$\mathsf{ZF}$ is not finitely axiomatizable

Solution 1:

Suppose $\text{ZF}$ is consistent and $\text{ZF}$ is fnitely axiomatizable. Let $\Gamma \subset \text{ZF}$ be a finite subset such that $\Gamma \vdash \text{ZFC}$. Referring to Jech or Kunen, $\text{ZF} \vdash \text{Reflection Theorem}$. So $\text{ZF} \vdash \text{Con}(\Gamma)$. Since $\Gamma \vdash \text{ZF}$, $\text{ZF} \vdash \text{Con}(\text{ZF})$. If $\text{ZF}$ is consistent, this contradicts the Second Incompleteness Theorem.

Solution 2:

You can find the proof in Kunen's famous book "Set Theory: An Introduction To Independence Proofs" (North Holland, 1980). It is Corollary IV 7.7 on page 138, and it does make use of reflection.