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: $$ X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n. $$