# If a statement holds for all standard models of PA, then does it hold for all models?

Suppose that $\varphi$ is a consequence of every standard model of PA. Then is it provable from PA?

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Standard examples of sentences $\varphi$ showing that this is not the case: $\mathrm{Con}(\mathsf{PA})$, "Hercules wins the Hercules-Hydra game", $\mathrm{Con}(\mathsf{ZFC}+$"there is a measurable cardinal"$)$, etc. –  Andres Caicedo Jul 5 at 3:16
More interesting is to ask whether if a statement holds in all nonstandard models of $\mathsf{PA}$, then it also holds in the standard model. (And yes, that's the case.) –  Andres Caicedo Jul 5 at 3:17
@Andres: (... trivially, because thanks to the upward Löwenheim-Skolem theorem some of the nonstandard models are elementarily equivalent to the standard one) :P –  Henning Makholm Jul 5 at 14:03

## 1 Answer

There is (up to isomorphism) only one standard model of PA. And there are sentences true in the natural numbers that are not theorems of PA.

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Is the 2nd part a hint at Goedel's theorem? –  Nikos M. Jul 5 at 3:55
Yes, the theorem on the Incompleteness of PA is being referred to. –  André Nicolas Jul 5 at 4:00