I know that ZF is not finitely axiomatizable, but what about Z (i.e. ZF without Replacement)?
Is Zermelo set theory finitely axiomatizable?
7
$\begingroup$
logic
set-theory
model-theory
1 Answers
9
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)
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1@Sumac - see Hao Wang, [A Survey of mathematical logic](https://books.google.it/books), Chapter XVII: Relative Strength and Reducibility, page 438. – 2017-05-10