Timeline for How dangerous are set-size assumptions?
Current License: CC BY-SA 4.0
30 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2019 at 2:28 | vote | accept | Pace Nielsen | ||
| Jun 21, 2019 at 15:49 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
added 88 characters in body
|
| Jun 21, 2019 at 2:50 | comment | added | Noah Schweber | Re: Fermat's last theorem, you're conflating the issue of universes (which can only be removed by actually looking at the proof - no general metatheorem applies) with choice (where we have a metatheorem, namely absoluteness, which automatically removes choice based purely on the syntactic complexity of the theorem itself). | |
| Jun 21, 2019 at 2:40 | answer | added | Timothy Chow | timeline score: 12 | |
| Jun 20, 2019 at 15:06 | comment | added | Asaf Karagila♦ | You make it sound as if students are being taught Ferge's set theory. I've never seen a course, as dull (in set theory) as it was where they didn't at least mention Russell's paradox. | |
| Jun 20, 2019 at 13:42 | comment | added | user21820 | Even before the philosophical issues concerning universes, we would have to deal with those for ZFC. See this post. And not all logicians are convinced that even just full second-order arithmetic is arithmetically sound, since after all there is no solid ontological basis for impredicative second-order comprehension. So if you are feeling philosophically suspicious, you would really have to examine your views on the powerset of the naturals very carefully, and perhaps you might decide that ZFC is not even the first dangerous point... | |
| Jun 20, 2019 at 13:32 | comment | added | Tim Campion | For a meatier example, ZF + AD and ZFC are both consistent, but at most one of them can be true, because they contradict each other. If you subscribe to some kind of mathematical pluralism, the way to think of it is that they can't both be true in the same context -- because they contradict each other! | |
| Jun 20, 2019 at 13:31 | comment | added | Tim Campion | @HagenvonEitzen Consistency and truth are two different things. For instance, $PA + Con(PA)$ and $PA + \neg Con(PA)$ are both consistent, but at most one of them can be true -- after all, one of them asserts $Con(PA)$ while the other asserts $\neg Con(PA)$! Depending on your philosophy, you might deny that it is ultimately meaningingful to ask whether / assert that a mathematical theory is "true". But in this case, you can still "model" the situation as follows: when somebody makes such assertions, interpret them as being relative to some (unspecified) "ultimate metatheory". | |
| Jun 20, 2019 at 10:30 | comment | added | Hagen von Eitzen | How is "could be inconsistent" a different danger than "could prove false things about natural numbers"? | |
| Jun 20, 2019 at 7:06 | comment | added | David Roberts♦ | @jon I should point out the lower bound with current technology: arxiv.org/abs/1207.0276 "The cohomology of coherent sheaves and sheaves of Abelian groups on Noetherian schemes are interpreted in second order arithmetic by means of a finiteness theorem. This finiteness theorem provably fails for the etale topology even on Noetherian schemes." | |
| Jun 20, 2019 at 6:58 | comment | added | David Roberts♦ | @jon étale cohomology (of a given scheme) is the derived functor cohomology of the étale topos attached to that scheme. Colin McLarty proved that relatively weak axiomatic systems equiconsistent with higher-order arithmetic can construct the relevant derived functors: arxiv.org/abs/1102.1773 "We formalize the practical insight by founding the theorems of EGA and SGA, plus derived categories, at the level of finiteorder arithmetic" (see also arxiv.org/abs/1207.6357) | |
| Jun 20, 2019 at 2:40 | comment | added | user40276 | Also, since others mentioned reflection, in terms of provability, ZF and ZF + "there exists a universe" are not really that different. Indeed, one can canonically add a new constant symbol which mimics the universe (this is due to Fefferman). This new theory is a conservative extension of ZF. One can iterate this a countable number of times and still get something conservative. | |
| Jun 20, 2019 at 2:23 | history | became hot network question | |||
| Jun 20, 2019 at 1:23 | comment | added | David Roberts♦ | "my understanding is that this dependence can be completely removed due to the fact that Fermat's Last Theorem has small quantifier complexity." No, the use of Grothendieck universes can be removed because they are unnecessary for treating the necessary topos theory for relatively small sites, not due to the complexity of the statement of FLT. | |
| Jun 19, 2019 at 22:44 | answer | added | Nik Weaver | timeline score: 22 | |
| Jun 19, 2019 at 22:26 | answer | added | Monroe Eskew | timeline score: 6 | |
| Jun 19, 2019 at 21:56 | answer | added | Tim Campion | timeline score: 7 | |
| Jun 19, 2019 at 21:45 | review | Close votes | |||
| Jun 20, 2019 at 10:44 | |||||
| Jun 19, 2019 at 21:11 | history | edited | YCor |
edited tags
|
|
| Jun 19, 2019 at 20:47 | comment | added | user40276 | Just some remarks. 1) By using second-order ZFC, the internal power set operation coincides with the external one (this is essentially why models second-order ZFC and Grothendieck universes are the same thing). 2) Consistency of PA is certainly a philosophical question. It boils down to whether the standard model is really a model. Now, universes are $V_{\kappa}$'s (for $\kappa$ strongly inaccessible) and these models are usually considered the intended models of ZFC when you think about a cumulative hierarchy. They have nice properties such as a "true" power set operation. | |
| Jun 19, 2019 at 19:56 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
added 719 characters in body
|
| Jun 19, 2019 at 19:45 | comment | added | Pace Nielsen | On question #2, I'll post an addendum to my question to clarify things. | |
| Jun 19, 2019 at 19:43 | comment | added | Pace Nielsen | @TimCampion Thank you for that distinction. In the context of ZFC vs. ZFC+Universes, the answer to #1 is no (assuming ZFC+Universes is consistent), but I am indeed interested in knowing the answer in the context of conceptualist (or predicativist) mathematics. I'm not knowledgeable enough to know if it just boils down to consistency issues. | |
| Jun 19, 2019 at 19:34 | comment | added | Tim Campion | Ok. So there are two things going on in your question: 1.) You're asking whether "conceptualist mathematics" might prove different arithmetical statements then ZFC, say -- a mathematical question. 2.) You're asking whether there are reasons to think that the arithmetical consequences of one theory are more "true" than another--a philosophical question. To answer either question would require a much more precise specification of which theories, exactly we're discussing. FWIW I've never heard of "conceptualist mathematics" but generally restricting the powerset axiom is called "predicativism". | |
| Jun 19, 2019 at 19:33 | comment | added | Pace Nielsen | @TimCampion I added a paragraph at the end of the question that I hope illuminates things further. | |
| Jun 19, 2019 at 19:30 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
added 585 characters in body
|
| Jun 19, 2019 at 19:21 | comment | added | Pace Nielsen | @TimCampion In most of mathematics $P(\mathbb{N})$ is treated as a set, but not in conceptualist mathematics. In (at least) one of the linked paper's by Nik Weaver, it is claimed that ZFC might prove false things about the natural numbers. I'm trying to get an idea of what that would mean---apart from the obvious point that if ZFC is inconsistent then it proves all things. You can think of the boxed question as a refinement of the question about Grothendieck universes, to the context of conceptualist mathematics vs. "ordinary mathematics" (as you put it). | |
| Jun 19, 2019 at 19:05 | comment | added | Tim Campion | I don't understand the question in the box -- in most of mathematics, $P(\mathbb N)$ is treated as a set and not a class. So it sounds like you're simply asking whether ordinary mathematics can prove false arithmetic statements. But I get the impression from the surrounding discussion that you're trying to ask something specific about Grothendieck universes... I don't know what you're asking though. | |
| Jun 19, 2019 at 18:49 | comment | added | Pace Nielsen | @jon The "category of all categories" runs into similar paradoxes as the "set of all sets", whereas the "category of all $U$-small categories" doesn't. We'd like to freely work with the standard categories (of groups, sets, monoids, etc...) as if they (together) can form a new collection which forms the object set of a category, and morphisms between them are functors. The standard way this is done is by working with the $U$-small categories instead. For this and other examples, look where textbooks make use of the axiom of universes. | |
| Jun 19, 2019 at 18:08 | history | asked | Pace Nielsen | CC BY-SA 4.0 |