axiom of power set

The axiom of power set is an axiom of Zermelo-Fraenkel set theory which postulates that for any set $X$ there exists a set $\mathcal{P}(X)$ , called the power set of $X$ , consisting of all subsets of $X$ . In symbols, it reads:

$\forall X\exists\mathcal{P}(X)\forall u(u\in\mathcal{P}(X)\leftrightarrow u% \subseteq X).$

In the above, $u\subseteq X$ is defined as $\forall z(z\in u\rightarrow z\in X)$ . By the extensionality axiom, the set $\mathcal{P}(X)$ is unique.

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

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

The Cartesian product is a set since

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

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

$X_{1}\times\cdots\times X_{n}=(X_{1}\times\cdots\times X_{n-1})\times X_{n}.$