Comments on: Constructive gem: juggling exponentials http://math.andrej.com/2009/09/09/constructive-gem-juggling-exponentials/ Mathematics for computers Mon, 30 Jan 2012 07:14:48 +0000 hourly 1 http://wordpress.org/?v=3.2-beta2-18055 By: Sridhar Ramesh http://math.andrej.com/2009/09/09/constructive-gem-juggling-exponentials/comment-page-1/#comment-12743 Sridhar Ramesh Sun, 25 Oct 2009 19:13:22 +0000 http://math.andrej.com/?p=283#comment-12743 (One tiny comment just for the sake of not leaving errors unremarked upon: I did make one mistake above in post #21: I asserted that the situations where N is isomorphic to 2^(2^N) were another class of intuitionistic counterexamples to CSB. But in doing so, I invoked an injection from 2^N into 2^(2^N), which needn't exist intuitionistically; indeed, in the relevant cases, it generally won't) (One tiny comment just for the sake of not leaving errors unremarked upon: I did make one mistake above in post #21: I asserted that the situations where N is isomorphic to 2^(2^N) were another class of intuitionistic counterexamples to CSB. But in doing so, I invoked an injection from 2^N into 2^(2^N), which needn’t exist intuitionistically; indeed, in the relevant cases, it generally won’t)

]]> By: Andrej Bauer http://math.andrej.com/2009/09/09/constructive-gem-juggling-exponentials/comment-page-1/#comment-12731 Andrej Bauer Fri, 23 Oct 2009 08:49:53 +0000 http://math.andrej.com/?p=283#comment-12731 I took the liberty to edit your post and fixed the broken math. Yes, I agree that ultimately we can show that [0,1] and (0,1) are not homeomorphic, constructively, but for our purposes it is easier to just take your example (0,1) vs `(0,1) union (2,3)`. By the way, it is interesting to think about what "connected" means constructively, and which of the several notions of connectedness (0,1) satisfies. I took the liberty to edit your post and fixed the broken math. Yes, I agree that ultimately we can show that [0,1] and (0,1) are not homeomorphic, constructively, but for our purposes it is easier to just take your example (0,1) vs `(0,1) union (2,3)`. By the way, it is interesting to think about what “connected” means constructively, and which of the several notions of connectedness (0,1) satisfies.

]]> By: Sridhar Ramesh http://math.andrej.com/2009/09/09/constructive-gem-juggling-exponentials/comment-page-1/#comment-12729 Sridhar Ramesh Fri, 23 Oct 2009 08:35:09 +0000 http://math.andrej.com/?p=283#comment-12729 Of course, in a topos where all functions are continuous, the two (bijection and homeomorphism) amount to the same thing. Anyway, that (0, 1) and [0, 1] are non-homeomorphic should follow from the fact that every point in (0, 1) is a cut-point, while the two endpoints of [0, 1] are not. [In satisfying myself of the last line, I invoke near-trichotomy principles like `(p != q) => (p < q or p > q)` whose validity I suppose could vary from particular construction/notion of reals to particular construction of reals, but I'd be shocked if the argument couldn't be ultimately made to go through for any natural notion of reals or real-like quantities]. Anyway, going back to what this was all originally being used for, the specific example of (0, 1) and [0, 1] isn't essential; I gave other, non-topological examples above of how to make the same ultimate point (about intuitionistic failure of Cantor-Schroeder-Bernstein), but even sticking to the topological approach, we can use (0, 1) and (0, 1) U (2, 3) instead, if it makes life easier. Clearly, each of these injects into the other, and clearly, they cannot be homeomorphic (the former is connected, the latter is not). Of course, in a topos where all functions are continuous, the two (bijection and homeomorphism) amount to the same thing. Anyway, that (0, 1) and [0, 1] are non-homeomorphic should follow from the fact that every point in (0, 1) is a cut-point, while the two endpoints of [0, 1] are not. [In satisfying myself of the last line, I invoke near-trichotomy principles like `(p != q) => (p < q or p > q)` whose validity I suppose could vary from particular construction/notion of reals to particular construction of reals, but I'd be shocked if the argument couldn't be ultimately made to go through for any natural notion of reals or real-like quantities].

Anyway, going back to what this was all originally being used for, the specific example of (0, 1) and [0, 1] isn’t essential; I gave other, non-topological examples above of how to make the same ultimate point (about intuitionistic failure of Cantor-Schroeder-Bernstein), but even sticking to the topological approach, we can use (0, 1) and (0, 1) U (2, 3) instead, if it makes life easier. Clearly, each of these injects into the other, and clearly, they cannot be homeomorphic (the former is connected, the latter is not).

]]> By: Andrej Bauer http://math.andrej.com/2009/09/09/constructive-gem-juggling-exponentials/comment-page-1/#comment-12728 Andrej Bauer Fri, 23 Oct 2009 05:33:35 +0000 http://math.andrej.com/?p=283#comment-12728 Showing that [0,1] and (0,1) are not homeomorphic does seem easier than showing there is no continuous bijection from one to the other. We're constructive here, so we just have to make sure we don't use a classical fact. I don't see a very easy argument at the moment. Showing that [0,1] and (0,1) are not homeomorphic does seem easier than showing there is no continuous bijection from one to the other. We’re constructive here, so we just have to make sure we don’t use a classical fact. I don’t see a very easy argument at the moment.

]]> By: Sridhar Ramesh http://math.andrej.com/2009/09/09/constructive-gem-juggling-exponentials/comment-page-1/#comment-12725 Sridhar Ramesh Thu, 22 Oct 2009 23:46:05 +0000 http://math.andrej.com/?p=283#comment-12725 Sorry, I should not have said "continuous bijection", but rather, continuous functions which are inverse to each other; i.e., a homeomorphism. Is it not true that there can be no homeomorphism between [0, 1] and (0, 1)? Sorry, I should not have said “continuous bijection”, but rather, continuous functions which are inverse to each other; i.e., a homeomorphism. Is it not true that there can be no homeomorphism between [0, 1] and (0, 1)?

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