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Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a question about structure of his proofs:

Did Nelson attempts try to, for some statement $\phi$, prove both $\phi$ and not $\neg\phi$?

Some of you might ask a question "What other possibility could there be?", and here is one such possibility: PA might just prove the statement "PA is inconsistent". What is the difference? The difference is that supposed "proof" of contradiction might have nonstandard length. You can think of this in terms of different theory, namely $PA+\neg Con(PA)$. This theory shows that PA is inconsistent (because one of its axioms says so), and, as extension of PA, it can show itself inconsistent. However, the theory itself is still consistent.

What the reasoning above shows is that PA is not $\omega$-consistent. So, slightly restating the question,

Did Nelson in his attempts really try to show PA inconsistent, or just $\omega$-inconsistent?

Also, if anyone knows a source where I could find Nelson's papers, I'd be thankful; this is the reason I added "reference-request" tag.

Thanks for all feedback!

Nate Eldredge
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Wojowu
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    Google it. He tried to use the proof of Kritchman-Raz of the 2nd incompleteness theorem to actually prove Con(PA) in PA and therefore get a contradiction. It didn't work. – Monroe Eskew Oct 25 '14 at 10:47
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    Before I couldn't find the outline, but now, when I looked it up with "Kritchman-Raz" keyword, I was able to find it. Thanks! – Wojowu Oct 25 '14 at 11:06
  • If you have a MathSciNet subscription, Nelson's MR Author ID is 212680, and you can get a list (link) of all his indexed papers. Of course, many (all?) of these attempts are unpublished since they didn't work. – Nate Eldredge Oct 26 '14 at 16:12
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    Related question: Nelson's program to show inconsistency of ZF – Nate Eldredge Oct 26 '14 at 16:14
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    For those who may not know, I would like to note that Nelson passed away this September. – Steven Gubkin Oct 26 '14 at 16:41
  • Nelson removed his paper claiming $\neg Con(PA)$ from his website, but it would be among his Nachlass - my hope is that these will be available at some point. – David Roberts Oct 27 '14 at 00:08
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    There is still, of course, some online papers here: https://web.math.princeton.edu/~nelson/papers.html – David Roberts Oct 27 '14 at 00:11
  • If PA proves the statement "PA is consistent", that would for sure establish the inconsistency of PA. If PA proves "PA is inconsistent", that only proves the inconsistence of PA + Con(PA) or stronger theories such as ZF. – bof Nov 26 '14 at 00:35

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See the discussion here: https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html. As Monroe says, he tried to prove that PA was inconsistent, not just $\omega$-inconsistent.

Noah Schweber
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