axiom of pairing

The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.

In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing.

Together with the axiom of empty set, the axiom of pairing can be generalised to the following schema:

Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, and specification.

The axiom of regularity (with the axiom of pairing) also prohibits such a universal set, however this prohibition is redundant when added to the rest of ZF.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory.

The axioms of pairing, union, replacement, and power set are often stated so that the members of the set x whose existence is being asserted are just those sets which the axiom asserts x must contain.

Nevertheless, in the standard formulation of the Zermelo-Fraenkel set theory, the axiom of pairing follows from the axiom schema of replacement applied to any given set with two or more elements, and thus it is sometimes omitted.