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:

${\displaystyle \forall x\,\exists y\,\forall z\,[z\in y\iff \forall w\,(w\in z\Rightarrow w\in x)]}$

where y is the power set of x, ${\displaystyle {\mathcal {P}}(x)}$.

In English, this says:

Given any set x, there is a set ${\displaystyle {\mathcal {P}}(x)}$ such that, given any set z, this set z is a member of ${\displaystyle {\mathcal {P}}(x)}$ if and only if every element of z is also an element of x.

More succinctly: for every set ${\displaystyle x}$, there is a set ${\displaystyle {\mathcal {P}}(x)}$ consisting precisely of the subsets of ${\displaystyle x}$.

Note the subset relation ${\displaystyle \subseteq }$ is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership, ${\displaystyle \in }$. By the axiom of extensionality, the set ${\displaystyle {\mathcal {P}}(x)}$ is unique.

The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.