Zermelo-Fraenkel set theory: Encyclopedia II - Zermelo-Fraenkel set theory - The Axioms

Zermelo-Fraenkel set theory - The Axioms

The axioms of ZFC are:

• Axiom of extensionality: 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 also use {} to denote this empty set.

• Axiom of pairing: If x, y are sets, then there exists a set containing x and y as its only elements, which we denote by {x,y} or {x} ∪ {y}.

• 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 y ∪ {y}.

• 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 separation (or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds. (This is an axiom schema.)

• 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 choice: Given any set of mutually exclusive non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets.

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

Adapted from the Wikipedia article "The Axioms", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki