TITLE: Is Zermelo set theory finitely axiomatizable?
QUESTION [7 upvotes]: I know that ZF is not finitely axiomatizable, but what about Z (i.e. ZF without Replacement)?

REPLY [9 votes]: No. It is not.
You can find the proof as Theorem 8 in:

Mathias A. R. The Strength of Mac Lane Set Theory, Annals of Pure and Applied Logic, 110 (2001) 107--234.

(The article also appears on Mathias' homepage without the need for a paywall)