von Neumann-Bernays-Gödel set theory (abbreviated "NBG") is a version of set theory which was designed to give the same results as Zermelo-Fraenkel set theory, but in a more logically elegant fashion. It can be viewed as a conservative extension of Zermelo-Fraenkel set theory in the sense that a statement about sets is provable in NBG if and only if it is provable in Zermelo-Fraenkel set theory.
Zermelo-Fraenkel set theory is not finitely axiomatized. For example, the axiom of
replacement is not really a single axiom, but an infinite family of axioms, since
it is preceded by the stipulation that it is true "for any set-theoretic formula
."
Montague (1961) proved that Zermelo-Fraenkel
set theory is not finitely axiomatizable, i.e., there is no finite set of axioms
which is logically equivalent to the infinite set of Zermelo-Fraenkel
axioms. In contrast, von Neumann-Bernays-Gödel set theory has only finitely
many axioms, and this was the main motivation in its construction. This was accomplished
by extending the language of Zermelo-Fraenkel
set theory to be capable of talking about set classes.
