Concept in axiomatic set theory

The elements of the power set of the set {

*x*,

*y*,

*z*}

ordered with respect to

inclusion.

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:

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

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.