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!
Best Answer
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.