# Is “PA has no non-standard models” consistent with ZF?

I have seen several proofs that there exist nonstandard models of arithmetic, but they all seem to rely on the compactness theorem, which is not implied by ZF. So are there any proofs in ZF that there's exists a nonstandard model of PA? Tennenbaum's theorem, which states that every nonstandard model is uncomputable seems to hint at the possibility that there are models of ZF that contain no nonstandard models of PA.

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Ah sorry, my mistake. –  Qiaochu Yuan Jul 27 '14 at 5:01