Recall the definition of a Natural Numbers Object in a topos, and the first order axioms for Peano Arithmetic. I am more familiar with the first definition than the second, so I cannot tell from the (obviously infallible) Wikipedia page whether first order PA is equivalent to 'second order' PA -- PA with the induction axiom scheme replaced by a similar one involving inductive subsets $A \subset \mathbb{N}$.

The equivalence between 'second order' PA and a NNO in $Set$ in one direction (NNO $\Rightarrow$ PA) is easy, and the other (PA $\Rightarrow$ NNO) is in MacLane's book *Mathematics, form and function* which I haven't seen (bonus question: another reference for the proof would be nice).

But I would like to see a proof of NNO = (first order) PA, if possible.

My motivation is to consider possibly weaker forms of arithmetic, and it is how to deal with (versions of) the induction axiom schema as usually presented from a logic point of view on which I would like a bit of background.

Edit: As Andreas points out, I really should be asking about *models* of PA and NNOs.