Axiom of power set

In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set ${\displaystyle x}$ the existence of a set ${\displaystyle {\mathcal {P}}(x)}$, the power set of ${\displaystyle x}$, consisting precisely of the subsets of ${\displaystyle x}$. 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.