Timeline for Why not adopt the constructibility axiom $V=L$?
Current License: CC BY-SA 4.0
32 events
| when toggle format | what | by | license | comment | |
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| Jun 8, 2019 at 22:52 | answer | added | Master | timeline score: 1 | |
| Jun 3, 2019 at 15:21 | comment | added | Monroe Eskew | @NotMike That’s a rather specific aesthetic. | |
| Jun 3, 2019 at 11:02 | comment | added | Not Mike | "why hasn’t there been historically a stronger push to adopt it as a foundational axiom for mathematics?" Because leveraging the slimness of $L$ (along with it's version of the reflection theorem) to prove results by establishing any instance of reflection of the failure leads to a contradiction; is not a particularly natural way of proving results. My favorite example is the construction of a Kurepa tree without an A-subtree using $\Pi^1_2$-reflection below $\omega_1$ | |
| May 23, 2019 at 11:47 | comment | added | Noah Schweber | @AlexKruckman Or the existence of saturated models. Although in each case I think that's rude. | |
| May 20, 2019 at 14:22 | answer | added | Joel David Hamkins | timeline score: 18 | |
| May 20, 2019 at 13:49 | answer | added | Timothy Chow | timeline score: 28 | |
| May 20, 2019 at 13:23 | answer | added | Andreas Blass | timeline score: 20 | |
| May 20, 2019 at 13:18 | comment | added | Alex Kruckman | One axiom that goes beyond ZFC but is often assumed without explicit mention in papers is the axiom of universes / a proper class of inaccessibles. | |
| May 20, 2019 at 12:13 | answer | added | Nik Weaver | timeline score: 16 | |
| May 20, 2019 at 11:11 | comment | added | Andrej Bauer | @AndrésE.Caicedo: good point, thanks for bringing those up. | |
| May 20, 2019 at 11:03 | comment | added | Andrés E. Caicedo | @Andrej Many papers in general topology are like that. There are also quite a few (current) examples in real analysis, group theory, model theory, etc. | |
| May 20, 2019 at 8:26 | comment | added | Mohammad Golshani | From time to time people argue we should “believe” or “adopt” as an axiom the statement “$V = L$”; my own inclination is strongly against this. This universe looks like a very special thin and uncharacteristic case, and adopting it would kill many interesting theorems; we shall return to this issue below. In any case, I do not think anybody takes it seriously. In spite of some rumors to the contrary, Jensen flatly does not “believe” in $V = L$ (though it would certainly be to his personal advantage) but he thinks a proof under $V = L$ is significantly better than a consistency result. | |
| May 20, 2019 at 8:26 | comment | added | Mohammad Golshani | In The future of set theory Shelah writes: | |
| May 20, 2019 at 6:02 | comment | added | Andrej Bauer | @NoahSchweber: Sure, I just wanted to give an example of a paper using a non-standard axiom of set theory, outside the area of set theory. Such paper seem to be few and far between. | |
| May 20, 2019 at 4:13 | history | edited | YCor | CC BY-SA 4.0 |
added name in the title
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| May 20, 2019 at 3:35 | comment | added | Noah Schweber | @AndrejBauer Yes, but right there in the abstract it states "The proof is carried out in ZFC set theory extended with Martin’s Axiom at an uncountable cardinal." So there's no pretension on the part of the authors to treat Martin's Axiom equally to the other axioms of ZFC (e.g. no general math or cs paper that I'm aware of in the last 60 years has opened with "we will use the axiom(s) of separation"). I think the OP is talking about a paper which used V=L without disclosing it. Certainly if I were refereeing I would demand that they do so .. | |
| May 20, 2019 at 3:33 | answer | added | Noah Schweber | timeline score: 49 | |
| May 20, 2019 at 1:19 | comment | added | user44143 | There's not much point arguing for $ZFC+V=L$ over $ZFC$ if you prefer $PA^3$ to either. That preference is better called predicativist than contructivist, and it is @NikWeaver's position as I understand it. | |
| May 20, 2019 at 1:08 | history | made wiki | Post Made Community Wiki by Todd Trimble♦ | ||
| May 20, 2019 at 1:02 | answer | added | Pace Nielsen | timeline score: 12 | |
| May 20, 2019 at 0:45 | answer | added | Rodrigo Freire | timeline score: 19 | |
| May 20, 2019 at 0:20 | review | Close votes | |||
| May 20, 2019 at 20:12 | |||||
| May 19, 2019 at 23:09 | comment | added | Andrej Bauer | I know of a paper in theoretical computer science which got accepted and it uses Martin's axiom: arxiv.org/pdf/1211.1511v2.pdf | |
| May 19, 2019 at 23:04 | comment | added | Gro-Tsen | A perhaps less contentious way of phrasing the question would be to ask why $V=L$ does not have the same epistemological status in ordinary mathematical practice as the axiom of choice (see here and here for parallels between the two). | |
| May 19, 2019 at 22:58 | comment | added | Monroe Eskew | @AndrejBauer Well does anyone know of a paper that used the axiom without claiming the theorem as conditional? | |
| May 19, 2019 at 22:54 | comment | added | Andrej Bauer | Does anyone know of a journal rejecting a proof on the grounds that it assumed $V = L$? | |
| May 19, 2019 at 22:44 | comment | added | Monroe Eskew | @bof I mean that mainstream mathematics journals would accept a proof that uses the axiom just as much as any other proof. One could still study alternative theories of course, but this would be considered akin to studying models where AC fails. | |
| May 19, 2019 at 22:41 | comment | added | bof | What does it mean to "adopt" $V=L$? Does it mean that all research into alternative theories (with large cardinals, determinacy, etc.) should be halted, and papers about such research no longer be published? Otherwise, if people are going to continue investigating consequence both of $V=L$ and its alternatives, how is that different from the present situation? | |
| May 19, 2019 at 22:37 | comment | added | MyNinthAccount | A cynical viewpoint: It's "effective" in settling questions. But mathematicians would rather have open problems (gives them busy work). | |
| May 19, 2019 at 22:05 | comment | added | Alec Rhea | I've always gotten the impression that $V=L$ is too restrictive for me to want to live in that universe, akin to working in a topos where all functions on the reals are continuous or something like this -- nice as a toy model for the real mathematical universe and some interesting things can be said, but not actually where mathematics takes place since it knocks out large cardinals that I find interesting etc. (mathoverflow.net/questions/190614/…). | |
| May 19, 2019 at 21:57 | comment | added | Asaf Karagila♦ | Forcing axioms? Large cardinals? | |
| May 19, 2019 at 21:54 | history | asked | Monroe Eskew | CC BY-SA 4.0 |