I apologize that this question is fairly philosophical and not purely mathematical. For the purposes of this question, I would like to take the point of view that that natural numbers are "real" and so every statement made about them is either true or false, while the universe of sets is a convenient "fiction". Basically it's Platonism for numbers and formalism for sets. From this point of view, is there a convincing argument that every theorem proven by ZFC about the natural numbers is actually true?
I'm guessing that it is only possible to make this convincing argument for certain statements about the natural numbers. For example if ZFC proves $\forall x\in \mathbb{N}\ \phi(x)$ for some quantifier-free $\phi$, then the consistency of ZFC should be enough to justify the truth of $\forall x \ \phi(x).$ If there were some counterexample $y$ where $\phi(y)$ fails then ZFC would prove $\exists x\in \mathbb{N} \ \neg \phi(x),$ an inconsistency. This argument should work with sentences that have any number of universal quantifiers, that is $\Pi_1$ formulas.
Edit: I have read that ZFC and ZF prove the same arithmetical sentences, so this should be equivalent to asking whether ZF is arithmetically sound.