Von Neumann–Bernays–Gödel set theory
This theory is not finitely axiomatized. ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced.
The following axioms and definitions are needed for the proof of the class existence theorem.
The following definitions specify how formulas define relations, special classes, and operations:
The axioms of pairing and regularity, which were needed for the proof of the class existence theorem, have been given above. NBG contains four other set axioms. Three of these axioms deal with class operations being applied to sets.
In set theory, the definition of a function does not require specifying the domain or codomain of the function (see Function (set theory)). NBG's definition of function generalizes ZFC's definition from a set of ordered pairs to a class of ordered pairs.
Von Neumann worked on the problems of Zermelo set theory and provided solutions for some of them:
Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC (or vice versa). For a survey of known results of this nature, see Pudlák 1998.