I know that closure under the Gödel operations is equivalent to $\Delta\_0$-separation (plus extensionality, union, pair, foundation). This is finitely axiomatizable. But when we add $\Delta\_0$-collection, to obtain KP, is it still finitely axiomatizable? And in this case, how to prove it?



Is Kripke Platek theory finitely axiomatizable?
The answer to the question depends on how "foundation" is formulated, i.e., as a *single axiom* or as a *full scheme* (one which makes sure that every nonempty *parametrically definable class* has a minimal element, equivalently: the principle of $\in$-induction holds). **Emil Jeřábek 's answer pertains to the latter formulation, but not the former.** 


In the old days, beginning with Barwise, KP was defined so as to include the full scheme of foundation, perhaps because of the fact that KP was designed to study the class of well-founded models of a well-behaved fragment of ZF. But this has changed somewhat, for example in Adrian Mathias' paper [The Strength of MacLane Set Theory](https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf) (Annals of Pure and Applied Logic, 2001) KP only includes $\Pi\_1$-Foundation, asserting that every nonempty $\Pi\_1$-definable class (parameters allowed) has a minimal element (see p.111 of the above hyperlinked text).


**It is well-known that KP as defined by Mathias is finitely axiomatizable.** The proof uses the fact that there is a definable truth definition for $\Pi\_1$-formulae in KP.


