The First-Order axiomatisation of PA is:

1. $\forall x. x = x$
2. $\forall x, y. x = y \rightarrow y = x$
3. $\forall x, y, z. x = y \land y = z \rightarrow x = z$
4. $\forall x. 0 \ne S(x)$
5. $\forall x, y. S(x) = S(y) \rightarrow x = y$
6. $\forall x. x + 0 = x$
7. $\forall x, y. x + S(y) = S(x + y)$
8. $\forall x. x \* 0 = 0$
9. $\forall x, y. x \* S(y) = x\*y + x$
10. Axiom schema

Now a thing is eating at me. Why is it that a model where $0$ is the only natural number and $S(x)$ is defined nowhere isn't a model of PA? If $S(0)$ doesn't exist, and furthermore $S(x)$ doesn't exist for any x, wouldn't that be also a nonstandard model?