Question about Godel and ZF

From: Acid Pooh (poohonlsd_at_yahoo.com)
Date: 06/09/04 

Date: 9 Jun 2004 14:12:41 -0700

Hi everybody,

I have a question about Godel Incompleteness theorem and how it
applies to ZF. In particular, as "classically" phrased, Godel's
theorem applies to PA. It's an easy corollary to show that it also
applies to ZF since one can model PA in ZF. But I was wondering: is
ZF incomplete only in this respect? That is, can every sentence which
has no "arithmetic interpretation" in ZF be proven? (i.e.: Let ZF_PA
be the subset of ZF which models PA. The question can now be
formulated more precisely. Is ZF_PA a proper subset? Is ZF\ZF_PA
complete?)

I suspect that the answer is negative, but can't really get my head
around the first question.

thanks,
'cid 'ooh