A version of set theory which is a formal system expressed in first-order predicate logic. Zermelo-Fraenkel set theory is based on the Zermelo-Fraenkel axioms. 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 A(u, v)." Montague 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. von Neumann-Bernays-Gödel set theory provides an equivalent finitely axiomized system.

logic | von Neumann-Bernays-Gödel set theory | Zermelo-Fraenkel axioms | Zermelo set theory