Timeline for Applications of set theory in physics
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 1, 2016 at 23:43 | comment | added | Gerry Myerson | Kristeva gets a thorough going-over in Chapter 3 of Bricmont and Sokal, Intellectual Impostures. | |
| Aug 1, 2016 at 23:09 | comment | added | Michael Greinecker | In their book "Probability and Finance: It's Only a Game!", Shafer and Vovk use Borel determinacy to prove Kolmogorov's strong law of large numbers, certainly mainstream mathematics. | |
| Aug 1, 2016 at 19:29 | comment | added | François G. Dorais | @BenCrowell It would definitely be less accurate to say that! Perhaps you meant to say that ZFC was extraordinarily successful at providing a common ground where mathematicians could work without impediment? Perhaps it was too successful and the reasons for seeking such a common ground have been forgotten. In any case, this is besides the point I was trying to make, which was that because set theory is tailored to live in the background, it is difficult to see exactly where it is relevant and it is very easy to be mistaken in such matters. | |
| Aug 1, 2016 at 18:22 | comment | added | user21349 | @FrançoisG.Dorais: mathematicians conventionally work within the framework of set theory Wouldn't it be more accurate to say that most mathematicians do work that is completely independent of any foundational framework, and that most mathematics papers would look exactly the same if ZFC had never been invented? | |
| Apr 4, 2015 at 3:43 | comment | added | Adam Epstein | In this connection, it bears mention that Coret showed that that the stratified instances of Replacement are already theorems of Zermelo set theory, whence "requires some Replacement" entails "requires some unstratifiable Replacement". | |
| Apr 3, 2015 at 18:47 | comment | added | Adam Epstein | By necessary I mean unprovable, or at least not currently known to be provable, in Zermelo set theory. | |
| Apr 3, 2015 at 18:47 | comment | added | Adam Epstein | @FrançoisG.Dorais Reflection Principles are unquestionably important, but still somehow metamathematical. Mostowski Collapse is tremendously useful in set theory, but irrelevant in settings where isomorphism invariance, rather than identity, is the issue. Regarding "enough injectives" in homological algebra, McLarty's recent work has eliminated transfinite recursion from Baer's argument. Do you have any examples of necessary use of Replacement in, say, algebraic topology/geometry/number theory, combinatorics, functional analysis, nonlinear PDE,...? | |
| Apr 3, 2015 at 14:21 | comment | added | François G. Dorais | Replacement has tons of applications! From "having enough this-and-that" to the Reflection Principle which allows one to blindly work with proper classes in ZFC without resorting to Grothendieck universes and similar tools. The fact that replacement is invisible doesn't mean it's not used. This is a natural side effect that, while mathematicians conventionally work within the framework of set theory, most never work with bare axioms. In fact, that's the way set theory is intended to be used... | |
| Apr 3, 2015 at 8:17 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |
clarification
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| Apr 1, 2015 at 23:30 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Apr 1, 2015 at 19:47 | comment | added | Adam Epstein | I hope it's clear from what I wrote that I am not in any way casting aspersions on set theory, either as a discipline unto itself, or as an applicable field of mathematics. Rather, it is the rest of mathematics that needs to catch up. For starters, how about some applications of Replacement? Borel Determinacy is wonderful, but I would know people who might quibble about whether that result has implications for "mainstream mathematics". | |
| Apr 1, 2015 at 17:41 | history | answered | Adam Epstein | CC BY-SA 3.0 |