One can construct a proof by a small variant of the proof that a recursively axiomatized complete theory is decidable.
$1$. It is well-known that there is no algorithm that, given any sentence $\varphi$ of Number Theory, will determine whether that sentence is true in the natural numbers. More precisely, if the set of sentences of Number Theory is indexed as $(\varphi_n)$ in any one of the usual ways, then the set of $n$ such that $\varphi_n$ is true in $\mathbb{N}$ is not recursive. Indeed, the situation is much worse than that: the set is not even arithmetical.
$2.$ Suppose that $T$ is a recursively axiomatized extension of Peano Arithmetic. Then there is a sentence $\varphi$ of Number Theory such that $\varphi$ is neither provable nor refutable in $T$. The idea is that we can write a program that list the proofs in $T$, since the set of (indices of) proofs is recursively enumerable.
If for every sentence $\varphi$ of Number Theory, one of $\varphi$ or $\lnot\varphi$ is a theorem of $T$, then sooner or later one of $\varphi$ or $\lnot\varphi$ will show up as the final sentence of a proof in the list. This procedure would give an algorithm for determining truth of sentences in $\mathbb{N}$, contradicting ($1$).
There is a minor technicality that one needs to deal with, since ZFC is not an extension of Peano Arithmetic. To deal with that, for any sentence $\psi$ of Number Theory, we can mechanically produce a sentence $\psi'$ of Set Theory such that if $\psi'$ is a theorem of ZFC, then $\psi$ is true in $\mathbb{N}$. The sentence $\psi'$ is obtained from the usual construction of $\mathbb{N}$, and its addition and multiplication, in ZFC.
