Timeline for Why not adopt the constructibility axiom $V=L$?
Current License: CC BY-SA 4.0
11 events
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| Apr 14 at 12:31 | comment | added | Timothy Chow | Since the "truth" of these axioms can't be settled by traditional mathematical means (i.e., via proof, the way that questions such as Fermat's Last Theorem and the Poincare conjecture were settled), the mathematical community at large isn't too interested in debating it. | |
| Apr 14 at 12:29 | comment | added | Timothy Chow | @JesseElliott From a pluralistic standpoint, ZFC's "foundational status" is a matter of convention rather than truth. But I agree that if you think that people will remain indefinitely receptive to adopting new axioms, then there's no reason for a time limit. I just don't think the mathematical community at large has that attitude any more. The shift didn't happen by fiat; it happened because people realized that it's convenient to fix a standard by convention, while allowing you to state whatever axioms you feel like, without having to argue about their "truth." | |
| Jul 12, 2019 at 21:15 | comment | added | Jesse Elliott | ZFC having "foundational status" and pluralism seem to be at odds with one another. GCH needs to be settled, in my view, under the foundationalist view, but not under the pluralistic one. It makes sense that it would take more time to resolve GCH than it did the Axiom of Choice. I don't know why there should be a time limit when it comes to resolving a problem like GCH. | |
| Jul 12, 2019 at 16:01 | comment | added | Timothy Chow | @JesseElliott : That's an interesting (and of course controversial!) question, but one does not necessarily need to answer it to address Monroe Eskew's question. It's an empirical fact that the mathematical community has decided to accord the axioms of ZFC "foundational status." Eskew's question is, why hasn't V=L been granted the same status? We don't necessarily need to have a precise definition of "foundational status," let alone a theory of the relationship between "foundational status" and truth, as long as we agree that the community has granted ZFC that status. | |
| Jul 11, 2019 at 5:59 | comment | added | Jesse Elliott | Is an axiom true because the majority of mathematicians believe it? What makes an axiom true? | |
| May 26, 2019 at 16:18 | comment | added | Timothy Chow | @AsafKaragila : Regularity is sort of irrelevant to mathematics, but replacement is an interesting case. It makes me think that an axiom is much more likely to gain foundational status if mathematicians use it "unconsciously"; i.e., assuming it tacitly without fully realizing that they're invoking an "extra" assumption. If that is true, then V=L is at a disadvantage because it's not the sort of thing that anyone is going to use unconsciously. | |
| May 22, 2019 at 1:39 | comment | added | Noah Schweber | @MonroeEskew "It seems that the decision not to promote V=L was made by set theorists during the time when foundations were very unclear." I think there's a separate, and very interesting, question about the history of attempts on the logicians' side to argue for new axioms. But that's different from acceptance by the mathematical community. | |
| May 20, 2019 at 15:24 | comment | added | Monroe Eskew | I have to say, I still don't understand why. The Whitehead problem was considered important. If history had turned out differently and it had been announced that Shelah had solved it by using deep structural facts about sets, rather than that Shelah showed it to be independent of the axioms, then maybe people would have said, "Hmm so those set theorists figured out something useful after all!" It seems that the decision not to promote V=L was made by set theorists during the time when foundations were very unclear. | |
| May 20, 2019 at 13:59 | comment | added | Asaf Karagila♦ | I think that the axiom of choice made the cut prior to replacement and regularity. Both of which are arguably foundational axioms just as much as they are set theoretic axioms. | |
| S May 20, 2019 at 13:49 | history | answered | Timothy Chow | CC BY-SA 4.0 | |
| S May 20, 2019 at 13:49 | history | made wiki | Post Made Community Wiki by Timothy Chow |