TITLE: If a statement holds for all standard models of PA, then does it hold for all models?
QUESTION [5 upvotes]: Suppose that $\varphi$ is a consequence of every standard model of PA. Then is it provable from PA?

REPLY [5 votes]: 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.