TITLE: Finite axiomatization of first order logic theories
QUESTION [5 upvotes]: I recently have read that there are set theories (for example, NBG) which are based on the first-order logic and also are finitely axiomatizable.
I am trying to understand what "finitely axiomatizable" means. At first, I thought that it means that it is possible to write all of the axioms as a finite list. In a sense, no axiom schemata occur in the theory.
Then, after consulting book by Kleene "Introduction to Metamathematics" section 19 pages 80-81 where he writes that in propositional calculus $B \implies A \lor B$ is an axiom schema because for every metamathematical expressions $A$ and $B$ it constitutes an axiom, I got confused as how can theories based on first order logic can be finitely axiomatizable if the propositional calculus already has axiom schema.
I would appreciate any comments on this topic!

REPLY [3 votes]: Finite axiomatizability (and similar concepts, like recursive axiomatizability) are referring to what is needed beyond the usual logical rules. In particular, a theory $T$ is finitely axiomatizable if there is some finite set $A$ of sentences such that every sentence in $T$ is provable from $A$ using the logical rules/axioms.
Note that axiomatizable is different from axiomatized: it's possible that the "obvious" axiomatization of $T$ is infinite, but there's a clever alternate axiomatization which is finite. There are in fact natural examples of this (if you're interested I can say a bit about these).