# Axiom of power set

Template:No footnotes 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:

$\forall A\,\exists P\,\forall B\,[B\in P\iff \forall C\,(C\in B\Rightarrow C\in A)]$ where P stands for the power set of A, ${\mathcal {P}}(A)$ . In English, this says:

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

Subset is not used in the formal definition because the subset relation is defined axiomatically; axioms must be independent from each other. By the axiom of extensionality this set is unique, which means that every set has a power set.

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.

## Consequences

$X\times Y=\{(x,y):x\in X\land y\in Y\}.$ Notice that

$x,y\in X\cup Y$ $\{x\},\{x,y\}\in {\mathcal {P}}(X\cup Y)$ $(x,y)=\{\{x\},\{x,y\}\}\in {\mathcal {P}}({\mathcal {P}}(X\cup Y))$ and thus the Cartesian product is a set since

$X\times Y\subseteq {\mathcal {P}}({\mathcal {P}}(X\cup Y)).$ One may define the Cartesian product of any finite collection of sets recursively:

$X_{1}\times \cdots \times X_{n}=(X_{1}\times \cdots \times X_{n-1})\times X_{n}.$ Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the Kripke–Platek set theory.