In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory.

In the formal language of the Zermelo-Fraenkel axioms, the axiom reads:

Or in words:

Given any set A, there is a set PA such that, given any set B, B is a member of PA if and only if B is a subset of A.

By the axiom of extensionality this set is unique. We call the set B the power set of A, and denote it PA. Thus the essence of the axiom is:

Every set has a power set.

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