Axiom of power set

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:

<math>\forall A, \exists\; {\mathcal{P}A}, \forall B: B \in {\mathcal{P}A} \iff (\forall C: C \in B \implies C \in A)<math>

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.