[FOM] Strict reverse mathematics of incompleteness.
dmehkeri at gmail.com
Sat Jun 18 14:15:19 EDT 2011
Are there any reverse mathematical results relevant to Gödel's
incompleteness theorems? It has been mentioned that even Robinson's Q
does not prove its own consistency. But I am also wondering about
something orthogonal to that: not just what is required of a formal
system for the incompleteness theorems to apply to it, but also what is
required of a formal system to prove the incompleteness theorems.
This would be at the level of "strict" reverse mathematics as Friedman
calls it. I am vaguely aware that encoding of syntax becomes an issue
here, and so the question may not even be precise enough as stated.
Unfortunately I don't know how to state it more precisely than that.
This is relevant to the Con(PA) discussion of course. Briefly, if the
incompleteness theorems cause us to doubt Con(PA), then why not doubt
Con(Q)? But if Con(Q) is in doubt, then why believe the incompleteness
theorems? So it might be useful to have a better idea of what the
incompleteness theorems require.
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