Timeline for Is there a metamathematical $V$?
Current License: CC BY-SA 4.0
12 events
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| Mar 11, 2023 at 3:57 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Added another reference
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| Feb 18, 2021 at 19:03 | comment | added | Pace Nielsen | @AndreasBlass and Timothy: Interestingly, Saul Kripke just posted a preprint at the arXiv, arxiv.org/abs/2102.08346, which goes into the Hilbertian view of finiteness. | |
| Feb 13, 2021 at 15:58 | comment | added | Andreas Blass | @TimothyChow It has also been suggested (though I don't remember by whom) in the context of set theory that we should accept the law of the excluded middle only for formulas in which all quantification is bounded; when unbounded quantifiers are involved only intuitionistic logic would apply because the domain of such quantifiers is open-ended. This may be a good way to express the idea that V is not a completed totality. And an analogous proposal might be used in arithmetic to express that $\mathbb N$ isn't completed. | |
| Feb 13, 2021 at 15:54 | comment | added | Andreas Blass | @TimothyChow As far as I can see, Gentzen's proof is finitary except for the assumption that (the standard notation system for) $\epsilon_0$ is well-ordered. I think Hilbert's view of finitism allowed universally quantified statements (equivalently, quantifier-free formulas interpreted as general laws) but nothing more. A fairly common view nowadays seems to be that primitive recursive arithmetic is a reasonable formalization of the finitistic viewpoint. | |
| Feb 13, 2021 at 14:11 | comment | added | Timothy Chow | @AndreasBlass Interesting viewpoint. Do you think that there is some specific number of alternation of quantifiers that implicitly assumes a completed infinity? Also, if one accepts Gentzen's proof as being correct and finitary, then doesn't that suggest that PA doesn't presuppose a completed infinity? To put it another way, wouldn't it be surprising that we could establish the validity of infinitary reasoning by finitary means? I know that that was Hilbert's original hope, but hasn't experience taught us that that hope was too optimistic? | |
| Feb 13, 2021 at 0:24 | comment | added | Andreas Blass | I'm not convinced that PA doesn't regard $\mathbb N$ as a completed infinity. The reason is that the induction schema allows formulas with deep nesting of alternating unbounded quantifiers, so it implicitly treats such formulas as meaningful. I don't see how to think of truth values for big strings of quantifiers without thinking of the totality $\mathbb N$ as complete. (This may be related to Gentzen's regarding his proof as finitary; this proof doesn't require deep alternations of quantifiers.) | |
| Feb 12, 2021 at 15:57 | review | Suggested edits | |||
| Feb 12, 2021 at 16:45 | |||||
| Feb 5, 2021 at 0:21 | comment | added | Timothy Chow | @PaceNielsen I am not aware of any attempts to capture the "minimal" consequences of "$\mathbb{N}$ is a completed infinity." By the way, Gentzen's proof of the consistency of PA was regarded by Gentzen himself as being "finitary." So that illustrates how fuzzy this question is. | |
| Feb 4, 2021 at 19:35 | comment | added | Pace Nielsen | (2/2) However, if I understand things correctly, the usual axiom of infinity (perhaps with separation and other seemingly harmless set axioms) does seem to prove that PA is consistent, which can be turned into an arithmetic statement not provable in PA. So, does a potentialist believe in PA, but not PA+Con(PA)? Maybe so, because perhaps the Turing machine used to check the consistency of PA cannot really be built and run forever. If you have any additional thoughts, they'd be appreciated. | |
| Feb 4, 2021 at 19:35 | comment | added | Pace Nielsen | (1/2) Timothy, thank you for this precise formalization of "$\mathbb{N}$ is not a completed infinity". The next question would then be: What is the precise formalization of "$\mathbb{N}$ is a completed infinity"? ZF is too powerful for that, so the fact that there are theorems of ZF about the natural numbers that are not provable in PA doesn't quite answer my question. For instance, one might, a priori, blame this on the impredicative nature of the power set axiom rather than the axiom of infinity. Even ZF-Power might be too strong. | |
| Feb 4, 2021 at 18:01 | comment | added | Julia Williams | Oooh, thank you for the link to the FoM mailing list. I didn't know that had been discussed there | |
| Feb 4, 2021 at 5:49 | history | answered | Timothy Chow | CC BY-SA 4.0 |