Gödel's incompleteness theorem asserts the existence of an arithmetical formula $\Delta$ such that neither $\Delta$ nor $\lnot\Delta$ can be proved by the ZFC set theory. We even know that $\Delta$ is $\Pi_1$, which means of the form $\forall n, P(n)$, where $P$ is an arithmetical formula with only bounded quantifiers. Arithmetical closed terms like $SSSSO$ must be interpreted in ZFC by the corresponding natural numbers in the set $\mathbb{N}$. The interpretation of $P$ and its bounded quantifiers follows, and yields a formula in the language of ZFC. But how does ZFC interpret the first unbounded quantifier $\forall n$ ?
Since $\Delta$ is not refutable by ZFC, $P(0)$ must be true. Likewise for each $k\in\mathbb{N}$, $P(k)$ must be true. So it seems that ZFC proves $\forall n\in\mathbb{N}, P(n)$ and therefore this last formula is not the interpretation of $\Delta$ (which ZFC does not prove). As far as I understand, $\Delta$ is undecidable in Peano arithmetic, because we can have non standard numbers $n$ such as $P(n)$ is false. So the interpretation of $\Delta$ in ZFC should leave some space for non standard models, and that is another reason why it should not start by $\forall n\in\mathbb{N}$. But then what is the ZFC interpretation of $\Delta$ ?