Zermelo-Fraenkel set theory

The Zermelo-Fraenkel axioms of set theory, denoted 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 classes and sets.

The axioms of ZFC are:

• Axiom of extension: Two sets are the same if and only if they have the same elements.
• Axiom of empty set: There is a set with no elements. We will use {} to denote this empty set.
• Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
• Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
• Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.
• Axiom of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y₁) and P(x,y₂) implies y₁ = y₂, there is a set containing precisely the images of the original set's elements. (This is an axiom schema.)
• Axiom of power set: Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.
• Axiom of regularity: Every non-empty set x contains some element y such that x and y are disjoint sets.
• Axiom of choice: Any product of nonempty sets is nonempty.

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[?].