I often read that without second-order logic we can't have a theory of natural numbers because we wouldn't be able to express the well-ordering principle. OK, then there must be a flaw in the reasoning I am about to share. Could you help me find it?
Let's just stay in set theory. Then I can define the set of natural numbers, which will be just one of the possible models of Peano axioms, but I don't care... I have it, I can prove the Peano axioms on that set and I call it $\mathbb{N}$. Now that I can refer to $\mathbb{N}$, I can also refer to $\wp(\mathbb{N})$, whose existence is guaranteed by ZFC. Now, I can quantify over the subsets of natural numbers by saying $(\forall A \in \wp(\mathbb{N}))$ and I can write everything in FOL. I appreciate I have used a different approach from creating a natural number theory and then verify it has a unique (in the isomorphic sense) model, but why is it wrong?