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## Summary

In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. The elements of the power set of the set {x, y, z} ordered with respect to inclusion.

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

$\forall x\,\exists y\,\forall z\,[z\in y\iff \forall w\,(w\in z\Rightarrow w\in x)]$ where y is the Power set of x, ${\mathcal {P}}(x)$ .

In English, this says:

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

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

Note the subset relation $\subseteq$ 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, $\in$ . By the axiom of extensionality, the set ${\mathcal {P}}(x)$ 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.

## Consequences

The Power Set Axiom allows a simple definition of the Cartesian product of two sets $X$  and $Y$ :

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

Notice that

$x,y\in X\cup Y$
$\{x\},\{x,y\}\in {\mathcal {P}}(X\cup Y)$

and, for example, considering a model using the Kuratowski ordered pair,

$(x,y)=\{\{x\},\{x,y\}\}\in {\mathcal {P}}({\mathcal {P}}(X\cup Y))$

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.