Is there a weak set theory that can prove that the natural numbers is a model of PA?

Question

$ZFC$ proves that $\mathbb N$ is a model of $PA$. Even $ZF$ does. How weak can go?

In particular, is there some weak set theory that proves that $\mathbb N$ is a model of $PA$, but does not proof that $PA + Con(PA)$ is consistient?

EDIT: The theory should prove that $\mathbb N$ (along with its operations) actually exist.