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Zermelo-Fraenkel set theory

The Zermelo-Fraenkel axioms of set theory (ZF), are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based. When the axiom of choice is included, the resulting system is ZFC.

The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Adolf Fraenkel in the same year, which was based on the axiom system put forth by Ernst Zermelo in 1908 (Zermelo set theory).

The axiom system is written in first-order logic. The axiom system has an infinite number of axioms because an axiom schema is used. An equivalent finite alternative system is given by the von Neumann-Bernays-Gödel axioms (NBG), which distinguish between classeses and sets.

The axioms of ZFC are:

While most metamathematicians believe that these axioms are consistent (in the sense that no contradiction can be derived from them), this has not been proved. In fact, since they are the basis of ordinary mathematics, their consistency (if true) cannot be proved in ordinary mathematics; this is a consequence of Gödel's second incompleteness theorem. On the other hand, the consistency of ZFC can be proved by assuming the existence of an inaccessible cardinal.

Also see