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?

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    $\begingroup$ The moment you say "set theory", you are outside the "theory of natural numbers". For the latter you would ever only mention natural numbers themselves, not other sets. $\endgroup$ – Wojowu Oct 20 '17 at 14:10
  • $\begingroup$ I agree and that's my point. Why do I need a theory of natural numbers if I can do everything within set theory without even needing second-order logic? I either don't get what's the advantage or where the flaw is in my reasoning. $\endgroup$ – The curious amateur Oct 20 '17 at 14:25
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    $\begingroup$ Who says we need second-order logic? See math.stackexchange.com/questions/523259/… -- apparently it is well-know that ZFC+(first-order logic) allows you to express well-ordering. But you can express well-ordering using only second-order logic, without ZFC. I hope someone can explain this better than I can. $\endgroup$ – David K Oct 20 '17 at 14:30
  • $\begingroup$ Thanks David. I am no mathematician so some details escape me, but it looks to me that my view of math foundations is about building on top of set theory while the norm would be to build completely independent theory and maybe prove some "bridging" theorems. I suppose that after writing a second order number theory, proving that Peano Systems exist effectively means providing the semantics of the language. $\endgroup$ – The curious amateur Oct 20 '17 at 14:47

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