[FOM] Godel's Second/comment
friedman at math.ohio-state.edu
Tue May 9 14:35:25 EDT 2006
I just realized that In #284: Godel's Second, there is a much cleaner way of
stating Godel's Second for PA.
GODEL'S SECOND FOR PA. PA is not interpretable in any finite fragment of PA.
Why do I call this Godel's Second for PA?
COROALLRRY. ZF is not interpretable in PA. ZF\P is not interpretable in PA.
ACA0 is not interpretable in PA.
Proof: Obviously PA is interpretable in some finite fragment T of ZF, using
the standard model of PA. If ZF is interpretable in PA then T is
interpretable in PA, and so T is interpretable in some finite fragment of
PA. Hence PA is interpretable in some finite fragment of PA. This
contradicts Godel's Second for PA.
The same argument works for ZF\P. The same argument also works for ACA0
since ACA0 is a finitely axiomatizable extension of PA. QED
This Corollary is the essence of the collapse of Hilbert's program.
More later, replacing PA with other systems, and other matters, in the next
numbered posting #285.
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