Timeline for How to tell a paradox from a "paradox"?
Current License: CC BY-SA 4.0
18 events
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| S Nov 20, 2023 at 6:08 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to Wikipedia
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| Nov 19, 2023 at 20:17 | review | Suggested edits | |||
| S Nov 20, 2023 at 6:08 | |||||
| Dec 11, 2018 at 21:53 | comment | added | Toby Bartels | @LSpice : Yes, it is; although after 6 years, I can no longer remember whether or not that was really an accident. | |
| Dec 11, 2018 at 16:46 | comment | added | LSpice | @TobyBartels, is it acceptable to be amused that your 2-part comment on "Banach–Tarski" accidentally used the term twice consecutively? :-) | |
| Oct 1, 2012 at 5:35 | vote | accept | CommunityBot | moved from User.Id=10891 by developer User.Id=35352 | |
| Sep 28, 2012 at 1:24 | comment | added | Toby Bartels | … Banach–Tarski shows a serious contradiction in the formal system ZFC + LM (where LM is the axiom that every subset of the real line is Lebesgue-measurable, proposed by alternate-universe Lebesgue in alternate-1904, and quickly accepted by alternate-universe mathematicians as intuitively obvious). However, BT is overkill for this; the alternate-universe historians refer to the Vitali paradox instead. (If you replace the axiom of choice with dependent choice, then ZF + DC + LM is consistent, proved by Solovay in 1970, assuming the consistency of ZFC plus one inaccessible cardinal.) | |
| Sep 28, 2012 at 1:18 | comment | added | Toby Bartels | There's a definite historical difference between the kind of paradox that is the Russell one and the kind that is the Banach–Tarski one: The Russell paradox demonstrated an inconsistency in a formal system that Frege seriously proposed as a foundation for all mathematics, while the Banach–Tarski paradox did no such thing. But standing at the end of all of that history, we can interpret either in either way. So maybe the Russell paradox is the interesting but unsurprising theorem that the class of all sets is proper (not a real paradox), and Banch–Tarski … | |
| Sep 27, 2012 at 8:05 | comment | added | Qfwfq | Voting to close as "not a real question" (not even from a metamathematical or philosophical viewpoint) | |
| Sep 26, 2012 at 17:39 | answer | added | none | timeline score: 4 | |
| Sep 26, 2012 at 14:50 | answer | added | Andrej Bauer | timeline score: 10 | |
| Sep 26, 2012 at 14:34 | answer | added | Marco Caminati | timeline score: 9 | |
| Sep 26, 2012 at 11:52 | answer | added | Andreas Blass | timeline score: 18 | |
| Sep 26, 2012 at 10:02 | answer | added | Florian | timeline score: 6 | |
| Sep 26, 2012 at 9:43 | comment | added | Pietro Majer | Note that in mathematics, and even in the current language, paradox should refer to the former acception you mentioned, that is, a counter-intuitive fact, something that is contrary to the common opinion, thus a priori not dangerous. For the latter notion, I think you mean antinomy, something that leads to a contradiction. | |
| Sep 26, 2012 at 8:55 | comment | added | Yemon Choi | @WillieWong (1) quite correct, oops (2) I knew that (non-)amenability was the main issue here but I couldn't recall the precise details in haste | |
| Sep 26, 2012 at 8:44 | comment | added | Willie Wong |
@Yemon: I think you forgot finite additivity in your list of conditions. :-) Anyway, Terry Tao gave a good description of the difference between the dimensions on his blog: terrytao.wordpress.com/2009/01/08/…
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| Sep 26, 2012 at 7:57 | comment | added | Yemon Choi | The point of Banach-Tarski is not just that one ball equals two balls - it is that the disassembly and assembly maps are rigid motions of Euclidean space, and hence there is no non-zero functional that is defined on all subsets and is invariant under the isometry group of Euclidean space. This only works in dimensions 3 and above; there is no B-T paradox IIRC in dimensions 1 or 2. | |
| Sep 26, 2012 at 7:45 | history | asked | user10891 | CC BY-SA 3.0 |