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Note: This question is inspired by a paper being discussed in the FoM mailing list.

The formal statement Con(PA) is a statement about the existence of a proof code for the derivation of a contradiction is PA, which involves quantification over the numbers of PA. For something to be provable in PA, it would have to hold in all models of PA but as we know, there are nonstandard models. Thus to prove in PA no number codes for the proof of a contradiction in PA it would have to hold in nonstandard models as well. Could it be that some nonstandard model of PA has proof codes which code for proofs of a contradiction yet are not finite under our normal understanding? The paper linked above seems to suggest we can prove in PA that there are no properly finite proofs of a contradiction in PA, yet Timothy Chow in the FoM mailing list seems to think that the paper suggests nothing new.

Could it be that there are proof codes for a contradiction in nonstandard models? Can we gain insight about the consistency of PA by restricting ourselves to proofs of properly finite length?

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    $\begingroup$ The question is what does it mean "by finitary means", and as long as the definition of "finitary" is external to the model of PA, it does not really offer anything new. The goal would be to argue that PA proves that there are no codes for contradictions. Not just that no standard number is a code for a contradiction. The reason is that the coding of proofs into PA is such that from a standard code we can reconstruct "an actual proof" (i.e. external to the model's internalization of FOL). Since $\omega$ is a model of PA, no standard number codes a proof of inconsistency. $\endgroup$ – Asaf Karagila Mar 26 at 15:12
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    $\begingroup$ (The above is, of course, under the assumption that the meta-theory is consistent. You may replace "model" by some other notion, depending on your meta-theory. The point is that somehow you'd need to make some assumptions about your meta theory.) $\endgroup$ – Asaf Karagila Mar 26 at 15:13
  • $\begingroup$ So, assuming PA and ZF are consistent, could there still be proof codes which are "nonstandard numbers" (sorry not sure the correct terminology here, I mean elements of a nonstandard model) which "code for a proof of contradiction" (though it is not clear what this means, I intend something like "satisfies Proves(x, Code(0=1))", where Proves is a predicate saying x is a code for a proof of the statement which is coded for in the second place). If there is such a nonstandard number, does that mean anything intuitively? $\endgroup$ – Keefer Rowan Mar 26 at 15:19
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    $\begingroup$ If ZF is consistent, then PA is consistent, and PA+not Con(PA) is consistent. But a model of the latter theory has to be non-standard, and the proof of 0=1 has to have a non-standard code. $\endgroup$ – Asaf Karagila Mar 26 at 15:20
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    $\begingroup$ I think that math.stackexchange.com/questions/1189216/… and math.stackexchange.com/questions/882653/… and math.stackexchange.com/questions/1383286/… might already tell you what you want to know. $\endgroup$ – Asaf Karagila Mar 26 at 16:27

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