Comments on: The Axiom of Choice is Wrong http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ Geometry, Topology, Categories, Groups, Physics, . . . Everything Sat, 19 Jul 2008 10:26:21 +0000 http://wordpress.org/?v=MU By: Adib Ben Jebara http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-3074 Adib Ben Jebara Sun, 13 Jul 2008 03:42:33 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-3074 My previous web site became innaccessible after an attack. See my web page where I apply the negation of the axiom of choice to physics and philosophy. Adib Ben Jebara. adib.jebara at topnet.tn My previous web site became innaccessible after an attack.
See my web page
where I apply the negation of the axiom of choice to physics and philosophy.
Adib Ben Jebara.
adib.jebara at topnet.tn

]]> By: eliza http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-3059 eliza Sat, 07 Jun 2008 21:46:32 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-3059 hello! ist cool ...is chido me encanta de jonas brothers son fabulosos sobre todo nick jonas es un bue...esta requete bueno... hello! ist cool …is chido me encanta de jonas brothers son fabulosos sobre todo nick jonas es un bue…esta requete bueno…

]]> By: Gorkem Ozkaya http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-3029 Gorkem Ozkaya Wed, 30 Apr 2008 14:58:56 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-3029 I think, in a universe where people have countably infinite memory, the countable version of axiom of choice would no longer be an axiom, it would be a theorem. Because in their understanding of mathematics, proofs would be allowed to contain infinite lines. I think, in a universe where people have countably infinite memory, the countable version of axiom of choice would no longer be an axiom, it would be a theorem. Because in their understanding of mathematics, proofs would be allowed to contain infinite lines.

]]> By: The joy of “pathology” « Mathemusicality http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-2987 The joy of “pathology” « Mathemusicality Sat, 12 Apr 2008 20:46:45 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-2987 [...] are still some people who are pissed off about Cantor’s discoveries, and who would sooner overthrow the standard axioms of mathematics than confront the “paradoxes” of the infinite. Even Hilbert, who had the good sense to [...] [...] are still some people who are pissed off about Cantor’s discoveries, and who would sooner overthrow the standard axioms of mathematics than confront the “paradoxes” of the infinite. Even Hilbert, who had the good sense to [...]

]]> By: Peter http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-1656 Peter Wed, 21 Nov 2007 09:23:51 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-1656 In regards to many of the above comments, the point of this problem has nothing to do with physical prisoners or physical memories or physical hats, especially since it seems unlikely that there is even a physical infinity. The point is that the axiom of choice leads to some unintuitive consequences. For me, this is ok, because the existence of an infinite set is an axiom of ZF set theory, and it seems probable that this axiom does not coincide with the real world. If we assume one "unphysical" axiom, this probably ruins our intuition, so even if the axiom of choice seems intuitive in some choice of phrasings, it could easily lead to nonintuitive results. This is fine and interesting, and I think we shouldn't be scared of it, as long as we realize that it shouldn't necessarily lead to results which agree with the real world. In regards to many of the above comments, the point of this problem has nothing to do with physical prisoners or physical memories or physical hats, especially since it seems unlikely that there is even a physical infinity. The point is that the axiom of choice leads to some unintuitive consequences. For me, this is ok, because the existence of an infinite set is an axiom of ZF set theory, and it seems probable that this axiom does not coincide with the real world. If we assume one “unphysical” axiom, this probably ruins our intuition, so even if the axiom of choice seems intuitive in some choice of phrasings, it could easily lead to nonintuitive results. This is fine and interesting, and I think we shouldn’t be scared of it, as long as we realize that it shouldn’t necessarily lead to results which agree with the real world.

]]> By: Paul Crowley http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-1617 Paul Crowley Mon, 19 Nov 2007 10:25:48 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-1617 A couple of people have said that the prisoners will require a countably infinite memory to remember their agreement. I think it's one power set worse than that, since there are uncountably many equivalence classes. A couple of people have said that the prisoners will require a countably infinite memory to remember their agreement. I think it’s one power set worse than that, since there are uncountably many equivalence classes.

]]> By: Adib Ben Jebara http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-1084 Adib Ben Jebara Thu, 18 Oct 2007 09:44:05 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-1084 See my web page about the negation of the axiom of choice : http://jebara.topcities.com Adib Ben Jebara. See my web page about the negation of the axiom of choice :
http://jebara.topcities.com
Adib Ben Jebara.

]]> By: Math Bloggers - Today’s Top Blog Posts on Mathematics - Powered by SocialRank http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-890 Math Bloggers - Today’s Top Blog Posts on Mathematics - Powered by SocialRank Mon, 01 Oct 2007 09:45:13 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-890 [...] The Axiom of Choice is Wrong « The Everything Seminar [...] [...] The Axiom of Choice is Wrong « The Everything Seminar [...]

]]> By: Surfaces and Spinors « The Everything Seminar http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-854 Surfaces and Spinors « The Everything Seminar Fri, 28 Sep 2007 18:18:57 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-854 [...] geometry is over-representation. Manifolds are conventionally defined via absurdly large, Zorn’s-lemma’d maximal atlases, giving an enormous sea of possible coordinates which could represent an object. [...] [...] geometry is over-representation. Manifolds are conventionally defined via absurdly large, Zorn’s-lemma’d maximal atlases, giving an enormous sea of possible coordinates which could represent an object. [...]

]]> By: Functor Salad http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-848 Functor Salad Fri, 28 Sep 2007 01:25:49 +0000 http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/#comment-848 I agree with the people who say that infinity in general is the problem, not the AC in particular. Tail events are creepy. I think I find Terence Tao's AC-free version (only a finite number have black hats and everyone says "white") quite convincing. Apparently it's essentially the same thing, but with a fixed equivalence class. And I think we don't have to worry about being able to construct an appropriate product measure if we are just appealing to intuitions anyway :) I agree with the people who say that infinity in general is the problem, not the AC in particular. Tail events are creepy.

I think I find Terence Tao’s AC-free version (only a finite number have black hats and everyone says “white” ;) quite convincing. Apparently it’s essentially the same thing, but with a fixed equivalence class. And I think we don’t have to worry about being able to construct an appropriate product measure if we are just appealing to intuitions anyway :)

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