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:

$\backslash forall\; A,\; \backslash exists\backslash ;\; \{\backslash mathcal\{P\}(A)\},\; \backslash forall\; B:\; B\; \backslash in\; \{\backslash mathcal\{P\}(A)\}\; \backslash iff\; (\backslash forall\; C:\; C\; \backslash in\; B\; \backslash implies\; C\; \backslash in\; A)$

Or in words:

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

By the axiom of extensionality this set is unique. We call the set $\backslash mathcal\{P\}(A)$ the power set of A. Thus the essence of the axiom is:

Every set has a power set.

The axiom of power set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory.

Consequences

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

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

The Cartesian product is a set since

$X\; \backslash times\; Y\; \backslash subseteq\; \backslash mathcal\{P\}(\backslash mathcal\{P\}(X\; \backslash cup\; Y)).$

One may define the Cartesian product of any finite collection of sets recursively:

$X\_1\; \backslash times\; \backslash cdots\; \backslash times\; X\_n\; =\; (X\_1\; \backslash times\; \backslash cdots\; \backslash times\; X\_\{n-1\})\; \backslash times\; X\_n.$

Notice that the existence of the Cartesian product can be proved in Kripke–Platek set theory which does not contain the power set axiom.

Axioms of set theory

Axiome de l'ensemble des parties | Assioma dell'insieme potenza | Potensmängdsaxiomet