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    Zermelo-Fraenkel Set Theory

    Definition

    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.

    Associated people

    Adolf Abraham Halevi Fraenkel | Ernst Friedrich Ferdinand Zermelo

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