn]nitions of the proper class, there can only be one or none of them. The empty set is the proper class or ur-element, as is in[CapitalThorn]nity. Its a _set_ theory, founded on nothing and everything. Once you have an in[CapitalThorn]nity, you can do a lot of stuff with it. All the in[CapitalThorn]nite ordinals can be considered as ordinals, with Ord, the order type of all ordinals, being that original in[CapitalThorn]nity, one in[CapitalThorn]nite value among all the in[CapitalThorn]nitely many [CapitalThorn]nite positive integers, equal to zero. Maybe Kronecker had a point, there are only integers, all the rest is the work of man, although thats just a quote that survived. Kronecker was also probably abrasive and unfriendly, at least to Cantor, they clashed, whereas I like everyone personally. In a _very simple_ set theory, each ordinal is a set, and each set is an ordinal. The powerset is just the order type, which is just the successor. The function mapping ordinal to successor is f(x)=x+1, and as well the root set of an ornate ordinal is the predecessor. Being able to rationalize that Cantors strong and subtle mathematical results about in[CapitalThorn]nity do not prevent me from considering bijections between any two in[CapitalThorn]nite sets, and to responsibly forget about cardinal numbers, then the point is to immediately think of as many useful things as possible that people, in particular creative forward-thinking mathematicians, can not or would not consider because they are bound by their orthodoxy, because when the dam breaks, they wont be washed away, and I have a head start in developing mathematical theories supplanting them. Its a wager, as it were. One thing I named that is just ancient as hell is the natural/unit equivalency function. Im not a mathematical historian, I am quite con[CapitalThorn]dent the construction is broadly considered, for example by Banach and Tarski, in so few words. Hey, people besides Curt, why is it that the only thing that anybody on sci.math seriously argues _against_ is the diagonal theorem? Where theres smoke, theres often [CapitalThorn]re. Several on this thread are quick to defend mathematics that has no relation to any known aspect of physical reality. Trans[CapitalThorn]nite cardinals are mental masturbation, a house of cards, a valiant effort to make reason out of inconsistency that is doomed. I laugh about the pickled three-headed sheep comment. Counterintuitive is one thing, reasoning counter to the obvious in[CapitalThorn]nite nature of some in[CapitalThorn]nite sets is ßawed reasoning. Paradoxes are wrong! Pairs of dachshunds are great, although theyre yippy little dogs, tastes vary, paradoxes are wrong! Paradoxes are signs of insuf[CapitalThorn]cient knowledge or false pretenses. Curt, Im trying to _[CapitalThorn]x_ math, for myself and others. Some people are very attached to their notions. Ahem. Youve been fooled by Cantor. Thats OK, he was fooled too. mathematics and the foundations of mathematics, Ive posted thousands of them. So, you can have a post-Cantorian set theory where in[CapitalThorn]nite sets are equivalent. Ross F. === Subject: Re: Cantors diagonal proof wrong? > In a _very simple_ set theory, each ordinal is a set, and each set is > an ordinal. > That is arguably not simple enough. > The usual set-theoretical successor of x is xU{x}, NOT p(x). > There are GOOD reasons for this. > The powerset is just the order type, which is just the > successor. The function mapping ordinal to successor is f(x)=x+1, > That is de[CapitalThorn]nitional if you are going to have ordinal arithmetic at all. > Hey, people besides Curt, why is it that the only thing that anybody > on sci.math seriously argues _against_ is the diagonal theorem? > That is an oversimpli[CapitalThorn]cation. There is a FAQ. LOTS of things > get argued against. I would say that one reason why this one is > prominent is that we get a lot of arguers who dont know basic > [CapitalThorn]rst-order logic and who dont know what the relevant axioms are > if theyre going to be talking about powersets. They have > seen an intuitive proof and want to keep reasoning in that > vein. What Cantors Theorem REALLY says is you cant biject a > set with its powerset. Given any set theory, that proof is very > straightforward and neither you nor anyone else would know how > to attack it -- and it bears stressing that it applies not only > to every set, but every CLASS as well, NO MATTER HOW BIG. > AND, for that matter, no matter how small: Finite/in[CapitalThorn]nite > ISNT EVEN RELEVANT -- it holds for all [CapitalThorn]nite > sets too, no matter how small, including the empty set. > The problem is that people have only been exposed to the > one particular case, and they get distracted by the details > of that case. The REAL truth is simply that if you understand > why Russells paradox is paradoxical, then you understand the > proof of Cantors theorem. If you were cursed to have been originally > exposed to some proof that hides the connection, well, dont blame us. > Where theres smoke, theres often [CapitalThorn]re. > BUt not here. Here, there is just a lot of irrelevant distraction > because people think w or aleph_0 is special. With respect to > Cantors theorem, IT ISNT. It is JUST LIKE EVERY OTHER set in > being non-bijectible with its own powerset. > Several on this thread are quick > to defend mathematics that has no relation to any known aspect of > physical reality. Trans[CapitalThorn]nite cardinals are mental masturbation, a > house of cards, a valiant effort to make reason out of inconsistency > that is doomed. > HTF would you know?? Dont you see that the burden of proof IS ON YOU? > IF ZFC + large_cardinal_axiom_of_RAFs_choice REALLY IS > inconsistent, THEN THERE IS A FINITE *PROOF* of that. > If you are telling the truth then you have NO EXCUSE for not > PRODUCING this proof! No, we are not holding our breath. > I laugh about the pickled three-headed sheep comment. > Counterintuitive is one thing, reasoning counter to the obvious > in[CapitalThorn]nite nature of some in[CapitalThorn]nite sets is ßawed reasoning. > No, it isnt. Believing in obvious nature that you cant > axiomatize, and THEN calling THAT reasoning, is brute stupidity. > Paradoxes are wrong! Pairs of dachshunds are great, although theyre > yippy little dogs, tastes vary, paradoxes are wrong! Paradoxes are > signs of insuf[CapitalThorn]cient knowledge or false pretenses. > Of course, but there is nothing paradoxical about different orders > of in[CapitalThorn]nity. Indeed, it is your attempts to try to formalize the > contention that all in[CapitalThorn]nities are the same size that is ACTUALLY > paradoxical. > > Curt, Im trying to _[CapitalThorn]x_ math, for myself and others. > Feel no fret: you MUST, someday, succeed, at least in > your own small way -- for society advances > one funeral at a time. Hi George, I debate internally whether to reply to you. I respond better to rational discourse than to barking. (Snare snare cymbal.) I read your insults as jocular overfamiliarity, because I choose to be not offended. Please dont call me stupid. About the ordinals, and sets as ordinals and ordinals as sets, the notion is that each set represents an ordinal, the predecessor of its order type. Thus, multiple sets represent the same ordinal. In that way then, the powerset is the successor and order type of any set, being as well an ordinal. I coined the phrase ornate ordinal as contrasted to some plain ordinal, eg x U {x}. You loudly state that no set maps bijectively to its powerset. You might consider sharing that volume with Holmes and the NFU crowd, for which that is not always so, although they basically have large and small sets. Me, I just have that f(x)=x+1 maps all elements of x, the set itself, to all elements of x+1, the set itself, for [CapitalThorn]nite or in[CapitalThorn]nite sets, neatly compounding the notions of powerset, order type, and successor. Technically, where its an in[CapitalThorn]nite set and its powerset, then in[CapitalThorn]nity is dually represented as zero, the case of the [CapitalThorn]nite set is mechanically true in that way, but not for each element, unless you dont have empty set being a subset of every set. You say that there are no paradoxes with Cantorian theory, and that I am supposed to show ZF inconsistent. It is uncertain whether that will happen, Im thinking about it. You mention Russells paradox, you probably didnt catch my recent twist on it validating my zero vis-a-vis in[CapitalThorn]nity dual representation. There would be Burali-Forti, thats similar to what would be Cantors paradox, that the set of all sets would indeed be its own powerset. With the above notion about ordinals, they are combined into one single issue, where Ord is less than nothing, and it itself is its own successor, which would again perhaps be the empty set, which is the opposite and same as Ord. You might notice that that is the statement of the singular proper class and exluded excluded middle. About Skolem, in this thread again you see what there is about Skolem in ZF: backpedalling and handwaving, insensate to the in[CapitalThorn]nite character, nature, of two in[CapitalThorn]nite sets. If you think about zero, youre probably not the [CapitalThorn]rst person to ever do that. What comes around goes around. You read my posts to sci.math, you know I use excluded middle from nothing, or conversely, everything, to get the opposite. In that sense consider this synthesis from ur-element(s): null, U, 1, U-1, 2, U-2, etcetera and consider how that illustrates an enumeration of the positive and negative integers: 0, -1, 1, 2, -2, ... Alternately, perhaps it is as so: null, 1, 2, 3, ... from which is extracted the set of natural numbers: 0, 1, 2, 3, ... Notice how it is remarkably similar to a machine integer from the binary transistorized register of a modern computer, except its the in[CapitalThorn]nite word width, and you dont know whether binary one and zero is inverted, nor is that at issue. The idea here is to very mechanically symbolically implement [CapitalThorn]eld operations on the integers, and then map those to various underlying representations of the ordinals, for uniqui[CapitalThorn]cation as well as intrinsic composition identifers, to implement the number system as close to the machine(s) as possible. So anyways, about the naturals and reals and Cantors antidiagonal argument, there is the above resolution about any in[CapitalThorn]nite set and its in[CapitalThorn]nite powerset each being in[CapitalThorn]nite, but more speci[CapitalThorn]cally in analysis of the natural/unit equivalency function as the compositional building block of metri[CapitalThorn]ed mappings to and from the naturals and the unit interval of the reals, binary or even decimal expansions are not capable of representing these non-standard reals, each a real number. You mention the sci.math FAQ, the frequently asked questions, a guideline of regularly encountered material. I recommend the Math Forums Dr. Math, because they tend to discuss mathematics. I do have to give .999... and 1 some credit, and offer resolutions about it based upon these non-standard reals that .999... is a confusing statement of 1 - iota, and that .999... equals 1, and 1 - iota does not equal 1, because there is no expansion for 1 - iota, and between zero and one there are only real numbers. Besides that, though, maybe even a hundred people have, on their own volition, registered disagreements with trans[CapitalThorn]nite cardinals directly to sci.math as it is the premier open forum for mathematical discussion with the widest readership on the Internet with open [CapitalThorn]shbowl type group critique. They vary in sophistication. As objectively as possible, I have bias towards my own theory. Mathematics shouldnt have any contentious issues, philosophy, perhaps, mathematics, no. In mathematics, there are cut-and-dried true-or-false answers, and various unknowns in terms of conjectures with unproven truth values that are each decidable. Ross A Finlayson === Subject: Re: Cantors diagonal proof wrong? > About the ordinals, and sets as ordinals and ordinals as sets, the > notion is that each set represents an ordinal, > the predecessor of its order type. Thus, multiple sets represent the > same ordinal. What does it mean for a set to represent an ordinal? What is an ordinal? What is the predecessor relation? What is an order type? You are obviously not using these expressions as they are used in ZF (insofar as they are used at all), so you need to tell us how you are using them. Until you do, you might as well be howling at the moon. What youve got here is, at it stands, literal nonsense. Thats not meant to be insulting. Its just a fact. Without an explicit theory and explicit de[CapitalThorn]nitions of your terms, when your usage departs so dramatically from standard usage, you are just talking nonsense. Chris Menzel === Subject: Re: Cantors diagonal proof wrong? > About the ordinals, and sets as ordinals and ordinals as sets, the > notion is that each set represents an ordinal, > the predecessor of its order type. Thus, multiple sets represent the > same ordinal. > What does it mean for a set to represent an ordinal? What is an > ordinal? What is the predecessor relation? What is an order type? You > are obviously not using these expressions as they are used in ZF > (insofar as they are used at all), so you need to tell us how you are > using them. Until you do, you might as well be howling at the moon. > What youve got here is, at it stands, literal nonsense. Thats not > meant to be insulting. Its just a fact. Without an explicit theory > and explicit de[CapitalThorn]nitions of your terms, when your usage departs so > dramatically from standard usage, you are just talking nonsense. > Chris Menzel I did try to answer your questions as I saw them. Do you care to answer the question about N in V vis-a-vis N in V^G? If N bijects with P(N), does not N biject with P(N)? You might notice that it takes a complete sentence to answer that question. Quite seriously, the notion of dual representation of the ur-element and the in[CapitalThorn]nite element in the pseudoultra[CapitalThorn]nitist sense takes the endless hyperinßated stack of buck-passing, handwaving metatheory and compacts it to a regular notion of completion, to reconcile the pure and applied. Curt, if you want to have a function between the naturals and reals, then you need to consider not only the antidiagonal, but also Cantors proof, and then things along the lines of Megills proof, and similar notions, even though those apply to any sets dense in the reals, and then through Bernstein things along the lines of Daves argument. Bye. One way to consider that is to weigh the idea that in[CapitalThorn]nite sets are inexhaustible against Cantors concept that they are not. A possible resolution of that is the amorphous co-existence of zero as in[CapitalThorn]nity. That is perhaps the most direct solution. Other current methods are to tip-toe about metatheoretically looking from outside the model, then quickly back from inside, although thats recursive and only half a shadow play, and the persistence of vision is an illusion against their contradictory premises, only seeming natural because in[CapitalThorn]nite sets are equivalent. Consider a construction in ZF with classes that cannot be a set, in that supposed set theory, and cannot be a class. English readily runs out of unique group nouns. Does not that seem to demand in[CapitalThorn]nitely many different group nouns? Where that is so, it is a good idea to get rid of that notion. Mark a point on the paper to be zero, and another to be one, connect them with a line. The smudgy tower of kaolite crumbles over _itself_ to construct a continuous line with a pencil point. Thats about a non-standard model of the real numbers, because there are only real numbers between zero and one. My opinion is that positive integer multiples of the quantumized iota are the reals from zero to one, and that for any [CapitalThorn]nite interval that they must be enumerated in that way, inde[CapitalThorn]nite. Confront your misperceptions. Consider the several brand new words, and how they correspond to the very most ancient history, in explanation of the roots of logic and being. If you [CapitalThorn]nd an error in the reference, note it. It might be more than ephemera. Are you part of the problem or part of the solution? So, do you have a countable model of your theory? Is it because Skolem says so or you think so yourself? When your usage departs so dramatically from standard usage, you are just talking nonsense. Ross F. === Subject: Re: Cantors diagonal proof wrong? > I did try to answer your questions as I saw them. Do you care to > answer the question about N in V vis-a-vis N in V^G? If N bijects > with P(N), does not N biject with P(N)? You might notice that it > takes a complete sentence to answer that question. Sorry, you dont have a clue what youre talking about. === Subject: Re: Cantors diagonal proof wrong? > About the ordinals, and sets as ordinals and ordinals as sets, the > notion is that each set represents an ordinal, > the predecessor of its order type. Thus, multiple sets represent the > same ordinal. > > What does it mean for a set to represent an ordinal? What is an > ordinal? What is the predecessor relation? What is an order type? You > are obviously not using these expressions as they are used in ZF > (insofar as they are used at all), so you need to tell us how you are > using them. Until you do, you might as well be howling at the moon. > What youve got here is, at it stands, literal nonsense. Thats not > meant to be insulting. Its just a fact. Without an explicit theory > and explicit de[CapitalThorn]nitions of your terms, when your usage departs so > dramatically from standard usage, you are just talking nonsense. > > Chris Menzel > I did try to answer your questions as I saw them. Do you care to > answer the question about N in V vis-a-vis N in V^G? If N bijects > with P(N), does not N biject with P(N)? You might notice that it > takes a complete sentence to answer that question. > Quite seriously, the notion of dual representation of the ur-element > and the in[CapitalThorn]nite element in the pseudoultra[CapitalThorn]nitist sense takes the > endless hyperinßated stack of buck-passing, handwaving metatheory and > compacts it to a regular notion of completion, to reconcile the pure > and applied. > Curt, if you want to have a function between the naturals and reals, > then you need to consider not only the antidiagonal, but also Cantors > proof, and then things along the lines of Megills proof, and similar > notions, even though those apply to any sets dense in the reals, and > then through Bernstein things along the lines of Daves argument. > Bye. > One way to consider that is to weigh the idea that in[CapitalThorn]nite sets are > inexhaustible against Cantors concept that they are not. A possible > resolution of that is the amorphous co-existence of zero as in[CapitalThorn]nity. > That is perhaps the most direct solution. Other current methods are > to tip-toe about metatheoretically looking from outside the model, > then quickly back from inside, although thats recursive and only half > a shadow play, and the persistence of vision is an illusion against > their contradictory premises, only seeming natural because in[CapitalThorn]nite > sets are equivalent. > Consider a construction in ZF with classes that cannot be a set, in > that supposed set theory, and cannot be a class. English readily > runs out of unique group nouns. Does not that seem to demand > in[CapitalThorn]nitely many different group nouns? Where that is so, it is a good > idea to get rid of that notion. > Mark a point on the paper to be zero, and another to be one, connect > them with a line. The smudgy tower of kaolite crumbles over _itself_ > to construct a continuous line with a pencil point. > Thats about a non-standard model of the real numbers, because there > are only real numbers between zero and one. My opinion is that > positive integer multiples of the quantumized iota are the reals from > zero to one, and that for any [CapitalThorn]nite interval that they must be > enumerated in that way, inde[CapitalThorn]nite. > Confront your misperceptions. > Consider the several brand new words, and how they correspond to the > very most ancient history, in explanation of the roots of logic and > being. > If you [CapitalThorn]nd an error in the reference, note it. It might be more than > ephemera. Are you part of the problem or part of the solution? > So, do you have a countable model of your theory? Is it because > Skolem says so or you think so yourself? When your usage departs so > dramatically from standard usage, you are just talking nonsense. Wow! That was a post! -- Eray Ozkural === Subject: Re: Cantors diagonal proof wrong? > You might have heard of proper classes, where someone says in ZF, > the set of all sets does not exist, there is only the proper class of > all sets. Under some naive and correct de[CapitalThorn]nitions of the proper > class, there can only be one or none of them. Which naive and correct de[CapitalThorn]nitions would those be? > The empty set is the proper class or ur-element, as is in[CapitalThorn]nity. Its > a _set_ theory, founded on nothing and everything. What are the axioms of this set theory? > In a _very simple_ set theory, each ordinal is a set, and each set is > an ordinal. And what are the axioms of *this* set theory? > The powerset is just the order type, which is just the successor. You appear to be using power set to mean something other than what everyone else means by it. Chris Menzel === Subject: Re: Cantors diagonal proof wrong? Basically from the empty set all possible constructions, which means basically any group of balanced brackets, is a set. As well, as there is no foundation axiom, then you would be able to arbitrarily label a set and then insert that label anywhere within that or any other set de[CapitalThorn]nition. Each of these is formed because each is unique, and thus through excluded middle, not your axiom of difference, it is generated. The sets by themselves can be quite meaningless. Its just the set of all possible constructions of the empty set, the set of all sets. Then, each set is claimed to fall together in a consistent way to form ordinals. Take any literal, and assign each set to be a given ordinal. Thus, each set is unique, and also, each set is among many other sets representing the same ordinal. It doesnt matter how you select the ordinals, its just a sequence, any sequence. Then are ascribed certain properties of the ordinals that they Of the many-to-one mappings of sets to ordinals, one has the property that it is basically the von Neumann form, or another form that has each ordinal as least set being an element of its order type as ordinal. The direct sum of in[CapitalThorn]nitely copies of N is the empty set. So there arent any axioms, just an assertion of empty set and excluded middle, and a de[CapitalThorn]nition of ordinal. In a generic extension V^G of V, there exists the set N and a bijection between N and P(N). Is that not so? Why or why not? What elements of N are not elements of N? The answer to that is none. Ross F. === Subject: Re: Cantors diagonal proof wrong? > You might have heard of proper classes, where someone says in ZF, > the set of all sets does not exist, there is only the proper class of > all sets. Under some naive and correct de[CapitalThorn]nitions of the proper > class, there can only be one or none of them. > Which naive and correct de[CapitalThorn]nitions would those be? > The empty set is the proper class or ur-element, as is in[CapitalThorn]nity. Its > a _set_ theory, founded on nothing and everything. > What are the axioms of this set theory? > In a _very simple_ set theory, each ordinal is a set, and each set is > an ordinal. > And what are the axioms of *this* set theory? > The powerset is just the order type, which is just the successor. > You appear to be using power set to mean something other than what > everyone else means by it. > Chris Menzel Chris, There are no axioms in the theory. That way it can be complete. Alternately in[CapitalThorn]nitely many axioms assert existence, for completeness, completion. There are some fundamental assumptions: that there is anything, and that there is something else. Thats about it. A set is equal to itself. The de[CapitalThorn]nition of proper class, its the same one you use, interpreted roughly. That and these other de[CapitalThorn]nitions of words, youll notice that for the most part they are already in regular usage, and their selection was simple in that they share the same meanings. Im not talking about ZF, although you could call it ZFF, simply removing the axiom of foundation and adding perhaps an inversion axiom. About the multiple representation of ordinals as sets, for any set its order type is the successor of the ordinal value. In this way the powerset of a set X, P(X), with the script P to delineate it from the probability P, and the singleton containing that set, {X}, represent the same ordinal. To get you to accept it as consistent it would probably be very useful to succinctly symbolize these constructs towards brevity and disambiguity. I prefer to work in the realm of plain language explanation, because its a higher level construct and helps me present it in an obscure, yet completely open, way thus that I simply continue developing it while having presented its irrepudiable developments directly to you in a form you might not care to decipher, until later when you discover that all along I was quite right. Im spontaneous, yet meticulous, and have read and reread these arguments many times. For example, I read about Nam Duc Nguyens new axiom he wants that he hasnt stated, and Ive already determined it might be a theorem of the null-axiom theory, the axiom-free theory, as the axioms of ZFC are theorems. That enables me to validate much of logic with my non-standard theory, in much the same way standard analysis is validated in the regular non-standard analysis, where the powerset result is not a theorem. So, you might see that as a troll of sorts, it is not, because it involves remaking the foundations of mathematical logic, which has been made easy by the acceptance of others of inconsistency in the form of that in[CapitalThorn]nite sets are not equivalent, thus that they are not yet actively working towards resolution of those issues because they are convinced of a falsehood that would prevent them, so I can make advancements in this way. In[CapitalThorn]nite sets are equivalent because theres always one more. Im de[CapitalThorn]nitely not the only person working with theories where the powerset bijection result is not a theorem, Im just the only person who has put forth a theory to that effect to you and everyone else who reads this or other posts on the Internet newsgroup sci, for science, math, for mathematics, sci.math. Thus I have an advantage in presenting them as my own developments as opposed to another, who can present them to others as my developments, in that way insulating themselves from pedagogical backlash, in several ways. Instead of pushing off the resolution of ZFs problems into ever-higher order logics, with thus no resolution, they are resolved in what can be a [CapitalThorn]nitist logic, it is not, there is the completed in[CapitalThorn]nity. Chris, in terms of Skolem, is not the generic extension of N the same set as N? Chris, when you extend the reals to hyperreals, its the same set of reals. So, does the extension of N have any elements not in N? So, I say no, that previous post of mine is not nonsense. How do you care for the use of the word insensate? Please reread the preceding post. I mangle it to sensei, in, t. The sensei is in. One time I ran a quarter mile in a minute. Thats wordplay, more seriously I question your de[CapitalThorn]nition of those terms, I want to know your de[CapitalThorn]nitions, and their speci[CapitalThorn]c de[CapitalThorn]nitions, or more speci[CapitalThorn]cally in what way you feel that they have been misused. You claim to not be able to make sense of my words, why? Ross F. === Subject: Re: Cantors diagonal proof wrong? > There are no axioms in the theory. > That way it can be complete. The relevant hypotheses of the incompleteness theorem are about representability, not axiom-existence. > Alternately in[CapitalThorn]nitely many axioms assert existence, for completeness, > completion. That is irrelevant unless the in[CapitalThorn]nite set of axioms in question IS RECURSIVE. > There are some fundamental assumptions: Then there ARE AXIOMS, TOO, dumbass. > that there is anything, and > that there is something else. Thats about it. A set is equal to > itself. Everything is equal to itself wherever equality actually occurs. But if you havent picked a LANGUAGE yet, its no wonder you cant pick any axioms. I cant believe you still expect not to get called stupid. > The de[CapitalThorn]nition of proper class, its the same one you use, interpreted > roughly. Around here,NOTHING IS EVER interpreted roughly! AXIOMS are presented (PRECISELY) and you can interpret NOT roughly but is INSTEAD *clear*! > That and these other de[CapitalThorn]nitions of words, youll notice that for the > most part they are already in regular usage, and their selection was > simple in that they share the same meanings. In logic, in case you hadnt noticed, NOBODY GIVES A what the regular usage of anything is because WE HAVE AXIOMS DEFINING the use that WE are making!! And youd best get some as well, if you want to keep playing. > Im not talking about ZF, although you could call it ZFF, simply > removing the axiom of foundation and adding perhaps an inversion > axiom. Aczel et al have ALREADY done this. Anti-foundational set theories are even MORE common and investigated than set theories with a universal set. JUST GOOGLE Aczel AND SEE WHAT YOU GET. AND THEN get back to us. > About the multiple representation of ordinals as sets, for any set its > order type is the successor of the ordinal value. What you mean by the ordinal value is ITS (the sets) ordinality. It is a FUNCTION OF the set. > In this way the > powerset of a set X, P(X), with the script P to delineate it from the > probability P, IF you publish some axioms then the P that is relevant is the one that OCCURS IN those axioms and if your axioms are NOT DISCUSSING probability, the NO CONFUSION IS POSSIBLE! > and the singleton containing that set, {X}, represent > the same ordinal. Again, Ross, since you had the utter idiocy to invoke regular usage above, the verb represent ALREADY HAS a regular usage and you MAY not use represent in this way. What you MEAN is that every set HAS a particular ordinal that is the one it is associated with, somehow. But the set DOES NOT represent that ordinal! > To get you to accept it as consistent it would probably be very useful > to succinctly symbolize these constructs towards brevity and > disambiguity. No, it would just be necessary to write some axioms. ZFC gets by with 10. Fewer if you try (Replacement implies a few of the others). WHEN YOUVE DONE YOUR HOMEWORK, get back to us. > I prefer to work in the realm of plain language > explanation, because Because youre A ING IDIOT, or should that be maybe a virgin idiot.... > its a higher level construct I.e., too high for you. > and helps me > present it in an obscure, yet completely open, way THAT is paradoxical. THe mere fact that you confess that the obscurity has positive value to you -- THAT all by itself UN[CapitalThorn]ts you to be responded to, but I never expected you to admit it. > thus that I simply > continue developing it while having presented its irrepudiable > developments directly to you in a form you might not care to decipher, > until later when you discover that all along I was quite right. > Im spontaneous, yet meticulous, Oh, bull. You are WAY too stupid to be meticulous. > and have read and reread these arguments > many times. To no discernible educatory effect, which is how we know youre stupid. > For example, I read about Nam Duc Nguyens new axiom he wants that he > hasnt stated, and Ive already determined it might be a theorem of > the null-axiom theory, the axiom-free theory, as the axioms of ZFC are > theorems. OK, I take it back, youre not stupid; youre just a better-than- average crank/troll. That actually requires wit. > That enables me to validate much of logic with my > non-standard theory, in much the same way standard analysis is > validated in the regular non-standard analysis, where the powerset > result is not a theorem. It IS SO TOO, dumbass. There is nothing about non-standard analysis that in any way impacts whether you can or cant biject a set with its powerset. > So, you might see that as a troll of sorts, right. > it is not, because it > involves remaking the foundations of mathematical logic, No, thats how we know it IS trolling. > which has > been made easy by the acceptance of others of inconsistency in the > form of that in[CapitalThorn]nite sets are not equivalent, That is NOT *inconsistency*, dumbass! If you think it is, then we repeat, THE BURDEN OF PROOF IS ON YOU! THERE *MUST* exist a proof IF there is an inconsistency! So you canNOT claim this burden is unreasonable! > thus that they are not > yet actively working towards resolution of those issues because they > are convinced of a falsehood that would prevent them, How do YOU KNOW its a falsehood?? > so I can make > advancements in this way. In[CapitalThorn]nite sets are equivalent because > theres always one more. Again, because was ALREADY in the dictionary and its traditional meaning leaves what you say here meaningLESS. > Im de[CapitalThorn]nitely not the only person working with theories where the > powerset bijection result is not a theorem, Youre not EVEN ONE of the people workingwith theories where the powerset bijection result is not a theorem, because YOU ARE NOT working with ANYthing as well-de[CapitalThorn]ned as a THEORY! YOU are working with terminal logorrhea! > Im just the only person > who has put forth a theory You have NOT put forth ANY theory and we are NOT holding our breath. === Subject: Re: Cantors diagonal proof wrong? >> You might have heard of proper classes, where someone says in ZF, >> the set of all sets does not exist, there is only the proper class of >> all sets. Under some naive and correct de[CapitalThorn]nitions of the proper >> class, there can only be one or none of them. >> Which naive and correct de[CapitalThorn]nitions would those be? >> The empty set is the proper class or ur-element, as is in[CapitalThorn]nity. Its >> a _set_ theory, founded on nothing and everything. >> What are the axioms of this set theory? >> >> In a _very simple_ set theory, each ordinal is a set, and each set is >> an ordinal. >> And what are the axioms of *this* set theory? >> The powerset is just the order type, which is just the successor. >> You appear to be using power set to mean something other than what >> everyone else means by it. > There are no axioms in the theory. That way it can be complete. No, that way it will be meaningless. Utterly. Without axioms, there are absolutely no constrains on the interpretation of any of the nonlogical expressions in your language. They can mean whatever anybody wants them to mean -- which is to say, they are utterly m.beaningless. > Alternately in[CapitalThorn]nitely many axioms assert existence, for completeness, > completion. This makes no sense at all. > There are some fundamental assumptions: that there is anything, If you mean: that there is something, then that is a basic theorem of logic. > and that there is something else. That there are two things? Ok, so you have one axiom: (exists x)(exists y)x =/= y > Thats about it. A set is equal to itself. Everything is equal to itself. This again is just a basic logical theorem. It tells us nothing about sets over and above anything else. > The de[CapitalThorn]nition of proper class, its the same one you use, interpreted > roughly. It cant *possibly* be the one I use, as that de[CapitalThorn]nition depends on the background of ZF set theory (with class terms). > That and these other de[CapitalThorn]nitions of words, youll notice that for the > most part they are already in regular usage, and their selection was > simple in that they share the same meanings. Nope. They can only share the same meanings if you adopt the same axioms. As you use them, the terms mean *nothing*. > Im not talking about ZF, although you could call it ZFF, simply > removing the axiom of foundation and adding perhaps an inversion > axiom. I dont know what an inversion axiom is. You cant use quasi-technical terms like this use words without de[CapitalThorn]ning them. Its no different from barking. As for removing foundation, there is a very large literature on non-well-founded set theory which explores ZF without foundation together with a variety of anti-foundation axioms. So once you tell us what an inversion axiom is, does that mean this de[CapitalThorn]nitely is the theory you are adopting? ZF - Foundation + Inversion? If so, then most everything youve claims as a theorem of your theory is not a theorem of your theory. > About the multiple representation of ordinals as sets, for any set its > order type is the successor of the ordinal value. De[CapitalThorn]ne order type. De[CapitalThorn]ne successor. De[CapitalThorn]ne ordinal value. > In this way the powerset of a set X, P(X) ... and the singleton > containing that set, {X}, represent the same ordinal. De[CapitalThorn]ne powerset. De[CapitalThorn]ne represents an ordina. Then show how this follows as a theorem. Then, and only then, you wont be talking nonsense. > To get you to accept it as consistent it would probably be very useful > to succinctly symbolize these constructs towards brevity and > disambiguity. Not simply useful. Absolutely essential. > I prefer to work in the realm of plain language > explanation, because its a higher level construct and helps me > present it in an obscure, yet completely open, way Nope. You might *feel* like it makes sense, but it doesnt. Everything you say about your theory is *nonsense* without axioms and de[CapitalThorn]nitions. > thus that I simply continue developing it You have as yet developed nothing. > while having presented its irrepudiable developments There is no content to repudiate. So far you have only barked. > directly to you in a form you might not care to decipher, There is nothing to decipher. Youve not provided the code. Youve just scribbled meaningless glyphs. > until later when you discover that all along I was quite right. It is not possible for you to be right or wrong without providing a background theory and de[CapitalThorn]nitions that make your assertions meaningful. > I question your de[CapitalThorn]nition of those terms, I want to know your > de[CapitalThorn]nitions, and their speci[CapitalThorn]c de[CapitalThorn]nitions, Pick up any standard text on ZF set theory. > or more speci[CapitalThorn]cally in what way you feel that they have been misused. You are using them in a sense that clearly doesnt comport with their standard meanings without providing the least clue as to how you are using them. > You claim to not be able to make sense of my words, why? Because they dont make any sense for *anyone*. -cm === Subject: Re: Cantors diagonal proof wrong? > slrncpp4ms.2u4.cmenzel@philebus.tamu.edu>... > said: > There are some fundamental assumptions: that there is anything, > If you mean: that there is something, then that is a basic theorem of > logic. ... of classical logic, not of free logic. PH === Subject: Re: Cantors diagonal proof wrong? <2voi2mF2o0lbiU1@uni-berlin.de> at 05:27 PM, curt@kcwc.com (Curt Welch) said: >Why is it ok to write 0.111... but not ...11111 ? Itsa OK to write either one, as long as you de[CapitalThorn]ne what they mean and dont assum e properties beyond those implied by their de[CapitalThorn]nitions. In your case you are assuming that ...11111 is an integer, which is patently false. >Its just a name we use to talk about the >real value which is 0.1111 repeating forever. No. 0.111... is a name of limit n->oo 0.1 (n times), which happens to be 1/9. You havent de[CapitalThorn]ned ...11111 to be the name anything. >And I can just as easily >de[CapitalThorn]ne the integer of 1 repeating forver. Thats not a de[CapitalThorn]nition. >The only reason we do not do that is a matter of convention. No, the only reason that we dont do it is that there is no sensible mapping from such strings into the integers. >Its not (so I claim) in violation of what integers are. You can claim that 1+1=3 if you want, but that wont cause anybody to take you seriously. Claims must be backed up with sound reasoning. >If you start with 0, and continue to apply the +1 function to it, >and ignore all the values you come up with which does not have all >1s, you [CapitalThorn]nd you have the exact same type of de[CapitalThorn]ntion that gives >you 1/9 when you generate a string of ones running to the right, >instead of running to the left. No. Again, you do not understand what an integer is or what a number is. >Also, all integers have an implied in[CapitalThorn]nite string of 0s running to >left No. An integer is not a string of digits. >(just like reals have am implied in[CapitalThorn]nite string of 0s running to >the left and right). No. A real number is not a string of digits. There are partial mappings from strings of digits to numbers. >Just change the implied 0, to an implied 1, That gives you a string of digits. It does not give you a number. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantors diagonal proof wrong? >Nath is not something I specialize in (and I dont read this group >normally), but Ive been looking at a few things lately and Ive >decided that some very big mistakes have been made in math because >people started playing around the concept of in[CapitalThorn]nity without >realizing the trouble they were creating for themselves. In this case, that is a sign that you understand neither the vocabulary nor the reasoning. >But, like I said, Im not a math expert by any means. So Im >posting the idea here so you experts can have fun laughing at me. The problem is that the circle squarers[1] and the angle trisecters[1] ceased being funny centuries ago. >The reason I came to these conclusions is because Ive spent a lot >of time trying to uncover the mysteries of AI totally unrelated. >basic assumptions on what an idea is Mathematics is not psychology. Mathematics does not concern itself with ideas, but with proofs. >So you just reverse the digits in the integer to create the real. I >claim this mapping is one to one and covers all the reals in that >range. Your claim is false. You dont understand what an integer is. >So, this proves that D(n) for all values of n, from 0 to in[CapitalThorn]nity, >is in the table. Heres where you go wrong. In[CapitalThorn]nity is not an integer, and there is no D(oo). >Cantors proof says that D(in[CapitalThorn]nity) is not in the table, Thats not even wrong. There is no D(oo) to not be in the table, because you havent de[CapitalThorn]ned it. >D(n) for all values of n, including in[CapitalThorn]nity, You havent de[CapitalThorn]ned D(oo). Were you to de[CapitalThorn]ne D(oo) to be the limit of D(n), you would [CapitalThorn]nd that it is not in the table, but that would not support your conclusion. >Lets create a table of integers like this: Again, you dont understand what an integer is. A string of digits is just a string of digits, not an integer. You can certainly represent an integer as a string of decimal digits, but such a string will have only a [CapitalThorn]nite number of nonzero digits. >Its just a normal list of integers, but instead of following the >normal convention of leaving off the leading zeros (which we all >know are implied even if we dont write them) I include them in that >table. They arent implied, theyre irrelevant. Both Ô12 and Ô012 are valid decimal representations of twelve. Neither one is twelve. >So lets use Cantors logic on this table and see if we can construct >a number which is not in the table. We take the numbers from the >diagonal, and construct the number ...111111 just like we did above. Not only is Ô...111111 not a number, it isnt a decimal representation of a number. So you havent proved anything except your lack of understanding. >Ok, so if Cantor was wrong, He wasnt. >The answer is one already well known to mathematicians. Alas, yes ;-) >They just never realized how it applied here. Would that were so. Youre not the [CapitalThorn]rst to confuse a string of digits with a number. >in[CapitalThorn]nity is only a name for something which can not exist. An interesting claim. What is the largest integer? What do you get when you add one to it? >The contradiction that Cantor put into his assumptions in the >diagonal proof, was that something of in[CapitalThorn]nite size does exist. He made no such assumption. >The number I call D, There is no such number. >he declares does it exist. No. >If you think its ok to use in[CapitalThorn]nity like it was real, it becomes >possible to prove anything by contradiction. No. >I can easily for example prove that 1= 0 by making the same mistake >by playing with an in[CapitalThorn]nit series of 1 - 1 + 1 - 1 + 1 ..., No. That is not a convergent series. >or by using 1/0 in a proof as if it were a number that >existed. The issue there is more subtle; by de[CapitalThorn]nition 1/0 is unde[CapitalThorn]ned, since there is no x such that x*0=1. One could, of course, extend the reals in such a way as to allow it to be de[CapitalThorn]ned, but only by sacri[CapitalThorn]cing one or more of the expected properties of the arithmetic operations. The fallacies that you have seen were the result of simultaneously assuming that 1/0 was de[CapitalThorn]ned and assuming that the normal properties of the arithmetic operations still applied. >So, what Im saying is that in[CapitalThorn]nite sized sets dont exist at all, >and cant exist. Youre free to say it. Youll need to do more than just say it if you want Mathematicians to take you seriously. >And any time you start with an axiom which says in[CapitalThorn]nite sized sets >do exist, you have introduced an contradiction into your axioms >which guaranties contradictions in your results. You have failed to demonstrate such a contradiction. Youre not exactly plowing new ground here. >We dont have the set of all integers. What we have is a counting >algorithm that can generate as many integers as you need for any >application. No, what we have here is a formal system, not an algorithm. >Its perfectly valid to talk about what in[CapitalThorn]nite algorithms do as >they approach in[CapitalThorn]nity. What do you mean by approach in[CapitalThorn]nity? >But once you start to pretend they reach it What do you mean by reach it? Why are you pretending that anybody pretends whatever it is that you mean? >and a world which has nothing to do with the universe we exist in. An interesting claim, one that you have failed to justify. >This is because ideas are not magic. You seem to belive that they are, because you are applying nonlogical processes in order to sell your ideas. >They are the result of mechanical computation. I see that it is not only the Mathematicians who are not psychologists. To the best of our knowledge and understanding, mental processes are electrochemical, not mechanical. Of course, proofs are not ideas, so you are even further off base. >So any time you talk about computing an in[CapitalThorn]nite sized set You are the only one that has talked about such a computation. The Cantor diagonalization is not a computation. >fantasy world The only fantasy world involved is the one in which people have written the things that you imagine they have written. >If you start an algorithm The proof does not involve an algorithm. >And as I showed above, You showed nothing except your misunderstanding of what an integer is. >If you pretend the job of construction does end Why are you pretending that there is a job of construction? >Much other important work, such as Gdels, also fell prey to this >same mistake. There is no Gdel; there is only G.9adel; your news client is broken. And you have failed to identify any mistake made by G.9adel. >Oh, and if you want a mapping from the integers to all the reals, >heres one: No. Again you are confusing an integer with a string of digits. >So your integer which grew to in[CapitalThorn]nity in one direction Integers dont grow to in[CapitalThorn]nity. They just are. >Now, I know most (if not all of you), will tell me Im crazy. More like confused and ignorant. >How is my proof You had no proof. >any less valid Because you manipulated terms that you didnt understand in a fashion that had nothing to do with logic. >If you cant tell me that, We can, and have, told you that. We cant make you overcome your preconceptions. >Has any one else put forth this same argument (or others) that >Cantors proof is invalid? The world is and was full of people who deluded themselves into believing that they had proofs of all sorts of things. [1] Using only protractors and straightedges, of course. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w For some reason, your posts seems to have shown up two days late here. Most of what said already been well covered in many other messages, but there was a point or two I wanted to comment on... >They are the result of mechanical computation. > I see that it is not only the Mathematicians who are not > psychologists. To the best of our knowledge and understanding, mental > processes are electrochemical, not mechanical. I am using the word mechanical in a non conventional way above, but not one that violates the basic meaning of the word. Ive simply extended it to include all interaction of physical matter. Whether its a collection of atom formed into a wheel, or individual atoms pushing and pulling each other in a chemical reaction, or individual electrons pushing and pulling each other in an electrical circuit, its all still mechanical in the fact that physical matter is interacting with other physical matter to produce a behavior unique to that con[CapitalThorn]guration of the matter. The word mechanical is normally used to describe the macro level behavior of matter and I just extended it in my usage to cover micro level of mater because when you use separate words to describe the different types of behavior, we tend to start thinking that there is a fundamentally different processes at work when there is not. >If you pretend the job of construction does end > Why are you pretending that there is a job of construction? This is just more of how I choose to look (and talk) about things because Im trying to understand something larger than just math. Im trying to understand the brain that creates it and how all of our understanding of the physical world and mental world are related and connected. Everything must be brought into existence. And I call that act construction, because the only way anything can be brought into existence in this universe requires that physical matter be moved. And that always happens as a result of matter interacting with other matter. Math is a language that starts on a foundation well rooted in the physical world. But then, with the power of language, it creates self referencing loops and seems to cut the physical world out of the story, and then allows itself to wander into a fantasy world created by the manipulation of language. We all know how this happens when writing [CapitalThorn]ction, but math is interesting because it seems to be stronger than [CapitalThorn]ction. Its not completely arbitrary. Yet, it does seem to have separated itself from reality at some point. Im curious about just what has happened in the language of math in this regard. When I talk about construction as a physical act in connection with math concepts, Im trying to tie various mathematical ideas back to the physical foundation they grew out of. Im trying to see if at some point, we accepted some fact that seemed valid, but was actually a violation of the physical world the language started off explaining. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? <41992f15$4$fuzhry+tra$mr2ice@news.patriot.net> at 08:00 PM, curt@kcwc.com (Curt Welch) said: >The word mechanical is normally used to describe the macro level >behavior of matter and I just extended it in my usage to cover micro >level of mater The problem with that is that the term mechanical has a connotation of classical behavior that is inapplicable to the micro world. >This is just more of how I choose to look (and talk) about things >because Im trying to understand something larger than just math. You wont understand it if you dont make necessary distinctions. The mental process of devising a proof is not the same as the proof itself, nor is the mental process of verifying the proof. >Everything must be brought into existence. Helium must be brought into existence. But nucleosynthesis is not Helium, and a knowledge of one casts little light on the other. >fantasy world No. Mathematics claims that certain sentences can be derived from certain other sentences in a prescribed manner. There is no fantasy there. If you choose to impose referents on the terms in an inappropriate manner, the fault lies in you, not in the Mathematics. >Yet, it does seem to have separated itself from reality at some >point. Only in the sense that Music has. Not every composition has to have a story, and not every theorem has to have a practical application. ObWigner Of course, in the real world the process of abstraction in modern Mathematics turns out to have practical value, but Mathematics is not and doesnt claim to be a representational art form. >Im curious about just what has happened in the language of math in >this regard. What has happened is that Mathematicians discovered that several apparently distinct systems had the same or similar behaviors and structures, and abstracted them. The net result of that has been to produce far more powerful tools, both for use by other Mathematicians and for use by those outside of Mathematics. >Im trying to see if at some point, we accepted some fact that >seemed valid, but was actually a violation of the physical world the >language started off explaining. Mathematics never explained the physical world; in some cases it has been useful in describing our understanding of it. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantors diagonal proof wrong? > [Curt Welch, with some arguments that depend on the existence of > integers with non-trivial in[CapitalThorn]nite decimal representations] > ... >>Has any one else put forth this same argument > Yup, many times, over many years. >>(or others) > Those too. >>that Cantors proof is invalid? > If this isnt literally true, its close enough : look at sci.math on > any day over the last decade, and youll [CapitalThorn]nd the same basic argument in > some then-current thread. > Rather than repeat all this, how about going to > and entering > Cantor diagonal > in the search box? There are close to 5,000 hits on that today. Broaden > the search to sci.logic and comp.theory to get more. [ .. rest deleted ..] If this is true, and it *IS* true, then it certainly is a clear signal. I have always learned that a mathematical proof should be CONVINCING in the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough to prevent these heated discussions to be launched every time again. And whats worse, these counter arguments are continuing now for more than A WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in the foreseeable future. Think about it. How comes that Socrates could even convince a slave that the theorem by Pythagoras is valid? And how comes that even professional mathematicians (I have read about Skolem lately) do _not_ wholeheartedly agree on those Cantorian proofs? Not even after a 100 years have been elapsed, since they were presented for the [CapitalThorn]rst time? Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? >> [Curt Welch, with some arguments that depend on the existence of >> integers with non-trivial in[CapitalThorn]nite decimal representations] >> ... >Has any one else put forth this same argument >> Yup, many times, over many years. >(or others) >> Those too. >that Cantors proof is invalid? >> If this isnt literally true, its close enough : look at sci.math on >> any day over the last decade, and youll [CapitalThorn]nd the same basic argument in >> some then-current thread. >> Rather than repeat all this, how about going to >> and entering >> Cantor diagonal >> in the search box? There are close to 5,000 hits on that today. Broaden >> the search to sci.logic and comp.theory to get more. [ .. rest deleted ..] >If this is true, and it *IS* true, then it certainly is a clear signal. Uh, right. Finding thousands of bits of nonsense, along with thousands of simple refutations, proves what? >I have always learned that a mathematical proof should be CONVINCING in >the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough >to prevent these heated discussions to be launched every time again. And >whats worse, these counter arguments are continuing now for more than A >WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in >the foreseeable future. >Think about it. How comes that Socrates could even convince a slave that >the theorem by Pythagoras is valid? When was the last time you read that thing? Socrates _stated_ point was not the he could convince the slave of the truth of the theorem, it was that the slave actually knew it all along, without being aware of it. (iirc). The whole things hilarious, the slave simply answering yes to leading questions taken as proof that he knows something about the math. But never mind that... >And how comes that even professional >mathematicians (I have read about Skolem lately) do _not_ wholeheartedly >agree on those Cantorian proofs? Not even after a 100 years have been >elapsed, since they were presented for the [CapitalThorn]rst time? This is complete and utter bull. There are no professional mathematicians who dispute the validity of that proof. If you think youve read that Skolem doubted it youre very confused about the meaning of something you read (or you read something written by someone who was very confused, but the [CapitalThorn]rst seems more likely). >Han de Bruijn ************************ David C. Ullrich === Subject: Re: Cantors diagonal proof wrong? <10imp0p62opq0ur3svr3lkia9loe9tb5gl@4ax.com> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~( 5eZ41to5f%E@ELIi $t^ VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw [...] >>Think about it. How comes that Socrates could even convince a slave >>that the theorem by Pythagoras is valid? > When was the last time you read that thing? Socrates _stated_ point > was not the he could convince the slave of the truth of the theorem, > it was that the slave actually knew it all along, without being > aware of it. (iirc). The whole things hilarious, the slave simply > answering yes to leading questions taken as proof that he knows > something about the math. When was the last time you read that thing? Sokrates makes a case before the slave, and gives him options to choose, and brings up counterarguments for every wrong turn the slave takes. And he shows that once every wrong turn gets blocked by being explained, the slave is able to go the right way. Anyway, this was not about Pythagoras, but about something like a square with a side as long as the diagonal of another square has twice its area IIRC (I dont have it handy). And the idea more or less was (again IIRC) that mathematical truth does not get as much invented as discovered, being there before. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cantors diagonal proof wrong? Originator: grubb@lola >> When was the last time you read that thing? Socrates _stated_ point >> was not the he could convince the slave of the truth of the theorem, >> it was that the slave actually knew it all along, without being >> aware of it. (iirc). The whole things hilarious, the slave simply >> answering yes to leading questions taken as proof that he knows >> something about the math. >When was the last time you read that thing? Sokrates makes a case >before the slave, and gives him options to choose, and brings up >counterarguments for every wrong turn the slave takes. And he shows >that once every wrong turn gets blocked by being explained, the slave >is able to go the right way. Anyway, this was not about Pythagoras, >but about something like a square with a side as long as the diagonal >of another square has twice its area IIRC (I dont have it handy). >And the idea more or less was (again IIRC) that mathematical truth >does not get as much invented as discovered, being there before. Actually, Ullrich got it right. Socrates was trying to show that all learning was just memory. He used the slave as an example of someone who had no Ôprevious knowledge but was able to answer certain questions correctly none-the-less. Kastrup is right that there was nothing about Pythagorus there, but Ullrich didnt claim there was, as far as I can see. My reading of it would agree more with Ullrichs Ôleading questions scenario also than Kastrups Ôcounterargument scenario. Socrates even made comments about it taking time for the slave to Ôremember as an explanation to why the [CapitalThorn]rst questions were answered incorrectly. --Dan Grubb === Subject: Re: Cantors diagonal proof wrong? Discussion, linux) > I have always learned that a mathematical proof should be CONVINCING in > the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough > to prevent these heated discussions to be launched every time again. And > whats worse, these counter arguments are continuing now for more than A > WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in > the foreseeable future. There have been cranks for centuries. I suppose you believe that an arbitrary angle can be trisected? > And how comes that even professional mathematicians (I have read > about Skolem lately) do _not_ wholeheartedly agree on those > Cantorian proofs? What did Skolem say that is in conßict with Cantors theorem? Where did he say it? -- Jesse F. Hughes Now Ôpure math makes sense as well as clearly its a peacock game, where some of you see it as a way to show you as being highly intelligent and thus more desirable to women. -- James S. Harris === Subject: Re: Cantors diagonal proof wrong? > There have been cranks for centuries. I suppose you believe that an > arbitrary angle can be trisected? It can be, using a marked straight edge and sliding it. Google for the details. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? <873bz8x1x1.fsf@phiwumbda.org> <3013h2F2po1c4U4@uni-berlin.de> Discussion, linux) >> There have been cranks for centuries. I suppose you believe that an >> arbitrary angle can be trisected? > It can be, using a marked straight edge and sliding it. > Google for the details. I assumed that the reader would understand the reference, but perhaps I should have been more precise. -- So, at this time, Id like to assure you that I am not interested in Ill have prosecutors knocking on your doors. I have no problem with === Subject: Re: Cantors diagonal proof wrong? [Tim Peters] ... >> Rather than repeat all this, how about going to >> and entering >> Cantor diagonal >> in the search box? There are close to 5,000 hits on that today. >> Broaden the search to sci.logic and comp.theory to get more. ... [Han de Bruijn] > If this is true, and it *IS* true, then it certainly is a clear signal. While the presence of a strong signal is clear, what it signi[CapitalThorn]es is open to question. > I have always learned that a mathematical proof should be CONVINCING in > the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough > to prevent these heated discussions to be launched every time again. And > whats worse, these counter arguments are continuing now for more than A > WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in > the foreseeable future. The end of the human race should do the trick . > Think about it. How comes that Socrates could even convince a slave that > the theorem by Pythagoras is valid? If he had known about it, he could have-- and even more easily --convinced the poor fellow that a set cant be put into bijection with its power set. *That* proof of Cantors attracts little opposition, possibly because it takes more than one second to visualize, and possibly because its much harder to make horribly wrong assumptions about what its saying. As many others have said here, the primary problem with the single proof of Cantors that attracts endless confusion is that many people really dont understand what either integer or real *mean*, so launch into fantastic objections based on fantastic misconceptions. For example, this very thread is based on the fantastic whaddya mean an integer cant have an in[CapitalThorn]nite number of digits?! misconception. Similar confusions about real numbers are even more evident in the even more numerous debates about whether 0.999... equals 1. Cantors beautiful proof that any set has smaller cardinality than its power set only talks about sets, subsets, and the set membership relation, so doesnt leave much opportunity for highly creative confusion. > And how comes that even professional mathematicians (I have read about > Skolem lately) do _not_ wholeheartedly agree on those Cantorian proofs? > Not even after a 100 years have been elapsed, since they were presented > for the [CapitalThorn]rst time? Skolem had no argument with Cantors proofs. He had some remarkable results of his own, most famously that if a countable [CapitalThorn]rst-order theory has a model, then it must have a (at least one) countable model -- even if it wasnt *intended* that the theory have a countable model, and no matter how hard one may try to prevent it from having a countable model. This applies to the most common axiomatization of set theory, in which we can prove that the reals are uncountable, and yet for which Skolems theorem proves that theres a countable model of the reals anyway. There isnt an actual contradiction in that, but understanding why requires precise understanding of what all the words mean. As a hint, Cantors proofs remain valid even when you keep one of those countable models in mind, as Skolem certainly knew. In the end, its probably viewed most often today as an inherent limitation of [CapitalThorn]rst-order theories, although Skolem was admittedly inclined to read more into it than just that. In any case, Skolem could hardly have gotten this result if he didnt know what countable meant -- and agreed that not countable also has meaning. After all, if he actually maintained that *all* sets were countable (which he did not maintain, and could not, because he knew what the words mean), then his proof that countable models exist would have had no content at all -- it would have been as empty as proving that, say, every model is a model. === Subject: Re: Cantors diagonal proof wrong? : I have always learned that a mathematical proof should be CONVINCING in : the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough : to prevent these heated discussions to be launched every time again. And : whats worse, these counter arguments are continuing now for more than A : WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in : the foreseeable future. The proof is convincing to those who understand the ideas and de[CapitalThorn]nitions used by the proof. A common theme among the people who argue with the proof is that they do not really understand the basic ideas used in the proof. For example, in this thread the original poster did not know what an integer was. It is not surprising that he does understand the proof when he does not know what an integer is, and it is not surprising that he is not convinced by something he understands. Many other people who argue with the proof do not know what a real number is, or what a function is, or what a set is, or what in[CapitalThorn]nite means. The often think they know what a real number or a function is, but it is often some imprecise personal de[CapitalThorn]nition. Cantors proof draws a lot of attention for the same reason that Relativity does. It is fairly simple conceptually, it is counter to common sense, and it relies on precise de[CapitalThorn]nitions. Stephen === Subject: Re: Cantors diagonal proof wrong? > : I have always learned that a mathematical proof should be CONVINCING in > : the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough > : to prevent these heated discussions to be launched every time again. And > : whats worse, these counter arguments are continuing now for more than A > : WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in > : the foreseeable future. > The proof is convincing to those who understand the ideas and > de[CapitalThorn]nitions used by the proof. A common theme among the people > who argue with the proof is that they do not really understand > the basic ideas used in the proof. For example, [ ... ] Can you believe me if I tell you that I do really understand the basic ideas used in the proof and that, nevertheless, Im still reluctant to accept the consequences(?), like trans[CapitalThorn]nite cardinals and ordinals? > Cantors proof draws a lot of attention for the same reason > that Relativity does. It is fairly simple conceptually, > it is counter to common sense, and it relies on precise > de[CapitalThorn]nitions. Maybe that common sense could be translated as not measurable with any physical device. Which is quite a different matter for Relativity. The two are not comparable in that respect. Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? :> : I have always learned that a mathematical proof should be CONVINCING in :> : the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough :> : to prevent these heated discussions to be launched every time again. And :> : whats worse, these counter arguments are continuing now for more than A :> : WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in :> : the foreseeable future. :> The proof is convincing to those who understand the ideas and :> de[CapitalThorn]nitions used by the proof. A common theme among the people :> who argue with the proof is that they do not really understand :> the basic ideas used in the proof. For example, [ ... ] : Can you believe me if I tell you that I do really understand the basic : ideas used in the proof and that, nevertheless, Im still reluctant to : accept the consequences(?), like trans[CapitalThorn]nite cardinals and ordinals? Even if I believe you, you are just one person. You are claiming that the repeated discussions about Cantor are evidence that their must be something wrong with Cantor, but from what I have seen most of the people disputing Cantor really do not understand it. But given that you elsewhere have claimed that every number is computable, I do not think you do really understand the ideas used in Cantors proof. :> Cantors proof draws a lot of attention for the same reason :> that Relativity does. It is fairly simple conceptually, :> it is counter to common sense, and it relies on precise :> de[CapitalThorn]nitions. : Maybe that common sense could be translated as not measurable with : any physical device. Which is quite a different matter for Relativity. : The two are not comparable in that respect. As many, if not more, people refuse to accept Relativity. The web is full of people denouncing relativity. The newsgroup sci.physics.relativity was created in order to draw off the relativity nay sayers from sci.physics. This has been going on for a century. According to your logic this is evidence that something must be wrong with relativity. Stephen === Subject: Re: Cantors diagonal proof wrong? > The newsgroup > sci.physics.relativity was created in order to draw off > the relativity nay sayers from sci.physics. This has been > going on for a century. Wow! Thats a pretty old newsgroup :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Cantors diagonal proof wrong? :> The newsgroup :> sci.physics.relativity was created in order to draw off :> the relativity nay sayers from sci.physics. This has been :> going on for a century. : Wow! Thats a pretty old newsgroup :-) That did not come out quite the way I intended. :) STephen === Subject: Re: Cantors diagonal proof wrong? ... > I have always learned that a mathematical proof should be CONVINCING in > the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough > to prevent these heated discussions to be launched every time again. And > whats worse, these counter arguments are continuing now for more than A > WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in > the foreseeable future. In that case there are a whole lot of theorems for which the proof is not convincing. Like impossibility of the trisection of an angle under some conditions. -- 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: Cantors diagonal proof wrong? > In that case there are a whole lot of theorems for which the proof is > not convincing. Like impossibility of the trisection of an angle under > some conditions. Hmm, yeah, right ... :-( Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > I have always learned that a mathematical proof should be CONVINCING in > the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing enough > to prevent these heated discussions to be launched every time again. And > whats worse, these counter arguments are continuing now for more than A > WHOLE CENTURY. Thus it is highly unlikely that they will be stopped in > the foreseeable future. Of course they wont. There will always be tireless cranks in mathematics, and misguided people who feel compelled for some reason to enter into prolonged discussions with them. === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > I have always learned that a mathematical proof should be CONVINCING in > the [CapitalThorn]rst place. Obviously, Cantors arguments are not convincing > enough to prevent these heated discussions to be launched every time > again. And whats worse, these counter arguments are continuing now for > more than A WHOLE CENTURY. Thus it is highly unlikely that they will be > stopped in the foreseeable future. > Of course they wont. There will always be tireless cranks in > mathematics, and misguided people who feel compelled for some reason > to enter into prolonged discussions with them. I personally dont see this thread being so heated. But, ignoring that.. Its not just the cranks that cause this. Its just the fact that we all die and the young must constantly be reeducated. However, I think the CONVINCING point above is a good one. The problem is that the result of the proof is so surprising, that it is not convincing. Yet, there is nothing else anyone here has been able to put forth to explain why something so surprising exists. All you have been able to do is say, we dont know why it exists, we just know that it does exist, and this fact has been well accepted and proven for a very long time now.. You dont need to go any further than that to accept it as fact in the [CapitalThorn]eld of mathematics. But we all end up dealing with the surprise in our own way none the less. Some deal with it by rejecting it and turning into a crank. But what if there was a logical and simple way to explain why such an unexpected thing existed in math? Might that not help reduce future discussions of the subject because you had made the argument more CONVINCING? I think there probably are ways to not only logically prove this fact, as Cantor did, but to logical explain why something so counter intuitive exists. And that is what I feel the need to search for. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > But what if there was a logical and simple way to explain why such an > unexpected thing existed in math? There is - I call it the Weak Mathropic Principle: The only thing that makes mathematics interesting is the fact that you never quite know what you are going to discover (despite setting out from some intuitively derived axioms, rather than exploring jungles or doing experiments). Therefore if there were no surprises in mathematics, no-one would do it, and we wouldnt even know the surprises were there. > .... Might that not help reduce future > discussions of the subject because you had made the argument more > CONVINCING? With luck it has. Meanwhile, you really ought to read up the story about Pythagoras and the realisation that rational numbers just arent enough. No doubt newsgroups of the time were full of cranks with subject lines like Pythagoras an irrational nut, Proof of root(2) = a/b, etc etc. Brian Chandler http://imaginatorium.org === Subject: Re: Cantors diagonal proof wrong? > If all youre interested in is the physical world, then you dont need > in[CapitalThorn]nite sets, let alone the real numbers. Huh? Would you like to argue that physics doesnt need the real numbers? Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > If all youre interested in is the physical world, then you dont > need in[CapitalThorn]nite sets, let alone the real numbers. > Huh? Would you like to argue that physics doesnt need the real numbers? Ok, I like a challenge. It is most certainly not yet proven, but there is (IMO) a very good chance that physics needs integers, number theory, combinatorial analysis, statistics, set theory, group theory and various other branches of discrete mathematics but does not need real numbers for anything other than computational convenience. It is entirely possible, IMO, that spacetime is not continuous but quantized and that the universe is an automaton (whether [CapitalThorn]nite or in[CapitalThorn]nite I leave open) which evolves in discrete steps at discrete locations. Something like a giant game of Life but with rules that will probably turn out to be rather more complex. In this model of physics, the universe may well be a Turing machine, with all the limits on computability and continuity that implies. Paul -- Hanging on in quiet desperation is the English way. The time is gone, the song is over. Thought Id something more to say. === Subject: Re: Cantors diagonal proof wrong? > Ok, I like a challenge. Me too. See below. > It is most certainly not yet proven, but there is (IMO) a very good > chance that physics needs integers, number theory, combinatorial > analysis, statistics, set theory, group theory and various other > branches of discrete mathematics but does not need real numbers for > anything other than computational convenience. > It is entirely possible, IMO, that spacetime is not continuous but > quantized and that the universe is an automaton (whether [CapitalThorn]nite or > in[CapitalThorn]nite I leave open) which evolves in discrete steps at discrete > locations. Something like a giant game of Life but with rules that > will probably turn out to be rather more complex. In this model of > physics, the universe may well be a Turing machine, with all the > limits on computability and continuity that implies. Its evident from your argument that you have not a well founded idea about Applied Science and how it works. If you think that continuity means not discrete there, then you are simply mistaken. For example, in ßuid dynamics everything is built up from molecules. Yet everybody in that discipline is working with partial differential equations, as if the discrete substrate didnt exist at all! The secret behind this is the phenomenon that a continuum, in physics, is characterized by the fact that its numbers are inaccurate, thus blurring everything which is below a certain level of perception. One of the most striking examples in this context has been the discovery of the so-called Fluid-Tube Continuum. The idea behind this comes from the classical theory of Porous Media. It is virtually impossible, namely to apply the original Navier-Stokes / Heat Transfer equations, together with their boundary conditions, to a truly detailed model of the tubes in a heat exchanger. With help of the porous media theory, though, tube bundles become amenable to mathematical treatment. The trick is that all tubes are blurred in such a way that they arent even visible anymore. Resulting in truly continuous models for the governing equations, which actually become partial differential equations, as usual. Yeah, and then the numerical method comes in. And the continuized PD equations must be discretized again. If you are interested, here is more about it: http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#cz In conclusion about your statement that real numbers may not be needed in physics, because of the possible universal presence of a discrete substrate? Such a statement holds no water! It is merely an utterance of the inability to understand why and how continuity is employed in the applied sciences. Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > Ok, I like a challenge. > Me too. See below. > It is most certainly not yet proven, but there is (IMO) a very good > chance that physics needs integers, number theory, combinatorial > analysis, statistics, set theory, group theory and various other > branches of discrete mathematics but does not need real numbers for > anything other than computational convenience. [My second para deleted because you didnt respond directly to it.] > Its evident from your argument that you have not a well founded idea > about Applied Science and how it works. If you think that continuity Perhaps so. I am (or rather was) but a mere spectroscopist and not an Applied Scientist. > means not discrete there, then you are simply mistaken. For example, > in ßuid dynamics everything is built up from molecules. Yet everybody > in that discipline is working with partial differential equations, as > if the discrete substrate didnt exist at all! The secret behind this > is the phenomenon that a continuum, in physics, is characterized by the > fact that its numbers are inaccurate, thus blurring everything which > is below a certain level of perception. However, I was very careful to use the phrase computational convenience in my argument. [ Excellent example of computational convenience deleted.] > In conclusion about your statement that real numbers may not be needed > in physics, because of the possible universal presence of a discrete > substrate? Such a statement holds no water! It is merely an utterance > of the inability to understand why and how continuity is employed in > the applied sciences. Having had several years of exposure to chemistry at university level, I am well aware that using quantum [CapitalThorn]eld theory to calculate all but a very tiny proportion of the macroscopic properties of matter is intractable. Nonetheless, I see no reason why in principle it should not be possible. Paul -- Hanging on in quiet desperation is the English way. The time is gone, the song is over. Thought Id something more to say. === Subject: Re: Cantors diagonal proof wrong? > However, I was very careful to use the phrase computational > convenience in my argument. > [ Excellent example of computational convenience deleted.] Ah, I see, you want to play it that way ... Hmm, what is the difference then between computational convenience and a theory of some sort? >> [ .. ] the inability to understand why and how continuity is employed >> in the applied sciences. > Having had several years of exposure to chemistry at university level, > I am well aware that using quantum [CapitalThorn]eld theory to calculate all but a > very tiny proportion of the macroscopic properties of matter is > intractable. Nonetheless, I see no reason why in principle it should > not be possible. I didnt mean to make suggestions about *your* inability, eventually. I have no doubts about the adequacy of your scienti[CapitalThorn]c background, not anymore :-) I still have problems, though, with that in principle. I would like to defend the statement that *discrete* mathematics will *never* be appropriate as the sole vehicle for the applied sciences, not even in principle. How, for example, would you ever describe a ßuid in an adequate manner, with only such means at your disposal? And, while working with such discrete stuff, have you ever been exposed to a combinatorial explosion? (As a chemist, you should have experience with explosions :-)) Even with moderate discretizations, they regularly occur. I have plenty of examples. How would you deal with them? As it is well known that no computer in the whole universe can ever be made fast enough to resolve (many of) them. Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > It is most certainly not yet proven, but there is (IMO) a very good > chance that physics needs integers, number theory, combinatorial > analysis, statistics, set theory, group theory and various other > branches of discrete mathematics but does not need real numbers for > anything other than computational convenience. Solomon Feferman believes that most of the mathematics involved in physical theories can be formalized in a weak type theory W that is conservative over PA. But of course the possibility of formalization in weak theories is one thing, and what is needed in a practical sense another. === Subject: Re: Cantors diagonal proof wrong? > [CapitalThorn]rst obvious. It sounded very logical and I quickly embraced it as fact. > Lately however, Ive come to see things very differently. I now belief the > proof is totally bogus. And the huge body of work built on top of the > concept is likewise, totally bogus. If an in[CapitalThorn]nite number of people toss a coin in[CapitalThorn]nite times each, is it possible for you to toss a coin in[CapitalThorn]nite times in a new sequence? Of course not, thats the problem with uncountable in[CapitalThorn]nity, unlike other branches of mathematics the proofs dont support one another, they just all make the same mistake. Herc === Subject: Re: Cantors diagonal proof wrong? Discussion, linux) >> [CapitalThorn]rst obvious. It sounded very logical and I quickly embraced it as fact. >> Lately however, Ive come to see things very differently. I now belief the >> proof is totally bogus. And the huge body of work built on top of the >> concept is likewise, totally bogus. > If an in[CapitalThorn]nite number of people toss a coin in[CapitalThorn]nite times each, is it > possible for you to toss a coin in[CapitalThorn]nite times in a new sequence? Of > course not... Of course yes. It is possible that the outcome of your experiment is this: Person Coin sequence 1 HTTTTTTTTTTTTTT... 2 HHTTTTTTTTTTTTT... 3 HHHTTTTTTTTTTTT... 4 HHHHTTTTTTTTTTT... ... n HH..HTTTTTTTTTT... ___/ n heads, followed only by tails. There are an awful lot of sequences that are not represented in this list. > [...] thats the problem with uncountable in[CapitalThorn]nity, unlike other > branches of mathematics the proofs dont support one another, they > just all make the same mistake. Thats the problem with Hercs proofs. They always make the same mistake of substituting bald assertion (Of course not...) for justi[CapitalThorn]cation. -- Jesse F. Hughes She testi[CapitalThorn]ed they had sex near the Oval Of[CapitalThorn]ce, not in the famous room itself, because that `wouldnt be appropriate, you know. === Subject: Re: Cantors diagonal proof wrong? >> [CapitalThorn]rst obvious. It sounded very logical and I quickly embraced it as fact. >> >> Lately however, Ive come to see things very differently. I now belief the >> proof is totally bogus. And the huge body of work built on top of the >> concept is likewise, totally bogus. >> > If an in[CapitalThorn]nite number of people toss a coin in[CapitalThorn]nite times each, is it > possible for you to toss a coin in[CapitalThorn]nite times in a new sequence? Of > course not... > Of course yes. > It is possible that the outcome of your experiment is this: > Person Coin sequence > 1 HTTTTTTTTTTTTTT... > 2 HHTTTTTTTTTTTTT... > 3 HHHTTTTTTTTTTTT... > 4 HHHHTTTTTTTTTTT... > ... > n HH..HTTTTTTTTTT... > ___/ > n heads, followed only by tails. > There are an awful lot of sequences that are not represented in this > list. > [...] thats the problem with uncountable in[CapitalThorn]nity, unlike other > branches of mathematics the proofs dont support one another, they > just all make the same mistake. > Thats the problem with Hercs proofs. They always make the same > mistake of substituting bald assertion (Of course not...) for > justi[CapitalThorn]cation. If that is the argument uncountable in[CapitalThorn]nity stands on its laughable. You are forgetting Hercs famous million monkeys making pi proof. proof a random sequence contains pi : probability of segment of length n contains [CapitalThorn]rst n digits of pi = 0.1^n. given any n, the probalility > 0 limit as number of segments -> oo of probability of atleast one segment equaling pi = 1 > rate measured. A sample frame rate is established and the output > of a digitised poisson distribution is recorded, 0 for no emission, > sample 1 0101010010010100101010110111010101010.. > sample 2 010101011010101010101010101010101010.. > sample 3 1110110101110101011101011010101101010.. > After several million samples, the variations in the [CapitalThorn]rst 15 > frames are all covered. 000000000000000101010101010... 000000000000001101010101010... 000000000000010101010101010... 000000000000011101010101010... .. 111111111111111101010101010... -----------------> all permutations the size of all permutations covered tends to oo as the size of the list -> oo. > You have INFINITE samples with radioactivity readouts. Are you > 100% certain you can [CapitalThorn]nd a new sequence of 1s and 0s? This is called strictness analysis. > 4 HHHHTTTTTTTTTTT... > ... > n HH..HTTTTTTTTTT... > ___/ > n heads, followed only by tails. This is another example. odd + odd = even even + even = even even + odd = odd Without checking the actual addition, is 487474874849587487481 + 4879487949874879498742 = 83737323837373737382 correct? the strictness analysis supporting argument of Cantor does NOT, I repead it does NOT work against all countable schemes, only trivial examples like 0.1, 0.2, 0.3, ..0.9, 0.01, 0.02, ...0.99, 0.001, 0.002. Get any UTM, can we get a UTM from the audience of sci.math please, now what real does UTM(1,0) output? what real comes next UTM(2,0)? what blazing freaking number are you possibly going to miss fool? Objectify numbers and in[CapitalThorn]nity and of course you will [CapitalThorn]nd *something* bigger, but theyre not numbers. Herc === Subject: Re: Cantors diagonal proof wrong? <87actino5i.fsf@phiwumbda.org> Discussion, linux) > [CapitalThorn]rst obvious. It sounded very logical and I quickly embraced it as fact. > > Lately however, Ive come to see things very differently. I now belief the > proof is totally bogus. And the huge body of work built on top of the > concept is likewise, totally bogus. > >> >> If an in[CapitalThorn]nite number of people toss a coin in[CapitalThorn]nite times each, is it >> possible for you to toss a coin in[CapitalThorn]nite times in a new sequence? Of >> course not... >> Of course yes. >> It is possible that the outcome of your experiment is this: >> Person Coin sequence >> 1 HTTTTTTTTTTTTTT... >> 2 HHTTTTTTTTTTTTT... >> 3 HHHTTTTTTTTTTTT... >> 4 HHHHTTTTTTTTTTT... >> ... >> n HH..HTTTTTTTTTT... >> ___/ >> n heads, followed only by tails. >> There are an awful lot of sequences that are not represented in this >> list. >> [...] thats the problem with uncountable in[CapitalThorn]nity, unlike other >> branches of mathematics the proofs dont support one another, they >> just all make the same mistake. >> Thats the problem with Hercs proofs. They always make the same >> mistake of substituting bald assertion (Of course not...) for >> justi[CapitalThorn]cation. > If that is the argument uncountable in[CapitalThorn]nity stands on its laughable. > You are forgetting Hercs famous million monkeys making pi proof. > proof a random sequence contains pi : > probability of segment of length n contains [CapitalThorn]rst n digits of pi = > 0.1^n. > given any n, the probalility > 0 > limit as number of segments -> oo of probability of atleast one > segment equaling pi = 1 That is no counterproof at all. In fact, it is not any proof at all, since it is mere assertion that the probability pi is in the set is 1. But *even if* this false statement was true, it would not prove that every sequence of heads and tails are produced by a countable set of coin ßippers. It would instead prove that the probability that any particular sequence was in the set was one, not that every sequence was in the set. More lame proof-by-Hercs-infallible-assertion. Youve replaced the [CapitalThorn]rst proof-by-assertion with a weaker but still lame proof-by-assertion. If its famous, then you should be mighty embarrassed. -- Jesse F. Hughes Knowing about logic is not the same as being in touch with reality. -- David Kastrup === Subject: Re: Cantors diagonal proof wrong? >> > [CapitalThorn]rst obvious. It sounded very logical and I quickly embraced it as fact. > > Lately however, Ive come to see things very differently. I now belief the > proof is totally bogus. And the huge body of work built on top of the > concept is likewise, totally bogus. > >> >> If an in[CapitalThorn]nite number of people toss a coin in[CapitalThorn]nite times each, is it >> possible for you to toss a coin in[CapitalThorn]nite times in a new sequence? Of >> course not... >> >> Of course yes. >> >> It is possible that the outcome of your experiment is this: >> >> Person Coin sequence >> 1 HTTTTTTTTTTTTTT... >> 2 HHTTTTTTTTTTTTT... >> 3 HHHTTTTTTTTTTTT... >> 4 HHHHTTTTTTTTTTT... >> ... >> n HH..HTTTTTTTTTT... >> ___/ >> n heads, followed only by tails. >> >> There are an awful lot of sequences that are not represented in this >> list. >> >> [...] thats the problem with uncountable in[CapitalThorn]nity, unlike other >> branches of mathematics the proofs dont support one another, they >> just all make the same mistake. >> >> Thats the problem with Hercs proofs. They always make the same >> mistake of substituting bald assertion (Of course not...) for >> justi[CapitalThorn]cation. > If that is the argument uncountable in[CapitalThorn]nity stands on its laughable. > You are forgetting Hercs famous million monkeys making pi proof. > proof a random sequence contains pi : > probability of segment of length n contains [CapitalThorn]rst n digits of pi = > 0.1^n. > given any n, the probalility > 0 > limit as number of segments -> oo of probability of atleast one > segment equaling pi = 1 > That is no counterproof at all. In fact, it is not any proof at all, > since it is mere assertion that the probability pi is in the set is > 1. > But *even if* this false statement was true, it would not prove that > every sequence of heads and tails are produced by a countable set of > coin ßippers. It would instead prove that the probability that any > particular sequence was in the set was one, not that every sequence > was in the set. > More lame proof-by-Hercs-infallible-assertion. Youve replaced the > [CapitalThorn]rst proof-by-assertion with a weaker but still lame > proof-by-assertion. > If its famous, then you should be mighty embarrassed. Im the one putting mathematics to paper here, [CapitalThorn]nd a fault in my proof, actually back up YOUR negative presumptuous assertions HYPOCRITE! > proof a random sequence contains pi : > probability of segment of length n contains [CapitalThorn]rst n digits of pi = > 0.1^n. > given any n, the probalility > 0 > limit as number of segments -> oo of probability of atleast one > segment equaling pi = 1 Which clearly delineated step are you whining about? we can all assume everything you argue for which is in text books must be right and everything people prove that you assert is wrong must be faulty *somewhere* but this is sci.math, back up your assertion or shut up. TO THE POINT >> If an in[CapitalThorn]nite number of people toss a coin in[CapitalThorn]nite times each, is it >> possible for you to toss a coin in[CapitalThorn]nite times in a new sequence? Of >> course not... >> >> Of course yes. >> >> It is possible that the outcome of your experiment is this: >> You missed the implied RANDOM toss, this implies any speci[CapitalThorn]c structured result is incorrect. Make it simpler for you. In[CapitalThorn]nite people RANDOMLY toss a coin 3 times. Is it possible for you to toss a coin 3 times in a new sequence? OBVIOUSLY NOT Herc === Subject: Re: Cantors diagonal proof wrong? <87actino5i.fsf@phiwumbda.org> <87hdnpvu7p.fsf@phiwumbda.org> Discussion, linux) >> More lame proof-by-Hercs-infallible-assertion. Youve replaced the >> [CapitalThorn]rst proof-by-assertion with a weaker but still lame >> proof-by-assertion. >> If its famous, then you should be mighty embarrassed. > Im the one putting mathematics to paper here, [CapitalThorn]nd a fault in my proof, > actually back up YOUR negative presumptuous assertions HYPOCRITE! >> proof a random sequence contains pi : >> >> probability of segment of length n contains [CapitalThorn]rst n digits of pi = >> 0.1^n. >> given any n, the probalility > 0 >> limit as number of segments -> oo of probability of atleast one >> segment equaling pi = 1 > Which clearly delineated step are you whining about? None of this makes any particular sense to me, aside from the possible exception of the [CapitalThorn]rst statement. But just to be precise, where did that limit as number... claim come from? It doesnt follow anything above at all. > It is possible that the outcome of your experiment is this: > > You missed the implied RANDOM toss, this implies any > speci[CapitalThorn]c structured result is incorrect. My outcome is a logical possibility given random tosses. It isnt Chaitin-random, but Chaitin-random has nothing to do with what is a possible outcome of a sequence of random events. Indeed, the outcome I gave >> Person Coin sequence >> 1 HTTTTTTTTTTTTTT... >> 2 HHTTTTTTTTTTTTT... >> 3 HHHTTTTTTTTTTTT... >> 4 HHHHTTTTTTTTTTT... >> ... >> n HH..HTTTTTTTTTT... >> ___/ >> n heads, followed only by tails. is not more or less likely than *any* other outcome of your thought experiment. > Make it simpler for you. In[CapitalThorn]nite people RANDOMLY toss a coin > 3 times. Is it possible for you to toss a coin 3 times in a new > sequence? > OBVIOUSLY NOT Obviously. It is perfectly possible that the *every* person that tossed his coin three times got heads every toss. That is a logical possibility and indeed that outcome isnt more or less likely than any other outcome. Note: The probability that your toss matches another toss on the list is 1. I dont dispute that --- but it is nonetheless possible that your outcome is not on the list. And the difference between three tosses and countably many tosses is relevant, too. The set of all sequences of {H,T} of length 3 is countable. The set of all in[CapitalThorn]nite sequences of {H,T} is uncountable. -- Jesse F. Hughes [Lancelot] sighed, defeated. ÔIt is as practical to hurry an acorn toward treeness as to urge a damsel when her mind is set. -- John Steinbeck, /The Acts of King Arthur and His Noble Knights/ === Subject: Re: Cantors diagonal proof wrong? >> >> More lame proof-by-Hercs-infallible-assertion. Youve replaced the >> [CapitalThorn]rst proof-by-assertion with a weaker but still lame >> proof-by-assertion. >> >> If its famous, then you should be mighty embarrassed. > Im the one putting mathematics to paper here, [CapitalThorn]nd a fault in my proof, > actually back up YOUR negative presumptuous assertions HYPOCRITE! >> proof a random sequence contains pi : >> >> probability of segment of length n contains [CapitalThorn]rst n digits of pi = >> 0.1^n. >> given any n, the probalility > 0 >> limit as number of segments -> oo of probability of atleast one >> segment equaling pi = 1 > Which clearly delineated step are you whining about? > None of this makes any particular sense to me, aside from the possible > exception of the [CapitalThorn]rst statement. But just to be precise, where did > that limit as number... claim come from? It doesnt follow anything > above at all. >> probability of segment of length n contains [CapitalThorn]rst n digits of pi = >> 0.1^n. >> given any n, the probalility > 0 1/ forall n, P(random digit sequence of length n = 1st n digits of pi) = 0.1^n 2/ forall n, P(random digit sequence of length n = 1st n digits of pi) > 0 Do you follow step 2 now? Countable in[CapitalThorn]nity logic must be painful to comprehend for people here, takes a bit of prodding. > It is possible that the outcome of your experiment is this: > > You missed the implied RANDOM toss, this implies any > speci[CapitalThorn]c structured result is incorrect. > My outcome is a logical possibility given random tosses. It isnt > Chaitin-random, but Chaitin-random has nothing to do with what is a > possible outcome of a sequence of random events. Indeed, the outcome > I gave >> Person Coin sequence >> 1 HTTTTTTTTTTTTTT... >> 2 HHTTTTTTTTTTTTT... >> 3 HHHTTTTTTTTTTTT... >> 4 HHHHTTTTTTTTTTT... >> ... >> n HH..HTTTTTTTTTT... >> ___/ >> n heads, followed only by tails. > is not more or less likely than *any* other outcome of your thought > experiment. > Make it simpler for you. In[CapitalThorn]nite people RANDOMLY toss a coin > 3 times. Is it possible for you to toss a coin 3 times in a new > sequence? > OBVIOUSLY NOT > Obviously. It is perfectly possible that the *every* person that > tossed his coin three times got heads every toss. That is a logical > possibility and indeed that outcome isnt more or less likely than any > other outcome. > Note: The probability that your toss matches another toss on the list > is 1. I dont dispute that --- but it is nonetheless possible that > your outcome is not on the list. > And the difference between three tosses and countably many tosses is > relevant, too. The set of all sequences of {H,T} of length 3 is > countable. The set of all in[CapitalThorn]nite sequences of {H,T} is uncountable. So if an in[CapitalThorn]nite number of people all tossed a coin, either heads or tails, you claim its possible to [CapitalThorn]nd a new undiscovered side to the coin? And this is the exact logic you behold to ascertain the existence of higher in[CapitalThorn]nities is it not? Monte Carlo does play cards. So is Cantors diag argument reducable to this. Assume the real list is now a particular structured list we can choose: Real 1 : 0.1 Real 2 : 0.01 Real 3 : 0.001 ... Hence we have hyper in[CapitalThorn]nities and super hyper in[CapitalThorn]nities only us graduates can comprehend, we get all numbers to in[CapitalThorn]nity and change every digit to make a new number, super duper triple looper in[CapitalThorn]nity for the super iq elite, if you dont agree you must be stupid, even if every possible digit sequence still appears on the list of computable numbers?? Why dont you use a suitable random generator like the digit sequence of sqrt(2) to remove the logical absurdity of 0 probability you are forced to introduce. Herc A million monkeys cooking in a million kitchens, can one of them make pi? === Subject: Re: Cantors diagonal proof wrong? <87actino5i.fsf@phiwumbda.org> <87hdnpvu7p.fsf@phiwumbda.org> <878y8y89au.fsf@phiwumbda.org> Discussion, linux) > probability of segment of length n contains [CapitalThorn]rst n digits of pi = > 0.1^n. >> > given any n, the probalility > 0 > 1/ forall n, P(random digit sequence of length n = 1st n digits of pi) > = 0.1^n > -> > 2/ forall n, P(random digit sequence of length n = 1st n digits of pi) > > 0 > Do you follow step 2 now? No. How does it follow? Lets abbreviate (1) by saying P(f(n)) = 0.1^n, so that (2) asserts P(f(oo)) > 0. How does (2) follow from (1)? If I were to guess, Id guess that P(f(oo)) = lim_{n -> oo} P(f(n)). But, of course, the fact that P(f(n)) > 0 for all n does *not* imply that lim_{n -> oo} P(f(n)) > 0. Indeed, anyone that groks beginner calculus can see that in this case, the limit is 0. > Countable in[CapitalThorn]nity logic must be painful to comprehend for people > here, takes a bit of prodding. Herc, your paranoid delusions regarding The Truman Show are much more plausible than your mathematical proofs. >> And the difference between three tosses and countably many tosses is >> relevant, too. The set of all sequences of {H,T} of length 3 is >> countable. The set of all in[CapitalThorn]nite sequences of {H,T} is uncountable. > So if an in[CapitalThorn]nite number of people all tossed a coin, either heads or > tails, you claim its possible to [CapitalThorn]nd a new undiscovered side to the > coin? No. I claim that its logically possible that all of the tosses come up heads, regardless of how improbable it is. -- Jesse F. Hughes To be honest, I dont have enough interest in math to spend the time it would take to clean up the mess that I believe has been created in the past 100 or so years. -- Curt Welch lets the world down. === Subject: Re: Cantors diagonal proof wrong? <87actino5i.fsf@phiwumbda.org> <87hdnpvu7p.fsf@phiwumbda.org> <878y8y89au.fsf@phiwumbda.org> <878y8wqt8c.fsf@phiwumbda.org> posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L >> 1/ forall n, P(random digit sequence of length n = 1st n digits of pi) >> = 0.1^n >> -> >> 2/ forall n, P(random digit sequence of length n = 1st n digits of pi) >> 0 >> Do you follow step 2 now? >No. then you just publicly declared you are an idiot. is your uncountable supremecy nonsense so fallible you have to outright ignore anything that outlines your erronous stance. 1/ forall n, X = 0.1^n 2/ forall n, X > 0 make your inferences out of. BTW: if thats famous you should be embarrased is a Truman Company comment, think about it, it was obviously (from my point of view) implanted in your head. Herc === Subject: Re: Cantors diagonal proof wrong? [...] > 1/ forall n, P(random digit sequence of length n = 1st n digits of pi) > = 0.1^n > -> > 2/ forall n, P(random digit sequence of length n = 1st n digits of pi) > > 0 > Do you follow step 2 now? Countable in[CapitalThorn]nity logic must be painful to > comprehend for people here, takes a bit of prodding. [...] > A million monkeys cooking in a million kitchens, > can one of them make pi? Suppose one has monkey no. 1, monkey no.2, and so on: one monkey per positive integer. Suppose monkey no.1 types 1 digit/second, monkey no. 2 types 2 digits/second, and monkey no. k types k digits/second. After one second, monkey no. k has typed k digits. The probability that all k digits are the same as the k [CapitalThorn]rst digits after the decimal point in Pi/10 [0.314159...] is 1/(10^k). Monkey no. 1 has a 1/10 chance of getting all his digits right after a second, monkey no. 2 has a chance of 1/100 of getting his two digits right after a second and so on. Because the random digits typed by distinct monkeys are independent, on average, after one second, 1/10 + 1/100 + 1/1000 + .... = 0.1111111111... = 1/9 monkeys will have all their digits correct after one second. Now we look at the situation after 2 seconds: monkey no. 1 has typed 2 digits , monkey no. 2 has typed 4 digits and so on. Therefore on average 1/100 + 1/10000 + 1/(100^3) + ... = 0.01010101... = 1/99 monkeys will have all their digits right after two seconds. After 3 seconds, 0.001001001001001 ... = 1/999 monkeys will have their digits right. Continuing in this way, after 60 seconds, only 1/999999999999999999999999999999999999999999999999999999999999 monkeys will have all their digits correct on average. So as each additional second passes, the average number of monkeys having all their digits correct is divided by a factor just a bit larger than 10. So after a long enough time, the average number of monkeys having all their digits correct will be less than 1/(10^googol), i.e. less than 1/(10^(10^100)). Also note that the average goes down as time increases... David Bernier === Subject: Re: Cantors diagonal proof wrong? <87actino5i.fsf@phiwumbda.org> <87hdnpvu7p.fsf@phiwumbda.org> <878y8y89au.fsf@phiwumbda.org> posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L you could always throw in more monkeys? how about this.. a googol of monkeys at a googol of typewriters, one of them will come up with the 1st million digits of pi! Herc === Subject: Re: Cantors diagonal proof wrong? <87actino5i.fsf@phiwumbda.org> <87hdnpvu7p.fsf@phiwumbda.org> <878y8y89au.fsf@phiwumbda.org> Discussion, linux) >> Make it simpler for you. In[CapitalThorn]nite people RANDOMLY toss a coin >> 3 times. Is it possible for you to toss a coin 3 times in a new >> sequence? >> OBVIOUSLY NOT > Obviously. It is perfectly possible that the *every* person that > tossed his coin three times got heads every toss. That is a logical > possibility and indeed that outcome isnt more or less likely than any > other outcome. > Note: The probability that your toss matches another toss on the list > is 1. I dont dispute that --- but it is nonetheless possible that > your outcome is not on the list. Im afraid this is likely nonsense. Probability distributions are supposed to satisfy countable additivity. The vaguely de[CapitalThorn]ned distribution that comes to my mind in this situation is not countably additive. I think that these thought experiments with an in[CapitalThorn]nite number of coin-ßippers do not work well with the countable additivity requirement. Probably if one wants to talk about these situations, one would want to relax that requirement. I dont know much about probabilities and corrections to my misconceptions are welcome. -- I am a force of Nature. Time is a friend of mine, and We talk about things, here and there. And sometimes We muse a bit [...] and then We watch them go... in the meantime, Time and I, We play with some of them, at least for a little while. --- JSH and His pal, Time. === Subject: Re: Cantors diagonal proof wrong? > There is no such thing in the physical world as even a natural number. > One, two, three, etc. are all entirely conceptual, not physical. > If you insist on physicality, give up mathematics. > I am exploring things that you believe do not exist. And your outlook is > not uncommon in the world. Its by far most common view all of mankind > seems to like to share. I dont think this is true at all. In my experience, if you ask a roomful of non-mathematicians whether i, the square root of -1, exists they will mostly claim it doesnt. Ask Does 3 exist? and overwhelmingly they respond that yes, it does. Then you can play an amusing little game, trying to establish where their particular boundaries between existing and not actually existing numbers lie. Virgil, of course, or any other mathematician, will just look a bit blank, and say What do you mean ÔExist?, but I think this is a minority view. Of course, mathematicians will hedge their response, because in some cases, your Exist? will obviously apply to mathematical existence, or not, as in four-sided triangles, even primes greater than 63, and so on. > Its that very fact that makes me at times, believe Ive found something > that has been missed for 100s of years. Matematics, by design, limits > its focus to a scope which does not include the things Im investigating. Ah! Two points. a) You have misspelled its, which is a pretty serious blunder. b) Youve also missed quite a lot that was found 100s of years ago. Several people answered your proof that the integers cannot be put in 1-1 correspondence with themselves (or something like that), but I think the answers were not terribly good (a bit Micro$oft-like, if you know the helicopter joke). You have a somewhat ill-de[CapitalThorn]ned set of objects, including the integers, 1, 2, 3, 57, 264, etc., and some things like ...11111; you havent really said quite what else. Which of these is one of your cwintegers (as Ill call them, in honour of yourself!): ...2121212121 * recurring decimals backwards ...5356295141 * fractional part of pi backwards -...1111111111 * cwintegers with a minus sign While you are free to [CapitalThorn]nish off the de[CapitalThorn]nition, so we know exactly what is and isnt a cwinteger, then you have to start doing some grunt work, proving results that suggest that your cwintegers are useful. The integers used by mathematicians have at least the property that they possess the qualities children learn about informally at a _very_ young age. In particular: You can always add, subtract, or multiply two numbers. You cant always divide, but if you cant, its because there is no answer at all, not because there are lots of answers. You can compare two numbers, and it works: if a>b, b 5 ? Is ...2222 < 5 ? Rather obviously, the subset of cwintegers (without minus signs) that have either an in[CapitalThorn]nite succession of 0s or 9s on the left, form a representation of the integers in 2s complement (idealised to in[CapitalThorn]nite registers). So algebra Ôworks within that subset, where anything starting ...999 is negative. But if you try to include ...222, how are you going to decide if its positive or negative? And so on (I should really read about p-adics some time, as perhaps so might you). As for all this stuff about AI, I recommend Daniel Dennetts book Darwins Dangerous Idea, particularly the section beginning with Chapter 15, The Emperors New Mind, and Other Fables. (He has a nice start to the second paragraph: What Goedels Theorem promise the romantically inclined...) Sort of review here: http://imaginatorium.org/books/dennett.htm - of course this is philosophy, and not to everyones taste, but personally I [CapitalThorn]nd Dennett rather readable. Good at demolishing other philosophers, if thats anything special... Anyway, I think a bit of reading would be a good idea, before you announce your Great Discovery. > You do not believe the conceptual world and the physical world are one > in the same. I do. And once you believe that, everything starts to get > very interesting, and everything starts to look very different. But does it make any sense? Do you mean that numbers (and things) Really Exist in the physical universe, for you? Including i? Brian Chandler === Subject: Re: Cantors diagonal proof wrong? > I dont think this is true at all. In my experience, if you ask a > roomful of non-mathematicians whether i, the square root of -1, > exists they will mostly claim it doesnt. Nonsense. The imaginary number i is 1 turned 90 degrees counterclockwise on the complex plane. It has as much existence as any integer or real number. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > There is no such thing in the physical world as even a natural > number. One, two, three, etc. are all entirely conceptual, not > physical. If you insist on physicality, give up mathematics. > I am exploring things that you believe do not exist. And your outlook > is not uncommon in the world. Its by far most common view all of > mankind seems to like to share. > I dont think this is true at all. In my experience, if you ask a > roomful of non-mathematicians whether i, the square root of -1, > exists they will mostly claim it doesnt. Ask Does 3 exist? and > overwhelmingly they respond that yes, it does. Then you can play an > amusing little game, trying to establish where their particular > boundaries between existing and not actually existing numbers lie. Yeah, you are probably right. I have not tried to play that game with people. Im glad you at least see the issue Im getting at. People use many words like exist without trying to understand or justify what they really mean. They simply, like parrots, use them correctly like they have been taught. Most of what we all do works like that. And for sure, what I write is full of that, as you go on to demonstrate below. > Virgil, of course, or any other mathematician, will just look a bit > blank, and say What do you mean ÔExist?, but I think this is a > minority view. Of course, mathematicians will hedge their response, > because in some cases, your Exist? will obviously apply to > mathematical existence, or not, as in four-sided triangles, even > primes greater than 63, and so on. > Its that very fact that makes me at times, believe Ive found > something that has been missed for 100s of years. Matematics, by > design, limits its focus to a scope which does not include the things > Im investigating. > Ah! Two points. > a) You have misspelled its, which is a pretty serious blunder. Grammatical and spelling errors get much worse than that in my writing. :) > b) Youve also missed quite a lot that was found 100s of years ago. > Several people answered your proof that the integers cannot be put in > 1-1 correspondence with themselves (or something like that), but I > think the answers were not terribly good (a bit Micro$oft-like, if you > know the helicopter joke). You have a somewhat ill-de[CapitalThorn]ned set of > objects, including the integers, 1, 2, 3, 57, 264, etc., and some > things like ...11111; you havent really said quite what else. Which > of these is one of your cwintegers (as Ill call them, in honour of > yourself!): Yes, this sounds like fun. What was I thinking an integer was? Well, for starters, it clearly included the set of all numbers which anyone would call an integer. And that includes all the normal well known properties of said numbers. But then I called the in[CapitalThorn]nite string of digits which could be constructed from the diagonal of my table, an integer and thats where the trouble started. > ...2121212121 * recurring decimals backwards > ...5356295141 * fractional part of pi backwards > -...1111111111 * cwintegers with a minus sign Let me just for fun (not to try and [CapitalThorn]x my proof), just try to formalize what I was thinking with ...11111 Well, its the same thing as the in[CapitalThorn]nite series 1*10^0 + 1*10^2 + 1*10^3 + .... Now, this is not incompatible with the integers in that you can simply substitute 0s for all the leading numbers. > While you are free to [CapitalThorn]nish off the de[CapitalThorn]nition, so we know exactly > what is and isnt a cwinteger, then you have to start doing some grunt > work, proving results that suggest that your cwintegers are useful. > The integers used by mathematicians have at least the property that > they possess the qualities children learn about informally at a _very_ > young age. In particular: > You can always add, subtract, or multiply two numbers. You cant > always divide, but if you cant, its because there is no answer at > all, not because there are lots of answers. You can compare two > numbers, and it works: if a>b, b are odd and even, and even+even=even. > And lots and lots more. Here are some simple questions about > cwintegers you should ask yourself: > 1 + 2 = ? > 3 - 2 = ? Those are obvious with cwints because cwints are the same as integers when they are [CapitalThorn]nite in size. > ...999 + 2 = ? You just substitute ...999 for the correct in[CapitalThorn]nite series, and you have your answer. however, the answer can not be translated back into this same notation, so a new notation must be created. 1...01. This is 1, with an in[CapitalThorn]nite string of leading zeros, and an in[CapitalThorn]nite carry of a 1 value. :) Now, I have to deal with how everything works with this new notation. 1...01 + 1 = 1...02 1...01 - 1 = 1...0 1...0 - 1 = ...999 And it continues to get more complex, which I will not work through right now. :) > 1 - 2 = ? > Is ...2222 > 5 ? Yes, clearly. > Is ...2222 < 5 ? No, clearly. But what about ...222 > ...111? Hard to say what to do about that. But it seems like you could just decide that the answer is true. The other option would be to declare it unde[CapitalThorn]ned. In either case, you would then have to work though what would either choice would mean for all the other operations. Yeah, clearly, I didnt grasp how much work has been done over 100 years ago. :) > As for all this stuff about AI, I recommend Daniel Dennetts book > Darwins Dangerous Idea, recommendation. I tend not to read much philosophy. Im an engineer. > Anyway, I think a bit of reading would be a good idea, before you > announce your Great Discovery. I [CapitalThorn]nd that to be true only to the extent of helping me to communicate with others. If the great Discovery is a thinking machine, I dont need to announce it. Ill let it do the announcing. :) > You do not believe the conceptual world and the physical world are > one in the same. I do. And once you believe that, everything starts > to get very interesting, and everything starts to look very different. > But does it make any sense? Do you mean that numbers (and things) > Really Exist in the physical universe, for you? Including i? Yes. But what is the physical universe and what does exist really mean? Thats where it gets interesting and thats why Im able to answer the question yes when others would like to say you cant answer it that way. In other words, the physical universe I know of is created by the behavior of my brain. Its all in my head. It is a purely subjective understanding of the universe. But in that virtual creation of my brain, is multiple representations of my body, and your body. And in our bodies, are brains, which are information processing machines, which create independent virtual views of the same physical universe. So where is the physical universe in this loop? Is it that stuff out there which we normal think about when we use those words, or is, as it must be, only the collection of ideas computed by my brain? The only things that actually exist to us is the electrical activity in our brain. If its not electrical activity in our brain, it doesnt exist to us. Everything that exists to us, from our own body, to the body of others, to numbers, exist as electrical activity in our brains. Electrical activity is the physical movement of electrons. Its physical stuff. Its my brain, and all the atoms and electrons in it, changing shape in response to its interaction with other matter in the universe. Thats what the number 1 is. Its not just an abstract idea which exists in some other dimension from the physical stuff. Ideas are the actions of physical stuff in this universe. So, there are many ways to talk about all this, and if you dont follow which way Im talking at the moment, its easy to get lost. So, everything we know about, and sense, from our physical sensations of the physical world, to all the sensations of the mental world, are created by electrons and chemicals ßowing in our brains. And the electrons that allows us to sense the idea of the imaginary number i, is just as real, as the electrons that allows me to sense the keyboard Im typing on. The physical world we are able to understand, is not out there, its in our head, along with the stuff that creates the idea of i. So our understanding of the physical world is just as real, as our understanding of a concept like i. And all this ability is built out of material from the physical world. > Brian Chandler -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? <9ZGdnRVWNYDZNQrcRVn-og@giganews.com> at 11:46 PM, curt@kcwc.com (Curt Welch) said: >Well, its the same thing as the in[CapitalThorn]nite series 1*10^0 + 1*10^2 + >1*10^3 + That doesnt answer the question. If i and j are cwintegers, how do you de[CapitalThorn]ne their sum, difference, product and quotient? How do you de[CapitalThorn]ne iThose are obvious with cwints No they are not, and it appears that you havent yet tried to de[CapitalThorn]ne them. >because There is no because. >cwints are the same as integers when they are [CapitalThorn]nite in size. K3wl. What about the ones that are not? >You just substitute ...999 for the correct in[CapitalThorn]nite series, The sum of two in[CapitalThorn]nite series is not always a cwinteger. In particular, the sum of the in[CapitalThorn]nite series ...9 and ...02 is ...9{11}, which is not a cwinteger. >You just substitute ...999 for the correct in[CapitalThorn]nite series, and you >have your answer. however, the answer can not be translated back >into this same notation, so a new notation must be created. 1...01. >This is 1, with an in[CapitalThorn]nite string of leading zeros, and an in[CapitalThorn]nite >carry of a 1 value. :) Thats not a cwinteger as you have de[CapitalThorn]ned it, nor is it an alternative de[CapitalThorn]nition. You need to give an explicit de[CapitalThorn]nition of a cwinteger, with no hand waving, if you expect to be taken seriously. >The only things that actually exist to us is the electrical activity >in our brain. No. The only things that exist for us are our perceptions; all else is inferred, including the existence of electrical activity. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantors diagonal proof wrong? > at 11:46 PM, curt@kcwc.com (Curt Welch) said: >Well, its the same thing as the in[CapitalThorn]nite series 1*10^0 + 1*10^2 + >1*10^3 + > That doesnt answer the question. If i and j are cwintegers, how do > you de[CapitalThorn]ne their sum, difference, product and quotient? How do you > de[CapitalThorn]ne iThose are obvious with cwints >You just substitute ...999 for the correct in[CapitalThorn]nite series, I never managed to understand this sentence. > The sum of two in[CapitalThorn]nite series is not always a cwinteger. In > particular, the sum of the in[CapitalThorn]nite series ...9 and ...02 is ...9{11}, > which is not a cwinteger. Well, since the OP clearly intended 8 to mean eight, and 7 to mean seven, then the sum is naturally de[CapitalThorn]ned using the standard carry. So 8+7 = 15 ...999 + ...0002 = ...0001 De[CapitalThorn]nitions of multiplication and subtraction follow automatically (just imagine an inde[CapitalThorn]nitely long mechanical adding machine decimal register). Division (I think) has to be de[CapitalThorn]ned by p * x = y => x / y = p This is only partially de[CapitalThorn]ned (cant do 2/5 for example); Im not sure that it is always uniquely de[CapitalThorn]ned. I tried to [CapitalThorn]nd zero divisors and failed, which probably means someone cleverer will have to do it. Am I right, though, in saying that it is clear that these things at least form a ring? Brian Chandler http://imaginatorium.org === Subject: Re: Cantors diagonal proof wrong? <9ZGdnRVWNYDZNQrcRVn-og@giganews.com> <419ad65b$5$fuzhry+tra$mr2ice@news.patriot.net> >Well, since the OP clearly intended 8 to mean eight, and 7 to mean >seven, then the sum is naturally de[CapitalThorn]ned using the standard carry. The standard carry is only de[CapitalThorn]ned for decimal expansions of integers. It is not de[CapitalThorn]ned for in[CapitalThorn]nite sequences of integers. You can provide a de[CapitalThorn]nition, but if you do it will break some expected properties of the integers. >Am I right, though, in saying that it is clear that these things at >least form a ring? Not unless you de[CapitalThorn]ne + and * in such a way that the ring properties hold. In fact, it will take some work to even make them form a group. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantors diagonal proof wrong? >> at 11:46 PM, curt@kcwc.com (Curt Welch) said: >>Well, its the same thing as the in[CapitalThorn]nite series 1*10^0 + 1*10^2 + >>1*10^3 + >> That doesnt answer the question. If i and j are cwintegers, how do >> you de[CapitalThorn]ne their sum, difference, product and quotient? How do you >> de[CapitalThorn]ne i>Those are obvious with cwints > >>You just substitute ...999 for the correct in[CapitalThorn]nite series, > I never managed to understand this sentence. >> The sum of two in[CapitalThorn]nite series is not always a cwinteger. In >> particular, the sum of the in[CapitalThorn]nite series ...9 and ...02 is ...9{11}, >> which is not a cwinteger. > Well, since the OP clearly intended 8 to mean eight, and 7 to mean > seven, then the sum is naturally de[CapitalThorn]ned using the standard carry. So > 8+7 = 15 > ...999 + ...0002 = ...0001 > De[CapitalThorn]nitions of multiplication and subtraction follow automatically > (just imagine an inde[CapitalThorn]nitely long mechanical adding machine decimal > register). > Division (I think) has to be de[CapitalThorn]ned by > p * x = y => x / y = p > This is only partially de[CapitalThorn]ned (cant do 2/5 for example); Im not > sure that it is always uniquely de[CapitalThorn]ned. I tried to [CapitalThorn]nd zero divisors > and failed, which probably means someone cleverer will have to do it. > Am I right, though, in saying that it is clear that these things at > least form a ring? They form the 10-adic integers: an obvious generalization of the p-adic integers. As far as I know the text by Mahler on p-adic numbers is the only book that deals with n-adics on the same basis as p-adics. But little is gained, for instance the 10-adics are the direct product of the 2-adics and 5-adics, so they have nontrivial idempotents (which are zero-dvisors :-) ). (Hint: solve the simultaneous congruences x = 1 (mod 2^n), x = 0 (mod 5^n) .) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > Thats not a cwinteger as you have de[CapitalThorn]ned it, nor is it an > alternative de[CapitalThorn]nition. You need to give an explicit de[CapitalThorn]nition of a > cwinteger, with no hand waving, if you expect to be taken seriously. I think you missed the whole point of that section of my post. I was de[CapitalThorn]ning cwints as a joke, or just for the fun of it. It was just me laughing at the idea of ...1111 being in integer. I was not expecting anyone to take me serious on the subject of cwints. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > > There is no such thing in the physical world as even a natural > number. One, two, three, etc. are all entirely conceptual, not > physical. If you insist on physicality, give up mathematics. > > I am exploring things that you believe do not exist. And your outlook > is not uncommon in the world. Its by far most common view all of > mankind seems to like to share. > I dont think this is true at all. In my experience, if you ask a > roomful of non-mathematicians whether i, the square root of -1, > exists they will mostly claim it doesnt. Ask Does 3 exist? and > overwhelmingly they respond that yes, it does. Then you can play an > amusing little game, trying to establish where their particular > boundaries between existing and not actually existing numbers lie. > Yeah, you are probably right. I have not tried to play that game with > people. > Im glad you at least see the issue Im getting at. People use many words > like exist without trying to understand or justify what they really mean. > They simply, like parrots, use them correctly like they have been taught. I dont think this is right, either. All this stuff about numbers existing or not existing isnt anything anyone _learnt_, anywhere. No maths textbook would go on about i Ôexisting, since it doesnt really mean anything, as we seem to agree. The bits that people parrot are things like Any polynomial equation can be solved with enough work, or if I could remember the formula. > Yes, this sounds like fun. What was I thinking a [cw]integer was? > Well, for starters, it clearly included the set of all numbers which anyone > would call an integer. And that includes all the normal well known > properties of said numbers. > But then I called the in[CapitalThorn]nite string of digits which could be constructed > from the diagonal of my table, an integer and thats where the trouble > started. Certainly did. Try this for a proof: Theorem There are an in[CapitalThorn]nite number of people in the world. Proof Well, suppose there were a [CapitalThorn]nite number. Then we could write all their names on a large enough piece of paper, and the list would be complete. But then we could write Chocolate sundae under the last persons name, and given that we regard Chocolate sundae as a person for the purposes of the argument, this shows the list was incomplete. Contradiction. For reasons I cant quite place, your argument reminds me of what must be one of the battiest academic books ever written - The vastness of natural language, by Langendoen and Postal (both Ôproper linguists, btw) pub. Blackwell. Their argument more or less begins with the claim that the English language (for that is the one linguists study) includes in[CapitalThorn]nite objects (sentences). I suppose its the way you keep adding bits in to your set of integers each time you consider the next problem. You might actually [CapitalThorn]nd Chomskys ideas on things like this interesting. If you could see why some of them are a bit loopy, it might help with reconsidering your own ideas. Chomsky asserts that one of the pieces of primitive biology (or was it physics?) in the brain is a mechanism for grokking countable in[CapitalThorn]nity, which he claims (without any obvious evidence) is something babies do without needing to learn about it. > ...2121212121 * recurring decimals backwards > ...5356295141 * fractional part of pi backwards > -...1111111111 * cwintegers with a minus sign > Let me just for fun (not to try and [CapitalThorn]x my proof), just try to formalize > what I was thinking with ...11111 > Well, its the same thing as the in[CapitalThorn]nite series 1*10^0 + 1*10^2 + 1*10^3 + > .... > Now, this is not incompatible with the integers in that you can simply > substitute 0s for all the leading numbers. Yes, but in some reasonable sense, chocolate sundaes are not incompatible with integers. That isnt enough. > ...999 + 2 = ? > You just substitute ...999 for the correct in[CapitalThorn]nite series, and you have > your answer. however, the answer can not be translated back into this same > notation, so a new notation must be created. 1...01. This is 1, with an > in[CapitalThorn]nite string of leading zeros, and an in[CapitalThorn]nite carry of a 1 value. :) Ho hum. Well, had you stopped with the idea of an idealised Ôin[CapitalThorn]nite-bit register, holding positive integers with 000 to the left, and negative ones with 999, chosen 2s complement (or 1s complement, if you prefer), you could have gone off and looked at allowing other inde[CapitalThorn]nite leftward continuations. I dont know quite what you get, but at least its presumably possible to de[CapitalThorn]ne addition and multiplication. > Now, I have to deal with how everything works with this new notation. > 1...01 + 1 = 1...02 > 1...01 - 1 = 1...0 > 1...0 - 1 = ...999 And ...999 * 1...0 ? Perhaps ...999...0 (One could, btw, knock up a quick crank [CapitalThorn]lter, by writing a regular expression for ... followed by [digits] followed by ....) > And it continues to get more complex, which I will not work through right > now. :) > 1 - 2 = ? > Is ...2222 > 5 ? > Yes, clearly. > Is ...2222 < 5 ? > No, clearly. > But what about ...222 > ...111? > Hard to say what to do about that. But it seems like you could just decide > that the answer is true. The other option would be to declare it > unde[CapitalThorn]ned. A better example would have been ...121212 and ...212121 > In other words, the physical universe I know of is created by the behavior > of my brain. Its all in my head. It is a purely subjective understanding > of the universe. ... Yeah, yeah, having tried empiricism and found it wanting, Ive switched to solipsism, but I cant [CapitalThorn]nd anyone else who thinks like I do... > The only things that actually exist to us is the electrical activity in our > brain. If its not electrical activity in our brain, it doesnt exist to > us. Everything that exists to us, from our own body, to the body of > others, to numbers, exist as electrical activity in our brains. Yes, but this is not the *idea*, its the representation of the idea. A score of Beethovens [CapitalThorn]fth is not the same as the (abstract) idea of the music. You dont have to be a mystic to realise that. Sorry, cant see much to say about the rest. Lets be blunt: the bad news is that I dont think you really have anything very much to write about. But theres good news too! You certainly have lots to read about. Brian Chandler http://imaginatorium.org === Subject: Re: Cantors diagonal proof wrong? >> >> There is no such thing in the physical world as even a natural >> number. One, two, three, etc. are all entirely conceptual, not >> physical. If you insist on physicality, give up mathematics. >> >> I am exploring things that you believe do not exist. And your outlook >> is not uncommon in the world. Its by far most common view all of >> mankind seems to like to share. >> I dont think this is true at all. In my experience, if you ask a >> roomful of non-mathematicians whether i, the square root of -1, >> exists they will mostly claim it doesnt. Ask Does 3 exist? and >> overwhelmingly they respond that yes, it does. Then you can play an >> amusing little game, trying to establish where their particular >> boundaries between existing and not actually existing numbers lie. >Yeah, you are probably right. I have not tried to play that game with >people. >Im glad you at least see the issue Im getting at. People use many words >like exist without trying to understand or justify what they really mean. >They simply, like parrots, use them correctly like they have been taught. >Most of what we all do works like that. And for sure, what I write is full >of that, as you go on to demonstrate below. >> Virgil, of course, or any other mathematician, will just look a bit >> blank, and say What do you mean ÔExist?, but I think this is a >> minority view. Of course, mathematicians will hedge their response, >> because in some cases, your Exist? will obviously apply to >> mathematical existence, or not, as in four-sided triangles, even >> primes greater than 63, and so on. >> Its that very fact that makes me at times, believe Ive found >> something that has been missed for 100s of years. Matematics, by >> design, limits its focus to a scope which does not include the things >> Im investigating. >> Ah! Two points. >> a) You have misspelled its, which is a pretty serious blunder. >Grammatical and spelling errors get much worse than that in my writing. :) >> b) Youve also missed quite a lot that was found 100s of years ago. >> Several people answered your proof that the integers cannot be put in >> 1-1 correspondence with themselves (or something like that), but I >> think the answers were not terribly good (a bit Micro$oft-like, if you >> know the helicopter joke). You have a somewhat ill-de[CapitalThorn]ned set of >> objects, including the integers, 1, 2, 3, 57, 264, etc., and some >> things like ...11111; you havent really said quite what else. Which >> of these is one of your cwintegers (as Ill call them, in honour of >> yourself!): >Yes, this sounds like fun. What was I thinking an integer was? >Well, for starters, it clearly included the set of all numbers which anyone >would call an integer. And that includes all the normal well known >properties of said numbers. >But then I called the in[CapitalThorn]nite string of digits which could be constructed >from the diagonal of my table, an integer and thats where the trouble >started. >> ...2121212121 * recurring decimals backwards >> ...5356295141 * fractional part of pi backwards >> -...1111111111 * cwintegers with a minus sign >Let me just for fun (not to try and [CapitalThorn]x my proof), just try to formalize >what I was thinking with ...11111 >Well, its the same thing as the in[CapitalThorn]nite series 1*10^0 + 1*10^2 + 1*10^3 + >.... >Now, this is not incompatible with the integers in that you can simply >substitute 0s for all the leading numbers. Regardless, you are _changing_ what the word integer _means_. Lets call these things curtigers instead, to prevent confusion. You have given a correct proof of the utterly obvious fact that the curtigers are in one-to-one correspondence with the reals between 0 and 1. But that has no relevance whatever to Cantors proof, because he didnt say anything about curtigers, the statement is about integers. Oh - I see someone has already named them cwintegers, [CapitalThorn]ne. Questions about whether cwints should count as integers, whether its possible to de[CapitalThorn]ne arithmetic operations on cwints, etc, have no relevance to the truth of Cantors theorem. Because words mean what the de[CapitalThorn]nitions say they mean, not what you think they should mean: even if you convinced everyone that cwints _should_ be included from now on as integers, they are _not_ included in the de[CapitalThorn]nition of the word integer as it appears in Cantors theorem. Regarding the irrelevant question of whether its possible to make cwints work like integers: Let x = ...212121, y = ...121212. Which is larger, x or y? We need to know, because it has some bearing on what x - y should be. >> While you are free to [CapitalThorn]nish off the de[CapitalThorn]nition, so we know exactly >> what is and isnt a cwinteger, then you have to start doing some grunt >> work, proving results that suggest that your cwintegers are useful. >> The integers used by mathematicians have at least the property that >> they possess the qualities children learn about informally at a _very_ >> young age. In particular: >> You can always add, subtract, or multiply two numbers. You cant >> always divide, but if you cant, its because there is no answer at >> all, not because there are lots of answers. You can compare two >> numbers, and it works: if a>b, b> are odd and even, and even+even=even. >> And lots and lots more. Here are some simple questions about >> cwintegers you should ask yourself: >> 1 + 2 = ? >> 3 - 2 = ? >Those are obvious with cwints because cwints are the same as integers when >they are [CapitalThorn]nite in size. >> ...999 + 2 = ? >You just substitute ...999 for the correct in[CapitalThorn]nite series, and you have >your answer. however, the answer can not be translated back into this same >notation, so a new notation must be created. 1...01. This is 1, with an >in[CapitalThorn]nite string of leading zeros, and an in[CapitalThorn]nite carry of a 1 value. :) >Now, I have to deal with how everything works with this new notation. >1...01 + 1 = 1...02 >1...01 - 1 = 1...0 >1...0 - 1 = ...999 >And it continues to get more complex, which I will not work through right >now. :) >> 1 - 2 = ? >> Is ...2222 > 5 ? >Yes, clearly. >> Is ...2222 < 5 ? >No, clearly. >But what about ...222 > ...111? >Hard to say what to do about that. But it seems like you could just decide >that the answer is true. The other option would be to declare it >unde[CapitalThorn]ned. >In either case, you would then have to work though what would either choice >would mean for all the other operations. >Yeah, clearly, I didnt grasp how much work has been done over 100 years >ago. :) >> As for all this stuff about AI, I recommend Daniel Dennetts book >> Darwins Dangerous Idea, >recommendation. >I tend not to read much philosophy. Im an engineer. >> Anyway, I think a bit of reading would be a good idea, before you >> announce your Great Discovery. >I [CapitalThorn]nd that to be true only to the extent of helping me to communicate with >others. If the great Discovery is a thinking machine, I dont need to >announce it. Ill let it do the announcing. :) >> You do not believe the conceptual world and the physical world are >> one in the same. I do. And once you believe that, everything starts >> to get very interesting, and everything starts to look very different. >> But does it make any sense? Do you mean that numbers (and things) >> Really Exist in the physical universe, for you? Including i? >Yes. But what is the physical universe and what does exist really >mean? Thats where it gets interesting and thats why Im able to answer >the question yes when others would like to say you cant answer it that >way. >In other words, the physical universe I know of is created by the behavior >of my brain. Its all in my head. It is a purely subjective understanding >of the universe. But in that virtual creation of my brain, is multiple >representations of my body, and your body. And in our bodies, are brains, >which are information processing machines, which create independent virtual >views of the same physical universe. >So where is the physical universe in this loop? Is it that stuff out >there which we normal think about when we use those words, or is, as it >must be, only the collection of ideas computed by my brain? >The only things that actually exist to us is the electrical activity in our >brain. If its not electrical activity in our brain, it doesnt exist to >us. Everything that exists to us, from our own body, to the body of >others, to numbers, exist as electrical activity in our brains. Electrical >activity is the physical movement of electrons. Its physical stuff. Its >my brain, and all the atoms and electrons in it, changing shape in response >to its interaction with other matter in the universe. Thats what the >number 1 is. Its not just an abstract idea which exists in some other >dimension from the physical stuff. Ideas are the actions of physical >stuff in this universe. >So, there are many ways to talk about all this, and if you dont follow >which way Im talking at the moment, its easy to get lost. >So, everything we know about, and sense, from our physical sensations of >the physical world, to all the sensations of the mental world, are created >by electrons and chemicals ßowing in our brains. And the electrons that >allows us to sense the idea of the imaginary number i, is just as real, as >the electrons that allows me to sense the keyboard Im typing on. >The physical world we are able to understand, is not out there, its in >our head, along with the stuff that creates the idea of i. So our >understanding of the physical world is just as real, as our understanding >of a concept like i. And all this ability is built out of material from >the physical world. >> Brian Chandler ************************ David C. Ullrich === Subject: Re: Cantors diagonal proof wrong? ... > Regarding the irrelevant question of whether its possible > to make cwints work like integers: Let > x = ...212121, > y = ...121212. > Which is larger, x or y? We need to know, because it has some > bearing on what x - y should be. But that is obvious: x - y = ...0909090909, and y - x = ...9090909091. So (y - x) = 10 * (x - y) + 1, and so 11 * (y - x) = 1, and so (y - x) = 1/11. Eh? Oh, wait, there must be an error somewhere. -- 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: Cantors diagonal proof wrong? :> And lots and lots more. Here are some simple questions about :> cwintegers you should ask yourself: :> ...999 + 2 = ? : You just substitute ...999 for the correct in[CapitalThorn]nite series, and you have : your answer. however, the answer can not be translated back into this same : notation, so a new notation must be created. 1...01. This is 1, with an : in[CapitalThorn]nite string of leading zeros, and an in[CapitalThorn]nite carry of a 1 value. :) What does ...999*10 + 9 equal? Stephen === Subject: Re: Cantors diagonal proof wrong? > :> And lots and lots more. Here are some simple questions about > :> cwintegers you should ask yourself: > :> > :> ...999 + 2 = ? > : You just substitute ...999 for the correct in[CapitalThorn]nite series, and you have > : your answer. however, the answer can not be translated back into this same > : notation, so a new notation must be created. 1...01. This is 1, with an > : in[CapitalThorn]nite string of leading zeros, and an in[CapitalThorn]nite carry of a 1 value. :) > What does ...999*10 + 9 equal? I didnt really understand CWs response, but yours is easy: ...999. So its the solution to 10x+9=x. Not surprisingly, since in an obvious interpretation its -1 (2s complement, though I realise I mean tens complement[?]). Similarly, since ...333 * 3 + 1 = 0, ...333 = -1/3 etc Brian Chandler http://imaginatorium.org === Subject: Re: Cantors diagonal proof wrong? :> :> And lots and lots more. Here are some simple questions about :> :> cwintegers you should ask yourself: :> :> :> :> :> ...999 + 2 = ? :> :> : You just substitute ...999 for the correct in[CapitalThorn]nite series, and you have :> : your answer. however, the answer can not be translated back into this same :> : notation, so a new notation must be created. 1...01. This is 1, with an :> : in[CapitalThorn]nite string of leading zeros, and an in[CapitalThorn]nite carry of a 1 value. :) :> What does ...999*10 + 9 equal? : I didnt really understand CWs response, but yours is easy: ...999. : So its the solution to 10x+9=x. Not surprisingly, since in an obvious : interpretation its -1 (2s complement, though I realise I mean tens : complement[?]). But that is clearly not how the OP was interpreting ...999999. Given the above interpretation ...9999999 = -1. Then following the same logic ...1111111 = -1/9. Going back to the diagonalization argument applied to the positive integers, the OP produced ...1111111 as a integer not on the list, and claimed that this was a contradiction. However -1/9 is neither positive nor an integer, so there is clearly no contradiction if the OP thought that ...11111111 = -1/9. He has since realized that ...111111111 is not an integer. Stephen === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > :> And lots and lots more. Here are some simple questions about > :> cwintegers you should ask yourself: > :> > :> ...999 + 2 = ? > : You just substitute ...999 for the correct in[CapitalThorn]nite series, and you > : have your answer. however, the answer can not be translated back into > : this same notation, so a new notation must be created. 1...01. This > : is 1, with an in[CapitalThorn]nite string of leading zeros, and an in[CapitalThorn]nite carry > : of a 1 value. :) > What does ...999*10 + 9 equal? > Stephen Something that shows how non-integer like cwintergers can be? :) -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > Could you please tell me then which integer corresponds to the real > number 1/9 (or 0.11111111111111111111... if you prefer)? > Jose Carlos Santos The array that Cantor devised for demonstrating the countability of rationals permit us to assign an integer N to each fraction n/d. Be S = n + d . If S is odd then: N = (S-1)(S-2)/2 + d If S in even then: N = (S-1)(S-2)/2 + n. For your fraction 1/9 we have: N = 9x8/2 + 1 = 37 The famous fraction 355/113 occupy the position N = 109166. === Subject: Re: Cantors diagonal proof wrong? > Could everything I believe be wrong? Of course. Time will tell if this > belief turns out to be useful or just a silly waste of time. About 90 percent of it. You spelled wrong right though. It is a silly waste of time. Take up a useful occupation. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? > What I have seen with a few seconds of looking at web pages on those > subjects is that the basic axioms used to de[CapitalThorn]ne math from always seem to > include the belief that in[CapitalThorn]nite sets are allowed to exist. In[CapitalThorn]nite sets do not need permission. There is one in[CapitalThorn]nite set we all deal with, the set of natural integers 1,2,...... . The set can be shown to be in[CapitalThorn]nite in a number of ways. One way is to notice there is no largest integer. If the set were [CapitalThorn]nite there would be a largest. Another to to show the in[CapitalThorn]nite cardinality of the integers is to note that the correspondence n <->2*n maps matches up the integers with a proper subset of the integers. In fact, the existence of a mapping one to one onto a proper subset can be taken as a de[CapitalThorn]nition of in[CapitalThorn]nte. So we know in[CapitalThorn]nite sets exist. This is not cotroversial. > My thought is that once you do that, you have created problems in the world > of math that do not exist in this universe. My thought is that you should > be able to use a set of axioms that does not include in[CapitalThorn]nite sets, yet > still, do all the math we do today. There are plenty of [CapitalThorn]nite structures (along with axioms). But they are not the integers. How about the set of permuations on N elements 1,2,...N. There are N! of those. They form a group. > That is, we de[CapitalThorn]ne the natural numbers not as an in[CapitalThorn]nite set, but as a > counting algorithm which can never complete. Peano has already done that. Gooogle Peano Axioms. > I believe a lot of my ideas parallels work done with computability and > Turing machines. But thats more that I dont know enough about yet. Actually you dont know what you are talking about. Do not quit your day job. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? > In[CapitalThorn]nite sets do not need permission. There is one in[CapitalThorn]nite set we all > deal with, the set of natural integers 1,2,...... . The set can be > shown to be in[CapitalThorn]nite in a number of ways. One way is to notice there is > no largest integer. If the set were [CapitalThorn]nite there would be a largest. > Another to to show the in[CapitalThorn]nite cardinality of the integers is to note > that the correspondence n <->2*n maps matches up the integers with a > proper subset of the integers. In fact, the existence of a mapping one > to one onto a proper subset can be taken as a de[CapitalThorn]nition of in[CapitalThorn]nte. > So we know in[CapitalThorn]nite sets exist. This is not cotroversial. > There is only one world. > In[CapitalThorn]nity is just a fancy way of saying and so on.... In[CapitalThorn]nity is > just inde[CapitalThorn]nite repetition of some operation. Potential in[CapitalThorn]nite? > All mathematical concepts are abstractions on actual experience. So far so good. But now we are going to make calculations with those trans[CapitalThorn]nite ordinals and cardinals. And prove Goodsteins theorem too. Then, do you still think, Bob, that the latter are abstractions on actual experience? Either you are a realist or an idealist, but not both. Or: how can you possibly reconcile such quite different points of view in a consistent manner? Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > Then, do you still think, Bob, that the latter are abstractions on > actual experience? Either you are a realist or an idealist, but not > both. Or: how can you possibly reconcile such quite different points > of view in a consistent manner? Abstraction and metaphor is how we think. The in[CapitalThorn]nite is just a fancy way of saying and so on..... All you need is some experience with a process that does not have a prinicpled way of being terminated. Like adding one to an integer. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? You can certainly de[CapitalThorn]ne the set of in[CapitalThorn]nite integers to be the set of all integers plus all numbers that have an in[CapitalThorn]nite progression of digits to the left of the decimal. This set, though, is larger than the set of integers since there is a 1 to 1 correspondence with the reals ... this set is not used much in standard mathematics either. Ôwhy cant I just write 111111111... ... and you can, but you no longer have an integer, you have a member of my set of in[CapitalThorn]nite integers. >> Could you please tell me then which integer corresponds to the real >> number 1/9 (or 0.11111111111111111111... if you prefer)? > ....1111111 > Why is it ok to write 0.111... but not ...11111 ? > My point is that 1/9 is in fact an algorithm for generating a real value. > It is not the real value itself. Its just a name we use to talk about > the > real value which is 0.1111 repeating forever. And I can just as easily > de[CapitalThorn]ne the integer of 1 repeating forver. The only reason we do not do > that is a matter of convention. Its not (so I claim) in violation of > what > integers are. > If you start with 0, and continue to apply the +1 function to it, and > ignore all the values you come up with which does not have all 1s, you > [CapitalThorn]nd you have the exact same type of de[CapitalThorn]ntion that gives you 1/9 when you > generate a string of ones running to the right, instead of running to the > left. > Also, all integers have an implied in[CapitalThorn]nite string of 0s running to left > (just like reals have am implied in[CapitalThorn]nite string of 0s running to the > left and right). So when you write 123, you are really writing > ....00000123. > Just change the implied 0, to an implied 1, and there you have it. The > integer pair for the real 1/9. > -- > Curt Welch > http://CurtWelch.Com/ > curt@kcwc.com > http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? >> Could you please tell me then which integer corresponds to the real >> number 1/9 (or 0.11111111111111111111... if you prefer)? > ....1111111 >Why is it ok to write 0.111... but not ...11111 ? Its ok to _write_ ...11111 . But ...11111 is not an _integer_. >My point is that 1/9 is in fact an algorithm for generating a real value. No, your point _was_ that that list gives a mapping that maps the integers onto the reals. Yesterday you speculated that people would call you crazy, and I pointed out that that depends on how you reply to explanations that youre wrong. The above is a very simple explanation of why youre simply wrong here, and youre not saying oops. Instead below you attempt to rede[CapitalThorn]ne what the word integer mneans. So: Youre crazy. There, happy now? >It is not the real value itself. Its just a name we use to talk about the >real value which is 0.1111 repeating forever. And I can just as easily >de[CapitalThorn]ne the integer of 1 repeating forver. The only reason we do not do >that is a matter of convention. Its not (so I claim) in violation of what >integers are. >If you start with 0, and continue to apply the +1 function to it, and >ignore all the values you come up with which does not have all 1s, you >[CapitalThorn]nd you have the exact same type of de[CapitalThorn]ntion that gives you 1/9 when you >generate a string of ones running to the right, instead of running to the >left. >Also, all integers have an implied in[CapitalThorn]nite string of 0s running to left >(just like reals have am implied in[CapitalThorn]nite string of 0s running to the >left and right). So when you write 123, you are really writing >....00000123. >Just change the implied 0, to an implied 1, and there you have it. The >integer pair for the real 1/9. ************************ David C. Ullrich === Subject: Re: Cantors diagonal proof wrong? !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~( 5eZ41to5f%E@ELIi $t^ VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Could you please tell me then which integer corresponds to the real > number 1/9 (or 0.11111111111111111111... if you prefer)? >> ....1111111 >>Why is it ok to write 0.111... but not ...11111 ? > Its ok to _write_ ...11111 . But ...11111 is not an _integer_. The beautiful thing is that one can _prove_ it is not an integer, by induction. You can prove that for every integer n there exists an integer K(n) (well, you actually can generously choose K(n)=n) so that for all k>K(n), the kth digit counted from the right is zero. And that means that the ultimate left is zero as long as we are talking about integers de[CapitalThorn]ned by the Peano axioms. For other de[CapitalThorn]nitions (like p-adic numbers), the bets regarding the ultraleftist digits are off. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > So: Youre crazy. There, happy now? Yes, it gives me that warm fuzzy feeling I was looking for. :) -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? Dave, How do you, year after year, reply to people who do not have the basic tools of mathematics again and again, just to hear youre wrong, you just blindly follow the rules of mathematics, over and over? For gods sake, youre a professor ... you can do this in class all day - do you just have an inner drive to Ôhelp people see? > Could you please tell me then which integer corresponds to the real > number 1/9 (or 0.11111111111111111111... if you prefer)? >> ....1111111 >>Why is it ok to write 0.111... but not ...11111 ? > Its ok to _write_ ...11111 . But ...11111 is not an _integer_. >>My point is that 1/9 is in fact an algorithm for generating a real value. > No, your point _was_ that that list gives a mapping that maps the > integers onto the reals. > Yesterday you speculated that people would call you crazy, and I > pointed out that that depends on how you reply to explanations > that youre wrong. The above is a very simple explanation of > why youre simply wrong here, and youre not saying oops. > Instead below you attempt to rede[CapitalThorn]ne what the word integer > mneans. > So: Youre crazy. There, happy now? >>It is not the real value itself. Its just a name we use to talk about >>the >>real value which is 0.1111 repeating forever. And I can just as easily >>de[CapitalThorn]ne the integer of 1 repeating forver. The only reason we do not do >>that is a matter of convention. Its not (so I claim) in violation of >>what >>integers are. >>If you start with 0, and continue to apply the +1 function to it, and >>ignore all the values you come up with which does not have all 1s, you >>[CapitalThorn]nd you have the exact same type of de[CapitalThorn]ntion that gives you 1/9 when you >>generate a string of ones running to the right, instead of running to the >>left. >>Also, all integers have an implied in[CapitalThorn]nite string of 0s running to left >>(just like reals have am implied in[CapitalThorn]nite string of 0s running to the >>left and right). So when you write 123, you are really writing >>....00000123. >>Just change the implied 0, to an implied 1, and there you have it. The >>integer pair for the real 1/9. > ************************ > David C. Ullrich === Subject: Re: Cantors diagonal proof wrong? Discussion, linux) > Extraordinary claims require extraordinary proof. So I offered > one. I agree. It was an extraordinary proof. -- I dont know why I live in a world with so many supposed mathematicians who are all so dumb AND rude. Why oh why couldnt someone like Gauss or Dedekind still be around? Shoot, Id even take someone like Hardy at this point. -- James S Harris compromises === Subject: Re: Cantors diagonal proof wrong? > Chairman of the David Hilbert Appreciation Society [...] > The fact that its reasonable in math, to talk about the size of one > in[CapitalThorn]nite set being larger than the size of another in[CapitalThorn]nite set, doesnt > seem to tie correctly to the physical world. I think the current attitude among most mathematicians is that mathematics doesnt need to have anything to do with the physical world. [...] > But, is there a point, where you use language to create a story, which can > never be true in this universe? I think there is, and I think the world of > math has done just that. There is a distinction between provability and truth. I think that it is philosophy (not strictly mathematics) that deals with notions of truth. Someone may want to correct me on that. > They have used language to create an imaginary > world, and at some point, they cross the line from possible, to impossible, > in the physical universe. It seems that you think mathematics is a fantasy game which has nothing to do with the physical universe and therefore is broken somehow. For mathematics to be not broken, you think, it must retain a connection with the physical world, or something like that. The following quote from Shapiro in _Thinking About Mathematics_, responds to the sentiment above more eloquently than anything I could think of offhand. ...one might argue that if mathematics gave serious pursuit only to those branches known to have applications in natural science, we would not have much of the mathematics we have today, nor would be have all of the /science/ we have today the history of science is full of cases where branches of Ôpure mathematics eventually found application in science. In other words, the overall goals of scienti[CapitalThorn]c enterprise have been well served by mathematicians pursuing their own disciplines with their own methodology So, if youre only interested in the physical world or natural science youve been well served by mathematicians doing what they do, however wrong or incomprehensible it may seem to you. [...] - Replace Roman numerals with digits to reply by email === Subject: Re: Cantors diagonal proof wrong? > The following quote from Shapiro in _Thinking About Mathematics_, > responds to the sentiment above more eloquently than anything > I could think of offhand. > ...one might argue that if mathematics gave serious pursuit only > to those branches known to have applications in natural science, > we would not have much of the mathematics we have today, nor > would be have all of the /science/ we have today the history > of science is full of cases where branches of Ôpure mathematics > eventually found application in science. In other words, the > overall goals of scienti[CapitalThorn]c enterprise have been well served by > mathematicians pursuing their own disciplines with their own > methodology > So, if youre only interested in the physical world or natural > science youve been well served by mathematicians doing > what they do, however wrong or incomprehensible it may > seem to you. Some artists wear a beard. Not everybody who wears a beard is an artist. Some parts of pure mathematics have been useful. But many have ... NOT. Its all a matter of ef[CapitalThorn]ciency. I [CapitalThorn]nd that contemporary mathematics, as far as its real world applications are concerned, is not as ef[CapitalThorn]cient and effective as it could have been, given the current state of the art. This has lead to situations where Applied has developed a mathematics of its own. An example is the Finite Element method, as has been developed by mechanical and civil engineers. Has someone ever read the _ugly_ book by O.C Zienkiewicz The Finite Element Method ? I would recommend it to every pure mathematician. As an eye-opener. Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > I think the current attitude among most mathematicians is that > mathematics doesnt need to have anything to do with the physical > world. All mathematical concepts are abstractions on actual experience. The fact that a mathematical idea needs to be expressed in a language tells you the mathematics has some connection to the real world. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w Chairman of the David Hilbert Appreciation Society > Chairman of the David Hilbert Appreciation Society > [...] > The fact that its reasonable in math, to talk about the size of one > in[CapitalThorn]nite set being larger than the size of another in[CapitalThorn]nite set, > doesnt seem to tie correctly to the physical world. > I think the current attitude among most mathematicians is that > mathematics doesnt need to have anything to do with the physical > world. Yes, that is exactly how I think most mathematicians think. And I believe that has allowed them to create a disconnect between the world of mathematical concepts and the physical world. I suspect most dont see why this is even an issue in my mind. > [...] > But, is there a point, where you use language to create a story, which > can never be true in this universe? I think there is, and I think the > world of math has done just that. > There is a distinction between provability and truth. I think that > it is philosophy (not strictly mathematics) that deals with notions > of truth. Someone may want to correct me on that. > They have used language to create an imaginary > world, and at some point, they cross the line from possible, to > impossible, in the physical universe. > It seems that you think mathematics is a fantasy game which has > nothing to do with the physical universe and therefore is broken > somehow. For mathematics to be not broken, you think, it must retain > a connection with the physical world, or something like that. > The following quote from Shapiro in _Thinking About Mathematics_, > responds to the sentiment above more eloquently than anything > I could think of offhand. > ...one might argue that if mathematics gave serious pursuit only > to those branches known to have applications in natural science, > we would not have much of the mathematics we have today, nor > would be have all of the /science/ we have today the history > of science is full of cases where branches of Ôpure mathematics > eventually found application in science. In other words, the > overall goals of scienti[CapitalThorn]c enterprise have been well served by > mathematicians pursuing their own disciplines with their own > methodology > So, if youre only interested in the physical world or natural > science youve been well served by mathematicians doing > what they do, however wrong or incomprehensible it may > seem to you. Yes, I agree completely with that point of view. It would be totally foolish to discontinue the pursuit of pure mathematics just because we had not yet found a scienti[CapitalThorn]c use for it. Scienti[CapitalThorn]c or practical use of the mathematical tools developed have for the most part, only come after they were developed in the study of pure mathematics. However, I believe that the world of ideas, and concepts, and math, are not disconnected from the physical world. I believe they are one and the same. I believe we simply just dont fully understand the connection yet. To understand that connection requires we understand what the brain does, and how it works. At some point in time, I believe we will close that gap by gaining the required understanding. When that happens, it should tells us some very interesting things about the limits of our reality and our ability to think, and create ideas, and use ideas, and our ability to understand the universe we exist in. Once we understand exactly how the physical worlds and the mental worlds are connected, then that will act as a bridge between the two worlds that currently does not exist. It will allow knowledge to ßow from our base of understanding about the physical universe, into the knowledge about our mental universe, and vice versa. Currently, the world of mathematics seemes to be built the best foundation that could be de[CapitalThorn]ned, yet it was probably a somewhat arbitrary foundation that seemed [CapitalThorn]tting for the job (there was nothing else to build it from). But what if, once we understand the connection between thought and the physical world, we [CapitalThorn]nd a very different looking foundation? What if we built a new type of mathematics de[CapitalThorn]ned from the realities of the physical foundation that the brain is actually built from? Would it be any different from what we already have created in mathematics? I suspect it would be, and using the ideas I already have about how the physical worlds and mental worlds connect, Im poking around with this idea to see if the foundation of modern mathematics seems to mesh with the physical reality I believe that thought is built on. What Im seeing, is that there seems to be a disconnect - especially in this use of ideas of in[CapitalThorn]nity. But clearly, it has been shown I dont understand the foundations of modern mathematics. So until I do, I cant [CapitalThorn]gure out if there is anything useful to be done here or not. Its also very likely that by the time I understand, and try to adjust the foundation, that it will make no change whatsoever in any [CapitalThorn]elds of mathematics. I will simply have used different words to de[CapitalThorn]ne the same functionally equivalent foundation. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > However, I believe that the world of ideas, and concepts, and math, are not > disconnected from the physical world. I believe they are one and the same. > I believe we simply just dont fully understand the connection yet. [ ... ] The world of ideas, and math, ARE disconnected from the physical world (though I have been reluctant to accept this myself, for a long time). At this moment, Im even convinced that it should be so: the world of mathematics and the physical world are NOT one and the same. And they never will be or should be. > Once we understand exactly how the physical worlds and the mental worlds > are connected, then that will act as a bridge between the two worlds that > currently does not exist. It will allow knowledge to ßow from our base of > understanding about the physical universe, into the knowledge about our > mental universe, and vice versa. Thats the right question: how does the connection work, the connection between the world of ideas and the world that contains the real stuff? There are two concepts which IMHO should be sharply distinguished here: 1. Abstraction 2. Idealization In a nutshell. Abstraction is something which done in _Physics_, by our measuring devices. Idealization, on the other hand, is something which typically gives rise to a kind of Mathematics. A good example of this is the idealization of common geometry to Euclids. More on ideal elements: http://huizen.dto.tudelft.nl/deBruijn/grondig/reconcyl.htm Now I think that your real concern, Curt, is in the _reverse_ process: once you have climbed that stairway to heaven, how to [CapitalThorn]nd a way back to earth? This reverse process could be called: MATERIALIZATION. Ive composed a web page on the issues of idealization and materialization: http://huizen.dto.tudelft.nl/deBruijn/grondig/klassiek.htm Hope this clari[CapitalThorn]es your desired connection a little bit further. Han de Bruijn === Subject: Re: On abstraction (was: Re: Cantors diagonal proof wrong?) * Han de Bruijn > There are two concepts which IMHO should be sharply distinguished here: > 1. Abstraction > 2. Idealization > In a nutshell. Abstraction is something which done in _Physics_, by our > measuring devices. Idealization, on the other hand, is something which > typically gives rise to a kind of Mathematics. A good example of this is > the idealization of common geometry to Euclids. More on ideal elements: In essence I certainly agree. However, I [CapitalThorn]nd the concept of _abstraction_ consiting of the _idealization_ and _extraction_. When you make a model in physics of some observed fenomena, you certainly extracts by pretending not to see a lot of things that are there, but that you dont want to include in your model. Moreover, you also do an idealization by pretending that certain characteristics are perfect, e.g. perfect sphere, no friction and so on. This de[CapitalThorn]nition is found in The Mathematicsl Experience by Davis and Hersh. -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@i[CapitalThorn].uio.no http://www.i[CapitalThorn].uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: Re: On abstraction (was: Re: Cantors diagonal proof wrong?) > * Han de Bruijn > There are two concepts which IMHO should be sharply distinguished here: > 1. Abstraction > 2. Idealization > In a nutshell. Abstraction is something which done in _Physics_, by our > measuring devices. Idealization, on the other hand, is something which > typically gives rise to a kind of Mathematics. A good example of this is > the idealization of common geometry to Euclids. More on ideal elements: > In essence I certainly agree. However, I [CapitalThorn]nd the concept of > _abstraction_ consiting of the _idealization_ and _extraction_. When > you make a model in physics of some observed fenomena, you certainly > extracts by pretending not to see a lot of things that are there, but > that you dont want to include in your model. Moreover, you also do > an idealization by pretending that certain characteristics are > perfect, e.g. perfect sphere, no friction and so on. > This de[CapitalThorn]nition is found in The Mathematicsl Experience by Davis and > Hersh. I like this nice de[CapitalThorn]nition. I cannot imagine abstraction without idealization either. That must be an interesting book. -- Eray Ozkural === Subject: Re: On abstraction (was: Re: Cantors diagonal proof wrong?) * Eray Ozkural exa > This de[CapitalThorn]nition is found in The Mathematicsl Experience by Davis and > Hersh. > I like this nice de[CapitalThorn]nition. I cannot imagine abstraction without > idealization either. That must be an interesting book. Indeed. It is about mathematics, and is easy to read, but is still quite advanced. -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@i[CapitalThorn].uio.no http://www.i[CapitalThorn].uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: Re: On abstraction >>* Han de Bruijn >There are two concepts which IMHO should be sharply distinguished here: >1. Abstraction >2. Idealization >In a nutshell. Abstraction is something which done in _Physics_, by our >measuring devices. Idealization, on the other hand, is something which >typically gives rise to a kind of Mathematics. A good example of this is >the idealization of common geometry to Euclids. More on ideal elements: >>In essence I certainly agree. However, I [CapitalThorn]nd the concept of >>_abstraction_ consiting of the _idealization_ and _extraction_. When >>you make a model in physics of some observed fenomena, you certainly >>extracts by pretending not to see a lot of things that are there, but >>that you dont want to include in your model. Moreover, you also do >>an idealization by pretending that certain characteristics are >>perfect, e.g. perfect sphere, no friction and so on. >>This de[CapitalThorn]nition is found in The Mathematicsl Experience by Davis and >>Hersh. > I like this nice de[CapitalThorn]nition. I cannot imagine abstraction without > idealization either. That must be an interesting book. Eray, please note that I [CapitalThorn]nd abstraction something that can be done by NON-HUMAN apparatus, like measuring devices, coupled with a computer of some sort eventually. This is essential, if you want to understand at least something about the role of mathematics in the applied sciences. But perhaps I should agree with Jon Haugsand and call this extraction. Idealization is something different anyway. Now, instead of reading books *about*, why not dig into the *real* thing for once in your life! Excellent examples of idealization in physics are found in The Theory of Heat Radiation by Max Planck (being my absolute favorite with respect to this). > -- > Eray Ozkural === Subject: Re: Cantors diagonal proof wrong? > Chairman of the David Hilbert Appreciation Society >,> But what if, once we understand the connection between thought and the > physical world, we [CapitalThorn]nd a very different looking foundation? What if we > built a new type of mathematics de[CapitalThorn]ned from the realities of the physical > foundation that the brain is actually built from? Would it be any > different from what we already have created in mathematics? My answer is no. Whether the origin of language is scienti[CapitalThorn]cally stablished, that would not change the way poetry, novels,essays etc will be written. The same for mathematics, the knowledge of brain operations will not change the form we make mathematics, because is a cultural product that is accepted as a game with its own laws. Think in the Theory of Prime Numbers. Is a theory founded on an algorithm for producing a special type of numbers and its developmnet will not be inßuenced by discoveries in the natural sciences. === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > Chairman of the David Hilbert Appreciation Society >,> But what if, once we understand the connection between thought and > the > physical world, we [CapitalThorn]nd a very different looking foundation? What if > we built a new type of mathematics de[CapitalThorn]ned from the realities of the > physical foundation that the brain is actually built from? Would it be > any different from what we already have created in mathematics? > My answer is no. > Whether the origin of language is scienti[CapitalThorn]cally stablished, that > would not change the way poetry, novels,essays etc will be written. > The same for mathematics, the knowledge of brain operations will not > change the form we make mathematics, because is a cultural product > that is accepted as a game with its own laws. Think in the Theory of > Prime Numbers. Is a theory founded on an algorithm for producing a > special type of numbers and its developmnet will not be inßuenced by > discoveries in the natural sciences. The way I look at it, you can start with natural numbers and their relationship to the physical world (how we pair them with physical objects for the purpose of counting, etc), and you can build complexity from there by de[CapitalThorn]ning addition, subtraction, multiplication, division, etc. As far as I know (which clearly isnt very far into the [CapitalThorn]eld of mathematics), this is how the [CapitalThorn]eld of mathematics evolved. Understanding the brain is not going to change what was built on top of the natural numbers. But then it seems that mathematicians wanted to try and reverse that building processes. Instead of building up from natural numbers, they wanted to try and identify any possible lower level foundation below natural numbers. And in this effort, as far as I can tell, a lot of this set theory work and a lot of stuff which has been touched on in this thread was created. In building down from natural numbers, they were in effect searching for the meaning of natural numbers. There were in effect searching for a lower level language to explain the natural numbers, just like the natural numbers can be a language to explain addition with. The question I wonder about, is how many different low level languages might there be for explaining the natural numbers? Clearly, from what Ive seen in only the past day, mathematicians have explored many variations on that question already over the years. But, Ive got a few languages of my own which I use to talk about how the brain works. And in that language, I can explain the natural numbers. Now, if my ideas about brain function have any validity to them, there should be some basic correlations or mappings from my language of the brain to the language of Mathematics which explains the nature of numbers. But currently, theres a big gap I cant connect. My language of the brain does not allow the existence of an in[CapitalThorn]nite number of abstract objects. Yet, the language of math always seems to start with an axiom that says a set of in[CapitalThorn]nite abstract objects does, and must, exist. The general feeling I get from what I read is that without that, you cant de[CapitalThorn]ne math. Now this is very natural to understand, because basically what they are saying is that the set of natural numbers is an in[CapitalThorn]nite set, and if you dont have them, you cant build everything that has been de[CapitalThorn]ned on top of the natural numbers. So what they are saying seems very logical. And if mathematics is all I was trying to understand, Id quickly accept this and move on. But because this basic foundation of math conßicts with my understand of the brain, Ive got a contradiction to resolve. Either my understand of the brain is wrong, or the [CapitalThorn]eld of mathematics has made some assumptions to de[CapitalThorn]ne math on which do not match the foundation of where math actually came from (the brain), or there is no contradiction and the only problem is I dont understand why these words that seem to create a contradiction does not in fact create a contradiction. So I struggle to resolve all of this. :) The biggest missing piece of the puzzle is that I dont have [CapitalThorn]rm grasp on the language of mathematics yet. I thought I had a better grasp than I obviously do. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? <9ZGdnRVWNYDZNQrcRVn-og@giganews.com> at 08:30 PM, curt@kcwc.com (Curt Welch) said: >But then it seems that mathematicians wanted to try and reverse that >building processes. Its called abstraction, and its important in a lot of discipleines, not just Mathematics. In particular, its important in CS. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantors diagonal proof wrong? > The way I look at it, you can start with natural numbers and their > relationship to the physical world (how we pair them with physical objects > for the purpose of counting, etc), and you can build complexity from there > by de[CapitalThorn]ning addition, subtraction, multiplication, division, etc. As far > as I know (which clearly isnt very far into the [CapitalThorn]eld of mathematics), > this is how the [CapitalThorn]eld of mathematics evolved. This ignores the whole of geometry as a study largely independent of arithmetic, which was the case prior to the invention/discovery of analytical geometry. === Subject: Re: Cantors diagonal proof wrong? > This ignores the whole of geometry as a study largely independent of > arithmetic, which was the case prior to the invention/discovery of > analytical geometry. Very much compliant with that! And a picture says more than a thousand words. Why is it that good old geometry does play such a minor role in mathematics these days? And why are pictures so scarcely used in modern mathematical publications? Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? > Very much compliant with that! And a picture says more than a thousand > words. Why is it that good old geometry does play such a minor role in > mathematics these days? And why are pictures so scarcely used in modern > mathematical publications? If you include diagrams as pictures, then just about any paper in Category Theory is [CapitalThorn]lled with pictures. Commutative mapping diagrams are ubiquitous. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? > However, I believe that the world of ideas, and concepts, and math, are not > disconnected from the physical world. I believe they are one and the same. > I believe we simply just dont fully understand the connection yet. To > understand that connection requires we understand what the brain does, and > how it works. There is only one world. > What Im seeing, is that there seems to be a disconnect - especially in > this use of ideas of in[CapitalThorn]nity. In[CapitalThorn]nity is just a fancy way of saying and so on.... In[CapitalThorn]nity is just inde[CapitalThorn]nite repetition of some operation. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > In[CapitalThorn]nity is just a fancy way of saying and so on.... In[CapitalThorn]nity is just > inde[CapitalThorn]nite repetition of some operation. Yes, thats how I understand it. But yet, in mathematics, there is this langauge which talks about one and so on... being able to produce more so ons than another type of and so on.... How is that possible? What does it mean in terms of your description of in[CapitalThorn]nity above? Thats the connection Im still unable to make. But I have a lot more reference material to study now on the language of mathematics. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > Chairman of the David Hilbert Appreciation Society > Chairman of the David Hilbert Appreciation Society > [...] > The fact that its reasonable in math, to talk about the size of one > in[CapitalThorn]nite set being larger than the size of another in[CapitalThorn]nite set, > doesnt seem to tie correctly to the physical world. > I think the current attitude among most mathematicians is that > mathematics doesnt need to have anything to do with the physical > world. > Yes, that is exactly how I think most mathematicians think. And I believe > that has allowed them to create a disconnect between the world of > mathematical concepts and the physical world. I suspect most dont see why > this is even an issue in my mind. The point is that any connection between mathematics and the physical world is dependent on the assumptions one makes about the nature of the physical world, and those assumptions are, of necessity, outside of mathematics. So that before a mathematician will agree to your application of mathematics to the physical world, he will require you to state clearly and unambiguously what you are assuming about the physical World, at least insofar as is relevant to your application of mathematics to that world. This is remarkably dif[CapitalThorn]cult to do, even for scientists who must do it on a regular basis. === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > The point is that any connection between mathematics and the physical > world is dependent on the assumptions one makes about the nature of the > physical world, and those assumptions are, of necessity, outside of > mathematics. > So that before a mathematician will agree to your application of > mathematics to the physical world, he will require you to state clearly > and unambiguously what you are assuming about the physical World, at > least insofar as is relevant to your application of mathematics to that > world. > This is remarkably dif[CapitalThorn]cult to do, even for scientists who must do it > on a regular basis. The nasty part is that it seems the brain that creates mathematics is part of the physical world it is trying use mathematics to describe. This creates a nasty little loop. The ultimate goal, if in fact this is a closed loop, is to describe the entire loop. But as you say, you must start somewhere by making some assumptions. You can make assumptions about the physical world, and then try to describe the physical world based on those starting assumptions. Then, from there, you try to describe how that physical world creates the mental world. If you are able to do that, then you describe in full, the mental world. When you are done, the description of the mental world should connect back, and justify, the original assumptions you made to describe the physical world. It should form a closed, and consistent loop. Or, you can start in the mental world and make assumptions about the nature of thought, and attempt to then describe the full nature of the mental world, and how in that world, the physical world is de[CapitalThorn]ned, and in turn, describe the full nature of the physical world, and then close the loop again by explaining how the mental world arises out of that physical world model. Its much like a Kline bottle in that it is not a simple loop. Its a loop that turns the nature of things inside out as you loop around it. Understanding the nature of this mind/body problem has been a central problem in philosophy since the beginning. If you pick the wrong staring assumptions in either case, you can [CapitalThorn]nd that by the time you try to close the loop, nothing [CapitalThorn]ts together. Or worse, that any starting assumption you pick, with the help of the loop, becomes a self ful[CapitalThorn]lling prophecy. Which leaves us to wonder which of the many ideas about the nature of the mental world, or the nature of the physical world, will be the right set of ideas to allows us to close the loop. We will not know if we have the correct set of assumptions until we are able to duplicate the power of the brain, and build a machine which duplicates our own mental powers. When we do that, we will have proved that the assumptions we were using to build the machine were suf[CapitalThorn]cient to close the loop. Until we do that, we wont know for sure if it even is a closed loop. The belief that it is a closed loop us just one of my starting assumptions. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > If all youre interested in is the physical world, then you dont need > in[CapitalThorn]nite sets, let alone the real numbers. > My interest is not in the physical world alone, its in the relationship > between the physical and mental worlds. To solve AI, we have to completely > understand that relationship. If those are necessary conditions for solving AI, youre screwed. Ôcid Ôooh === Subject: Re: Cantors diagonal proof wrong? > > If all youre interested in is the physical world, then you dont > need in[CapitalThorn]nite sets, let alone the real numbers. > > My interest is not in the physical world alone, its in the > relationship between the physical and mental worlds. To solve AI, we > have to completely understand that relationship. > If those are necessary conditions for solving AI, youre screwed. > :) > Thats exactly why AI is not an easy problem. It is the solution to how > the two worlds connect. If you can not explain the connection, you can not > solve AI. Unfortunately, no human will ever be able to explain this connection, if there even is one. >Its unproven if AI is a problem that we (humans) have the power > to solve. I do happen to belive we have the power to solve it, (or else it > would be kinda stupid for me to spend time on it), but I also know it is > just a leap of faith on my part to believe that. Sorry to burst your bubble, but if you expect to be able to verify that a machine is cognating, youre not going to be able to in any non-functionalist sense. And thats not going to satisfy many people. Ôcid Ôooh === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > Sorry to burst your bubble, but if you expect to be able to verify > that a machine is cognating, youre not going to be able to in any > non-functionalist sense. And thats not going to satisfy many people. It will satisfy them in the end. Ill let you [CapitalThorn]gure out why that must be true. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > Thats exactly why AI is not an easy problem. It is the solution to how > the two worlds connect. If you can not explain the connection, you can not > solve AI. Before you [CapitalThorn]gure out what Arti[CapitalThorn]cial Intelligence is, have you [CapitalThorn]gured out what the real thing is? > Its unproven if AI is a problem that we (humans) have the power > to solve. I do happen to belive we have the power to solve it, (or else it > would be kinda stupid for me to spend time on it), but I also know it is > just a leap of faith on my part to believe that. Yoda says: Do not hold your breath until you [CapitalThorn]nd AI, Young Curtis or purple will you turn and pass out you will. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w >Why is it ok to write 0.111... but not ...11111 ? > Because 0.111... is de[CapitalThorn]ned to be the limit of 0.1, 0.11, 0.111, ... > and ...11111 is not de[CapitalThorn]ned to be anything at all. > If you want to choose a de[CapitalThorn]nition for ...11111 thats ok, but make > sure you tell us all what it is before expecting us to discuss it. Yeah, I thought it was just as obvious that it was de[CapitalThorn]ned as much as the number 123 was de[CapitalThorn]ned. But I see these thigns are de[CapitalThorn]ned differently than I thought. So Im going to study more about just how these things have been de[CapitalThorn]ned. The reason I thought it was obvious is that the number 123 to me is de[CapitalThorn]ned as 123 preceeded with an in[CapitalThorn]nite number of leading zeros. If you are allowed to de[CapitalThorn]ne 123 that way, then why would it be wrong to de[CapitalThorn]ne a new number which is the same as 123, except where you change all the zeros to ones? It seemed obvious and valid to me. But apparently, my obvious logic does not mesh with the way it is de[CapitalThorn]ned in the greater community. So I have some work to do to understand both how it is de[CapitalThorn]ned and what the value is of de[CapitalThorn]ning it like that is. The obvious difference (as has been pointed out to me) is that 123 with leading zeros can be thought of as a series which converges on a single number. 123 with leading ones is a series which does not converge on a single number. But why is convergence a required part of what a natural number is? That is what I need to further understand because it conßicts with what I was thinking. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >The obvious difference (as has been pointed out to me) is that 123 with >leading zeros can be thought of as a series which converges on a single >number. 123 with leading ones is a series which does not converge on a >single number. But why is convergence a required part of what a natural >number is? Youre looking at it the wrong way round. The usual way is to start from a de[CapitalThorn]nition of the natural numbers in which they are 0 and its successors. The representation in a [CapitalThorn]nite number of decimal digits is just a more convenient notation. Once you have that, you can start considering extending the representation to an in[CapitalThorn]nite number of digits, and see what consistent interpretations you come up with. -- Richard === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w >The obvious difference (as has been pointed out to me) is that 123 with >leading zeros can be thought of as a series which converges on a single >number. 123 with leading ones is a series which does not converge on a >single number. But why is convergence a required part of what a >natural number is? > Youre looking at it the wrong way round. The usual way is to start > from a de[CapitalThorn]nition of the natural numbers in which they are 0 and its > successors. The representation in a [CapitalThorn]nite number of decimal digits > is just a more convenient notation. Once you have that, you can start > considering extending the representation to an in[CapitalThorn]nite number of > digits, and see what consistent interpretations you come up with. Yes, my problem was that I made some false assumptions about those ideas being equivalent. The differences however seem to turn out to be signi[CapitalThorn]cant. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? >Why is it ok to write 0.111... but not ...11111 ? > Because 0.111... is de[CapitalThorn]ned to be the limit of 0.1, 0.11, 0.111, ... > and ...11111 is not de[CapitalThorn]ned to be anything at all. > If you want to choose a de[CapitalThorn]nition for ...11111 thats ok, but make > sure you tell us all what it is before expecting us to discuss it. > Yeah, I thought it was just as obvious that it was de[CapitalThorn]ned as much as the > number 123 was de[CapitalThorn]ned. But I see these thigns are de[CapitalThorn]ned differently > than I thought. So Im going to study more about just how these things > have been de[CapitalThorn]ned. > The reason I thought it was obvious is that the number 123 to me is de[CapitalThorn]ned > as 123 preceeded with an in[CapitalThorn]nite number of leading zeros. If you are > allowed to de[CapitalThorn]ne 123 that way, then why would it be wrong to de[CapitalThorn]ne a new > number which is the same as 123, except where you change all the zeros to > ones? It seemed obvious and valid to me. But apparently, my obvious > logic does not mesh with the way it is de[CapitalThorn]ned in the greater community. > So I have some work to do to understand both how it is de[CapitalThorn]ned and what the > value is of de[CapitalThorn]ning it like that is. > The obvious difference (as has been pointed out to me) is that 123 with > leading zeros can be thought of as a series which converges on a single > number. 123 with leading ones is a series which does not converge on a > single number. But why is convergence a required part of what a natural > number is? That is what I need to further understand because it conßicts > with what I was thinking. The natural numbers are usually the starting point of arithmetic, and the standard basis for them is usually taken as something like the Peano axioms, which make no reference to the numerals by which they are represented. See, for example, http://en.wikipedia.org/wiki/Peano_axioms The numeral representation for the naturals comes later, and starts with 0 or 1, depending on whose version of the Peano axioms you use, and as the natural numbers themselves increase builds up gradually the number of digits used, but never to more that a [CapitalThorn]nite number of digits for a [CapitalThorn]nite natural number. === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > The natural numbers are usually the starting point of arithmetic, and > the standard basis for them is usually taken as something like the Peano > axioms, which make no reference to the numerals by which they are > represented. > See, for example, http://en.wikipedia.org/wiki/Peano_axioms intersting things which shows how much more I have to learn on the history of all the work done in these related areas. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > How many times are you going to repeat the same question without > bothering to read or respond to the many answers you have already > received? Your funny. I had already posted 3 replies addmiting that I understood the Do not get confused by the fact I post a lot and many times dont read all the replies ahead of time. Its midnight my time and Im just now the thread that come after this that I have not yet been able to read including two of yours. Im doing my best to get through them all. Ive been reading and posting for about 8 hours today already. Give me time. :) -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > How many times are you going to repeat the same question without > bothering to read or respond to the many answers you have already > received? Ive been reading and responding non stop for many hours today. Im sorry, but I cant go any faster. And Im sure the group thinks Ive posted way to much as it is. The thead will die soon, dont worry. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > I dont know what you mean by solve AI. I use it in the general sense of being able to build a machine with all the powers of cognition and behavior that humans have. I like to call that strong AI but that term actually means something different already in the AI argument. > Claude Shannon, the creator of > information theory, has an interesting argument that machine > intelligence is possible. He says, We are machines, Thats what you can not prove. It is something many of us accept on faith where as other reject it on faith and thats where we stand. Neither side can put forth a compelling argument which the other side will accept. The ultimate proof which I will present is a machine which duplicates all powers of human behavior and cognition. However, even that will not quickly convince the other side. They will call the machine a zombie which acts like a human, but is not conscious. Over time, that stance will vanish from the culture. > and we are > intelligent. Its hard to refute that. Actually its easy to refute and hard to defend. If I say, we are not machines, we are conscious and machines are not, and never can be, how can you prove me wrong? All you can really do is say no, thats not true. That AI debate is 50+ years old and no one has settled it. > A human being is a collection of > a [CapitalThorn]nite number of atoms, and we have self-awareness, consciousness, and > intelligence. In[CapitalThorn]nity is not required. Yeah, that is how I think. And I take it a step further and say in[CapitalThorn]nity is not even possible. But this is clearly not how everyone looks at this. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > Actually its easy to refute and hard to defend. If I say, we are not > machines, we are conscious and machines are not, and never can be, how can > you prove me wrong? All you can really do is say no, thats not true. > That AI debate is 50+ years old and no one has settled it. And no one will. In point of fact you cannot prove than anyone but you has a mind. The only things you can observe about other people are 1. Their behaviour. 2. Their physical makeup. 3. Some of the chemical and electrical processes. What do you notice about this list? Consciousness and mind is not on it. You cannot establish by empricial means that anyone but you has a mind. The rest of us can be very complicated zombies. Or maybe there is not difference between being a complicated zombie and having a mind. Maybe there is no such thing as a mind. After all, humans have been slicing each other open for over 10,000 years and no one has ever found a mind. > Yeah, that is how I think. And I take it a step further and say in[CapitalThorn]nity > is not even possible. But this is clearly not how everyone looks at this. Is there a largest integer? What is it? Add one and be ashamed. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? > I use it in the general sense of being able to build a machine with all the > powers of cognition and behavior that humans have. I like to call that > strong AI but that term actually means something different already in the > AI argument. I will tell you how to make an intelligent machine and have lots of fun doing it, too. Find a fertile female person, have sexual congress with her, and make babies. There are your intelligent machines. You dont have to be an engineer or a mathematician to do it, either. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? > I will tell you how to make an intelligent machine and have lots of fun > doing it, too. Find a fertile female person, have sexual congress with > her, and make babies. There are your intelligent machines. You dont > have to be an engineer or a mathematician to do it, either. Indeed. Thats what I call realism! Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? In sci.logic, Han de Bruijn : >> I will tell you how to make an intelligent machine and have lots of fun >> doing it, too. Find a fertile female person, have sexual congress with >> her, and make babies. There are your intelligent machines. You dont >> have to be an engineer or a mathematician to do it, either. > Indeed. Thats what I call realism! > Han de Bruijn The main problem with that method is that it requires an act of congress. (That, and 20 years of schooling, feeding, etc...) :o) -- #191, ewill3@earthlink.net Its still legal to go .sigless. === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > To show that ONE table of integers is missing some integers does not > show that EVERY table of integers must be missing some integers, and it > is the EVERY table which is essential to the issue. What I thought I was showing, is that by using the diagonal argument, I could prove that a table which included every natural number, was actually missing some natural numbers. If so, that creates a contradiction where there shouldnt be a contradiction. But, my de[CapitalThorn]ntion of natural numbers as it turns out didnt match what others use to de[CapitalThorn]ne natural numbers, so the contradiction which I saw wasnt a contradiction for them, and it didnt show anything useful or new for them. So now I need to further study how natual numbers and real numbers are de[CapitalThorn]ned and to futher understand why people choose to de[CapitalThorn]ne them like they do. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? >>To show that ONE table of integers is missing some integers does not >>show that EVERY table of integers must be missing some integers, and it >>is the EVERY table which is essential to the issue. > What I thought I was showing, is that by using the diagonal argument, I > could prove that a table which included every natural number, was > actually missing some natural numbers. This is a ßat out contradiction. A table is essentially a function whose domain is the natural integers. A function whose range includes the integers does not miss a thing (by de[CapitalThorn]ntion). A table which as all the integers clear is not missing any. Question: Are you capable of comprehending simple de[CapitalThorn]nitions? Are you capable of following a straightforward mathematical proof? If the answer to either of these questions is no, then forget about trying to understand mathematics and refrain from making judgements about matters in which you are demonstrably incompetent. Bob Kolker === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > So now I need to further study how natual numbers and real numbers are > de[CapitalThorn]ned and to futher understand why people choose to de[CapitalThorn]ne them like > they do. > > (Click on What are numbers?) some very basic set theory nomenclature which I am forced to guess at its meaning. Like Ive said many times now, I have my work cut out for me. :) -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > some very basic set theory nomenclature which I am forced to guess at its > meaning. Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house. Robert A. Heinlein Bob Kolker === Subject: Re: Cantors diagonal proof wrong? > Anyone who cannot cope with mathematics is not fully human. At best he > is a tolerable subhuman who has learned to wear shoes, bathe, and not > make messes in the house. Robert A. Heinlein Heinlein said a lot of silly things. === Subject: Re: Cantors diagonal proof wrong? > My interest is not in the physical world alone, its in the relationship > between the physical and mental worlds. To solve AI, we have to completely > understand that relationship. Completely? I suggest you change [CapitalThorn]elds. V. -- email: lastname at cs utk edu homepage: www cs utk edu tilde lastname === Subject: Re: Cantors diagonal proof wrong? > However, until you reach an in[CapitalThorn]nite number of digits, neither represents a > single point. What about 1/9? Is that a single point? And can you come up with a decimal representation other than 0.1111... for it? -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > However, until you reach an in[CapitalThorn]nite number of digits, neither > represents a single point. > What about 1/9? Is that a single point? And can you come up with a > decimal representation other than 0.1111... for it? Well, this is where it gets interesting. If is clearly a single point if you simply de[CapitalThorn]ne it to be so. And that is exactly what seems to be common practice. Its how I both was taught to think about the idea, and how I have always talked about it in the past. All the irrationals as well as the rationals exist as a precise, and single point on the real number line. But now Im starting to wonder if there may be value in looking at this differently. Might it be valid to think of division of 1 by 9 as an in[CapitalThorn]nite processes (algorithm) that produces closer and closer approximations, yet is always unable to produce the actual point on the line? When we write 1/9, are we making a reference to a single point on the line, or are we making reference to the algorithm which produces closer and closer approximations to the correct name for that point? Do all the points really exist if you can not name them? What do you gain or loose by talking like this? That is what Im interested in exploring. The reason I started to explore these ideas is because I think the mind is [CapitalThorn]nite. It can only name a [CapitalThorn]nite number of things. It can only recognize, and respond, in a [CapitalThorn]nite number of ways. It only has a [CapitalThorn]nite number of atoms, and neurons, to work with. It can not, do anything in[CapitalThorn]nite. To talk as if an in[CapitalThorn]nite set can exist (the set of all points on the real number line between 0 and 1 for example), can be related to believing the mind can create an in[CapitalThorn]nite set of ideas. But if the mind can not do it, then the in[CapitalThorn]nite set of ideas can not exist for us. Meaning, we can use language to de[CapitalThorn]ne the notion of an in in[CapitalThorn]nite set easy enough, but the idea we de[CapitalThorn]ne, cant actually exist in this universe because it would require a matching in[CapitalThorn]nite mind. Ideas are physical, and have [CapitalThorn]nite physical limits. Language gives us the power to talk about an imaginary universe where minds exist that can manipulate in[CapitalThorn]nite sets. But what is the advantage to exploring the properties of that imaginary universe if that universe isnt the one we exist in and doesnt have anything in common with the universe we exist in? The only thing that does seem in[CapitalThorn]nite in our universe is space and time. And maybe only time is really in[CapitalThorn]nite. So we can de[CapitalThorn]ne a processes, which runs forever - like counting. But to assume it will ever [CapitalThorn]nish takes us out of our universe of existence and into a universe that does not exist (at least for us). I dont know if any of the above logic is valid, or even useful, though my gut feeling is telling me its very useful to learn to talk and think in these ways. Im only throwing it out as food for thought, and trying to understand its implications. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? >>What about 1/9? Is that a single point? And can you come up with a >>decimal representation other than 0.1111... for it? > But now Im starting to wonder if there may be value in looking at this > differently. Might it be valid to think of division of 1 by 9 as an > in[CapitalThorn]nite processes (algorithm) that produces closer and closer > approximations, yet is always unable to produce the actual point on the > line? > When we write 1/9, are we making a reference to a single point on the > line, or are we making reference to the algorithm which produces closer and > closer approximations to the correct name for that point? Do all the > points really exist if you can not name them? > What do you gain or loose by talking like this? That is what Im > interested in exploring. Why does nobody point out that rational numbers are just de[CapitalThorn]ned as an _ordered pair_ of two whole numbers? There is nothing approximate in 1/9, as it is the ordered pair (1,9). The sum of two such ordered pairs, for example, is de[CapitalThorn]ned as (a,b) + (c,d) = (a.d + b.c, b.d) ; which is quite evident if you replace the comma separator by a slash. (On the other hand, how about the fact that 2 is approximated well by 1 + 1/2 + 1/4 + 1/8 + 1/16 + .... = 2 ? Or should I say 1 = 0.999999... thereby causing that good old thread to start again) :-) Irrational numbers are a different story altogether. They originated in geometry, through the well known theorem by Pythagoras: c^2 = a^2 + b^2. For a = b = 1 you get the equation c^2 = 2 , which can only be solved by an extension of the rationals. Curt, have you ever seen the _proof_ that the square root of 2 can not be written as a fraction m/n ? Here comes: http://www.homeschoolmath.net/other_topics/proof_square_root_ 2_irrational.ph p Han de Bruijn === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > Curt, have you ever seen the _proof_ that > the square root of 2 can not be written as a fraction m/n ? > Here comes: > http://www.homeschoolmath.net/other_topics/proof_square_root_ 2_irrational > .php > Han de Bruijn -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > But now Im starting to wonder if there may be value in looking at this > differently. Might it be valid to think of division of 1 by 9 as an > in[CapitalThorn]nite processes (algorithm) that produces closer and closer > approximations, yet is always unable to produce the actual point on the > line? Nonesense. One can always divide a line segment into N equal parts by a well known geometric construction. So getting the point on the segment corresponding to k/N for k = 0,1,...N is trivial. Or equivalently choose a unit length and lay out multiples of this length on an in[CapitalThorn]nite ray. Again one easily constructs points corresponding to k*N. Why do you complicate a very straightforward matter? Bob Kolker === Subject: Re: Cantors diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT{w > But now Im starting to wonder if there may be value in looking at this > differently. Might it be valid to think of division of 1 by 9 as an > in[CapitalThorn]nite processes (algorithm) that produces closer and closer > approximations, yet is always unable to produce the actual point on > the line? > Nonesense. One can always divide a line segment into N equal parts by a > well known geometric construction. So getting the point on the segment > corresponding to k/N for k = 0,1,...N is trivial. Or equivalently choose > a unit length and lay out multiples of this length on an in[CapitalThorn]nite ray. > Again one easily constructs points corresponding to k*N. Why do you > complicate a very straightforward matter? Because most things in life that seem straightforward turn out later to be anything but that. I enjoy [CapitalThorn]nding the exceptions, so I search. But like [CapitalThorn]shing, many times, there is just nothing to be caught. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantors diagonal proof wrong? > Because most things in life that seem straightforward turn out later to be > anything but that. I enjoy [CapitalThorn]nding the exceptions, so I search. But like > [CapitalThorn]shing, many times, there is just nothing to be caught. So far you have found nothing of signi[CapitalThorn]cance and you have revealed a great deal of your mathematical incompetence. Do not give up your day job and forget about a career in mathematics. You dont have the head for it. Bob Kolker === Subject: matrix diagolization In the diagonalization operation of a matrix A. [Q Gamma] = eig(A) Q= 1.0000 0 -1.0000 0 0 1.0000 0 -1.0000 0 0 0.0000 0 0 0 0 0.0000 Gamma= -0.0113 0 0 0 0 -0.0253 0 0 0 0 -0.0113 0 0 0 0 -0.0253 It seems that the eigenvalues are not different and the diagolization is actually failed. However, I was expecting the matix diagonalizable. I am wondering whether there is any techniques dealing with the matrix that can NOT be diagonalizable to simplify the math reasoning? -- -- Yan ZHANG http://www.geocities.com/iam_yanzhang/ === Subject: 2-manifold metric spaces with many symmetries A property of euclidean 2-D space is that: Given two congruent triangles with distinct edge-lengths, say triangle(A, B, C) and triangle(A, B, C) , then there is just one isometry that maps the [CapitalThorn]rst triangle to the second. ( A, B, C, A, B, C are points in euclidean 2-D space.) [ Property 1 ] I think the same is true for a torus, viewed as a [CapitalThorn]nite-height cylinder with vertical axis of rotation and where the upper boundary of the cylinder is glued to the lower boundary. For S^2 embedded in R^3 and where distances are measured along arcs of great circles, I think Property 1 also holds. For the hyperbolic plane, I dont know whether it has Property 1. For a small enough open neighborhood of a point on the torus (with metric as above), this will be isometric to some open set in R^2. For a 2-D metric space with Property 1, Im wondering what geometries are possible locally. I wish to exclude from consideration surfaces such as the surface of a cube, but I presently dont know how to say it precisely. David Bernier === Subject: how to [CapitalThorn]nd the autocorrelation and spectrum of the receiver signal in mobile communication? Hi all, I am wondering about the simplest model in mobile communicaiton, multipath. Suppose the received signal has random uniform [0, 2*pi] phase due to multipath fading, and also has Rayleigh distribution on its amplitude(assuming no direct line path), and also has doppler frequency shift in carrier frequency. The signal then can be modelled as r(t)=A*exp(j*2*pi*(f+delta_f)*t + THETA) where A and THETA are random variables... How to [CapitalThorn]nd its autocorrelation function and then Power Spectrum Density? === Subject: Monster of a problem...extensions of cyclic groups posting-account=4XHZpAwAAAB6b7qV0WqNfmj98QBHBa2z I have a problem that seems like an impossible monster. I want to describe all central extensions of a cyclic group Z_n by a cyclic group Z_m (obviously the answer depends on m and n). The more I work on this the more special cases I keep [CapitalThorn]nding. If m=n is prime, then there is just the abelian group Z_{m^2}. If n>m and both prime, then we get two cases: if m|n-1 --> Z_{nm}, semidirect product of Z_n and Z_m (I came up with a presentation of this group) otherwise --> Z_{nm} Then, it seems there is even more depending on whether or not (n,m)=1. I cant even imagine what can happen when m,n are not prime _and_ not relatively prime. I do not know much about central extensions except that the normal subgroup must lie in the center of the extension group. I guess I am hoping for some important theorem or idea about central extensions that will place very large restrictions on all possible extensions. Can anyone help? James === Subject: Re: Monster of a problem...extensions of cyclic groups > I have a problem that seems like an impossible monster. I want to > describe all central extensions of a cyclic group Z_n by a cyclic group > Z_m (obviously the answer depends on m and n). > The more I work on this the more special cases I keep [CapitalThorn]nding. If m=n > is prime, then there is just the abelian group Z_{m^2}. And also Z_m x Z_m. > If n>m and both > prime, then we get two cases: > if m|n-1 --> Z_{nm}, semidirect product of Z_n and Z_m (I came up with > a presentation of this group) ? In the semidirect product Z_n is not central. > otherwise --> Z_{nm} > Then, it seems there is even more depending on whether or not (n,m)=1. > I cant even imagine what can happen when m,n are not prime _and_ not > relatively prime. > I do not know much about central extensions except that the normal > subgroup must lie in the center of the extension group. In a central extension G of Z_n by Z_m, Z_n is contained in the centre of G and G/Z_n is isomorphic to Z_m. Its pretty clear from this that G has to be abelian .... -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ New job listings at http://jobs.phds.org - Jobs for PhDs List your job at no cost! http://jobs.phds.org/jobs/post * Junior PhD Quantitative Researcher .89 Hedge Fund: NJF Search International, London, New York or Chicago. Premier Hedge Fund is seeking talented individuals to join their growing team in the capacity of Junior Quantitative Researcher. They desire a candidate with strong quantitative abilities. The successful candidate will participate in the... * Junior PhD Quantitative Developers, C++ - Hedge Fund: NJF Search International, London, New York or Chicago. Premier Hedge Fund is seeking talented individuals to join their growing team in the capacity of Junior Quantitative Developers. They desire a candidate with strong quantitative abilities. The successful candidate will participate in the... * Teach Math or Science in NYC: The New York City Teaching Fellows, New York, NY. You remember your [CapitalThorn]rst grade teachers name. Who will remember yours? Become a NYC Teaching Fellow. The NYC Teaching Fellows program is a highly selective, innovative path to enter the classroom and make a difference in New York City s high-need... === Subject: Atan2(x, y) ?? How to implement?? sin(x) = x - x^3/3! + ... + (-1)^(n-1)*x^(2n-1)/(2n-1)! cos(x) = 1 - x^2/2! + ... + (-1)^(n)*x^(2n)/(2n)! Whats the formula of Atan() ???? === Subject: Re: Atan2(x, y) ?? How to implement?? posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs It looks like youve seen Taylor series for sin(x) and cos(x). What youve shown are truncated Taylor series, which are therefore approximations. How good the approximations are depend on how many terms you include and what value of x you use (these are expansions near x=0, so the approximation gets worse as x gets far from 0). Furthermore, these approximations dont converge well for any value of x, and dont converge at all for most. Pick a value of n and see what happens at x=0.5 and x = 1.5 for instance. Most computer implementations of the trig functions are based on what are called the CORDIC algorithms. Do a search for that and youll [CapitalThorn]nd plenty of material. === Subject: Re: Atan2(x, y) ?? How to implement?? > sin(x) = x - x^3/3! + ... + (-1)^(n-1)*x^(2n-1)/(2n-1)! In your opinion, what does this mean? Do you mean that for *each* x and *each* n you have that equality. If so, then it would still be true for each x and n = 1; that is, it would be true that sin(x) = x. You dont really believe that, do you?! Or do you mean that your equality is valid for *each* x and *some* n? Then the sinus function would be a polynomial function. > Whats the formula of Atan() ???? > Please give me a hint. Please start by providing some correct statements concerning sin(x) and cos(x). Jose Carlos Santos === Subject: Re: FINALLY THE TRUMAN PROBE GOES PUBLIC > Youre not helping Hercs recovery with replys like this. Recovery? Not enough meds in the world for that to happen. Ogie === Subject: Re: strange question about mortgage interest calculation > Ive got a mortgage problem that Im having trouble [CapitalThorn]nding a solution > for... I hope the brilliant minds here can help me out. :-) > Im trying to [CapitalThorn]nd a way to compare variable & [CapitalThorn]xed rate mortgages, > assuming a similar term and [CapitalThorn]xed payments throughout the term. > For example, for a $100,000 mortgage at 5% for 25 years, my monthly > payments will be $584.59. At the end of a 5-year term, my remaining > principal will be $88,580.19 and I will have paid $23,665.69 in > interest. > With a variable rate mortage at 3.5% (with monthly payments of $500.62) > at the end of the term, I end up owing $86,320.40 and paying $16,357.82 > in interest. > What Id like to determine is, at any given point in the variable-rate > term, how high can the interest rate rise and still give me either > lower total interest or lower outstanding principal (whichever is > easier to solve for) at the end of the term. > Im making a number of assumptions here, many of which wont be valid > in the real world: > - interest is calculated monthly (I live in Canada where the standard > is semi-annually) > - payments will be [CapitalThorn]xed throughout the term > - interest rates will only be adjusted once during the term, and only > at the beginning of a month > Does this make sense? It is even possible to solve without generating > amortization tables for all the different possible scenarios? The formulas that you need to start with: Given a capital K (in your case 100000) A monthly interest rate p (in your case 5% or 3.5%) An interest factor r = (1 + p/1200) Monthly payment S = 584.59 Outstanding principal after n months = K * r^n - S * (r^n - 1) / (r - 1) To make things comparable, compare the two scenarios: In each case assume K = 100000, S = 584.59. a. Fixed rate of 5% after 25 years = 300 months: Outstanding principal = K * (1 + 5/1200)^300 - S * ((1 + 5/1200)^300 - 1) / (5/1200) This is approximately zero. b. Rate of 3.5% for n months, followed by x% for (300-n) months: After n months: K1 = K * (1 + 3.5/1200)^n - S * ((1 + 3.5/1200)^n - 1) / (3.5 / 1200) After further 300-n months: K2 = K1 * (1 + x/1200)^(300-n) - S * ((1 + x/1200)^(300-n) - 1) / (x / 1200) You want K2 = 0, or more precise, K2 = same value as the result in (a). Easiest way in practice: Make a spreadsheet, enter the three formulas, depending on K, S, p, n and x. If you dont know S try values until p% after 300 months gives 0. Then try values for x. === Subject: Re: Feynmans fallacy > Ive been reading Richard Feynmans Six Not-So-Easy Pieces. > In his [CapitalThorn]rst chapter he makes the point that, because the laws of the > universe will be perceived the same from wherever the person observes > it, there is no way to [CapitalThorn]nd the center of the universe. > That is a fallacy. That any given point can provide the vintage point > of observation of the laws of the universe, means not that any point > can be the center of the universe but rather that the laws of the > universe are omnipresent within the universe - which, of course, they > would be expected to be. Let me try something : If there IS a center of the Universe, then there must also be an edge (otherwize there is no way to determine the center). Now, how can you determine the edge of the Universe, if the laws of physics are the same for all observers in the Universe ? If you are standing 1m from the edge of the Universe, and you kick a ball over the edge, and you see it Ôdisappear or explode or whatever else would happen at the edge, then your observations of the laws of physics are different from another observer 1 km away from the edge (who sees the ball disappear 1km away). If you would draft the laws of physics, you would draft them as Ôball behaves normal until 1m away from me, and the other observer the laws of physics would be drafted Ôball behaves normal until 1km away from me. Clearly different laws. So in itself, Feynmans statement is a self-ful[CapitalThorn]lling prophecy : If everyone sees the laws of physics the same, then there is no point of reference (such as an edge or a center of the Universe). > In no way does that state that there is no such thing as the center > of the universe, or the starting point for the laws of the universe, > or a substance or a creator or a divine essence from which the universe > comes. It DOES state that there is no center of the universe, but it does not say that there was no beginning : There could still be a Ôbeginning of time, but that can only have happened if it was also the beginning of space. Otherwize, the laws of physics would differ per observer. But also a Ôever-existing Universe, without a beginning is possible under this statement. The statement does not say anything about where the Universe came from. > Ilya Shambat. === Subject: Re: too much information! >> >> > 7) Pixels on a computer screen. 16,000,000 colors for each > pixel. > How many different pictures are possible? >> >>For a 640x480 screen, its 10^2219433, slightly larger than a > googol. >> >> Id say youre engaging in understatement, but that would be an >> understatement. Slightly larger than 10^100?? This number makes >> a googol look like one. That is, 1. >> Not even close, there are 32 bits for EACH pixel. > For 16 million colors its 24 bits, not 32. Actually, its > 16777216 = 2^24 >> 32 * 32 for 2 pixels >> 32* 32 * 32 for 3, and 32 ^ (1024 * 768) for a screenful, and thats >> just a 17 inch monitor. > You said _a_ computer screen. You didnt say _your_ computer screen. > And 640x480x24 is a valid screen setting. In fact, my old digital > camera > just happens to take 640x480 pictures. >> I have used 1200 * 1600 pixels. > So its 10^18495282 instead. Its still bigger than a googol. >> Total number of possible pictures = Invalid input for function > Whats the matter, dont know how to use logarithms? Sure do, bubba. Whats the matter, dont know how to raise 32 to the power (1024 * 768)? Androcles. >> Never underestimate yourself, let others do it for you. - Inspector >> Morse, Colin Dexter. > Heres a nickel, kid. Get yourself a better computer. > - the unix guru from Dilbert >> Androcles. === Subject: Re: too much information! >> |>> 7) Pixels on a computer screen. 16,000,000 colors for each >> pixel. >> |>> How many different pictures are possible? >> |>For a 640x480 screen, its 10^2219433, slightly larger than a >> googol. >> |> Id say youre engaging in understatement, but that would be an >> |> understatement. Slightly larger than 10^100?? This number >> makes >> |> a googol look like one. That is, 1. >> | Not even close, there are 32 bits for EACH pixel. >> | 32 * 32 for 2 pixels >> | 32* 32 * 32 for 3, and 32 ^ (1024 * 768) for a screenful, and >> thats >> | just >> | a 17 inch monitor. I have used 1200 * 1600 pixels. >> | Total number of possible pictures = Invalid input for function >> It doesnt matter what the size of the monitor is (how many inches). >> What matters is the screen resolution. >> My screen has 2048 x 1536 pixels. ___________________________Gerard >> S. > Well, bits of storage, anyway. > 2048 x 1536 x 32 is about 10^8 bits. Doesnt take much of a hard drive > to > outdo that these days. > -- > Timo Nieminen - Home page: > http://www.physics.uq.edu.au/people/nieminen/ > Shrine to Spirits: > http://www.users.bigpond.com/timo_nieminen/spirits.html ROFLMAO! Androcles === Subject: Re: Cantor K.O.d -- again! - Further explication <1105067703.01d8dc6ae69bd62db1e7610dc412b0a0@teranews> <1105333008.421cd7ce3aebd9d9b246b04a1ea8cc90@teranews> posting-account=zz1hbQ0AAACwD6SoADdUXo1mXFm4HYlV >>No. As you yourself stated, in the Emphasis Added section, >>one doesnt in fact reach the coordinates (1,-1), since no point >>(i.e. no natural number) is mapped to it. One can get arbitrarily >>close, but your one hour of travel wont get you there; there will >>always be another member of L* which you havent mapped yet. > I said no such thing, obfuscatory imbecile. > I politely request that you refrain from ad hominem arguments > of this sort. They dont add to the discussion. > The fact that (1,-1) > has no list symbol mapped to it doesnt mean that one doesnt > get there, any more than, in a calculus problem, the fact that > there are an in[CapitalThorn]nite number of intermediate points to be visited > in moving along a [CapitalThorn]nite distance from point A to point B prevents > one from arriving at point B. If you accept one you must accept > the other. Do you reject point-set geometry and the calculus? > In the [CapitalThorn]rst place, no, I dont reject point-set geometry and the calculus. In > the second place, whether one gets from point A to point B (Im not quite sure > how one would phrase this rigorously) in a calculus problem depends on the > environment in which one is working.If one wants to get from 0 to 1 while > working with the closed interval [0,1] (or any interval containing it), I > certainly agree that one can move through an in[CapitalThorn]nite number of points to > reach 1. On the other hand, if one wants to get from 0 to 1 in the half-open > interval [0,1), one still has an in[CapitalThorn]nite number of points to move through, but > one can never reach one, since it is not in the interval. (That is, one cannot > reach it because it is not there to be reached, although one can come > arbitrarily close - 1 is a limit point of the interval.) The same is true of > your construction. Since (1,-1) is not in your set (since it has no list symbol > mapped to it), one can come arbitrarily close to it but not reach it. Obfuscatory nonsense, since I stated quite explicitly in my original post that the journey began at the point (0,0), ended in the point (1,-1) and travelling along the diagonal line connecting these two points, stopping at each of the mapping points of the diagonal. msadkins04@yahoo.com Mark Adkins === Subject: interior of a nonplanar polygon? Is there a standard de[CapitalThorn]nition for the interior of a nonplanar polygon? I can think of more than one way to consider such a phrase, including the possibility that it is not conventionally de[CapitalThorn]ned. I Googled and failed to [CapitalThorn]nd anything on this. Ted Shoemaker === Subject: Re: just 5 quick answers then I can summarise and GO >> In this (simpli[CapitalThorn]ed) context, a number a is transcendental if there > no >> polynomial q(x) with rational coef[CapitalThorn]cients such that q(a) = 0. (In > the >> general case, we say that a number b is transcendental over a [CapitalThorn]eld F >> if there is no polynomial f with coef[CapitalThorn]cients in F so that f(b) = 0) >> pi, for instance, is transcedental *and* computable. I think youre >> looking for uncomputable numbers. ;-) >> Ôcid Ôooh > cool! Over what [CapitalThorn]eld is pi transcendental, then? It cant be Ôthe > reals because the polynomial is trivial: (1/(pi))(pi)**1 - 1 = 0 Thats why he speci[CapitalThorn]ed rational coef[CapitalThorn]cients. -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: ANSWER THE FCKING QUESTION GHOST >Results 1 - 10 of about 580 for david ullrich fool. this makes *me* look ridiculous? Youre the one saying HES RIGHT YOURE WRONG then trying to look smart still posting on the same thread. If your statements get challenged then you 1/ support them 2/ admit error 3/ go away Herc === Subject: Re: ANSWER THE FCKING QUESTION GHOST > It would appear that any [CapitalThorn]nite number of digits apears on the list. It > subtleties of the different perspectives that can be taken when no you moron, there is no subtelty to any [CapitalThorn]nite number. How is that different to It would appear any number of digits appears on the list. ? Herc === Subject: Re: ANSWER THE FCKING QUESTION GHOST >>It would appear that any [CapitalThorn]nite number of digits apears on the list. It >>subtleties of the different perspectives that can be taken when > no you moron, there is no subtelty to any [CapitalThorn]nite number. > How is that different to > It would appear any number of digits appears on the list. ? Thats not the subtlety. Things like determining whether you are focused on how many digits any given element shares with pi vs. how many digits of pi can be found somewhere on the list would be a subtlety. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false? ---------------------------------------------s-o-s----------- --------------- --------------- > Perhaps there may indeed be permanently unknowable truths. > Thats intriguing. > Consider the following. > The [CapitalThorn]rst six digits of Pi > 3.14159 > has [CapitalThorn]ve digits in the range 1 to 5 > (left hand) > and one in the range 6 to 9 and 0 > (right hand). > The following digit is a 2, that makes > it 6 to 1 for left to right. > Than a six, so right hand gets one more, > makes it 6 to 2. > One of the following statements is true: > A) left wins > B) right wins > C) It will change, sometimes > more digits in the lower range, and sometimes > more in the higher. > May be, we will never know the true > answer > Maybe. It is an open question. > Maybe we will [CapitalThorn]nd the answer next year (dont bet on it), or maybe never. > Event if there *is* a perfectly good answer, we might never [CapitalThorn]nd it. Or > maybe there is no proof either way using methods we would consider sound. > now i just need a proof of this. > Dont hold your breath. Exactly, they cross over with P=1 but expected time is oo. See http://www.ms.uky.edu/~mai/java/stat/brmo.html on random walks. Herc === Subject: Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false? Originator: joshp@xoxy.net (joshp) >PRA isnt strong enough to prove Goedels theorems, for example. According to Shankars _ Mathematics, Machines, and Goedels Proof _; ``Goedels incompleteness theorem represents a signi[CapitalThorn]cant landmark in mathematics. [...] Feferman [Fe82] has shown that the incompleteness proof can be carried out within PRA. (page 141) [Fe82] S. Feferman. Inductively presented systems and the formalization of mathematics. In D. van Dalen, D. Lascar, and J. Smiley, editors, _ Logic Colloquium Ô80 _. North-Holland, Amsterdam, 1982. -- Josh Purinton === Subject: Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false? <41e3ec76$0$575$b45e6eb0@senator-bedfellow.mit.edu> <41e40435$0$575$b45e6eb0@senator-bedfellow.mit.edu> posting-account=_-j7cgwAAADnQK9-r68QgRsgfV-jhA3A >That makes me think that the set of possible truths recognizable to >mathematicians is indeed r.e. Mathematicians may make wild leaps in >discovering their mathematical truth, but for their discoveries to be >accepted, they have to be polishable to something that is recognizable >as a mathematical proof. > The term under discussion was permanently unknowable *truths* (emphasis > mine). The problem here is not just that theoremhood in certain axiomatic > systems is undecidable. The problem is, at least in part, recognizing > which *axioms* are true. This seems silly to me. Suppose someone asks, Are the group axioms true? Obviously, they are true of any set of objects which satisfy them. But suppose he asks again, emphatically, No, but are they *true*? What is that to mean? > The axiom of choice is now accepted as true by most mathematicians. I dont think as true is necessary here. In fact, I think it obfuscates the issue. The acceptance of AC is as much a consequence of sociology as it is of logical consistency. Ôcid Ôooh === Subject: .999... stillstillstillstill =/= 1 I won the debate because I believe in God. I started this debate against an atheist 10 years ago. And I [CapitalThorn]nally won. ha ha ha.... Smarts Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: Difference between AN UNKNOWN and UNKNOWN > They arent things. Its UNKNOWN. > Having an unknown output doesnt make it not a thing. there is no such thing as unknown output. thats what it means > the result ISNT KNOWN. YOU DONT KNOW. > And that doesnt bother me. youre half way there. if you try to assign it to an_unknown then obviously you have some concern. > the program may halt IN THE FUTURE. > it has no HALT *value*. > Time is not a factor in the value of halt(1000). Why do you think it is? BUT IT IS! Stop taking determinism for granted, we are at its limits. What is the value of SE? SE = 1 iff our sun explodes, =1 otherwise > Things that are unknown are not objects. > Why do you think that? Perhaps I just have a different notion the > possible state of things than you. That is why you fail Herc