FOM: Finitely axiomatizable fragments of set theory

Charles Parsons parsons2 at
Tue May 15 17:28:45 EDT 2001

At 3:46 PM -0400 5/15/01, Richard G. Heck, Jr. wrote:
>>A final question: ZF proves the consistency of any finitely axiomatizable
>>subtheory of itself.  Do Z and PA also have this property?
>Certainly PA has this property, since PA proves Con(PA(n)), for all n,
>where PA(n) has induction for just the sigma-n formulae.

This was in effect proved in G. Kreisel and Hao Wang, "Some applications of
formalized consistency proofs," Fundamenta Mathematicae 42 (1955), 101-110.

I think similar arguments ought to work for Z.

Charles Parsons

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