I want to show that there is a finite conjunction $\phi$ of axioms of $ZF$, such that every transitive proper class $M$, which satisfies $\phi$, is already a model of $ZF$.
This is an exercise in Kunen's set theory. There is a hint, it seems to be useful to apply the Reflection principle to the union $M = \cup_{\alpha} M \cap R(\alpha)$. But I don't know with which axioms we can do that (we can only use finitely many!), and why this yields an ordinal which is independent from $M$. Please give me only a hint, because basically I want to solve this on my own, but I don't know how to start with the hint above.
Also, what is the "philosophical" reason that we cannot deduce from this, that $ZF$ is finitely axiomatizable (which is wrong)? I mean I cannot prove that this $\phi$ above proves every axiom, but is there also a deeper reason for this?
EDIT: There was an answer with some hints, but it was deleted... I still don't know how to produce this strange sentence $\phi$.