Editing Axiom of power set (section)

== Consequences ==

The power set axiom allows a simple definition of the [[Cartesian product]] of two sets <math>X</math> and <math>Y</math>:

:<math> X \times Y = \{ (x, y) : x \in X \land y \in Y \}. </math>

Notice that
:<math>x, y \in X \cup Y </math>
:<math>\{ x \}, \{ x, y \} \in \mathcal{P}(X \cup Y) </math>

and, for example, considering a model using the [[Ordered_pair#Defining_the_ordered_pair_using_set_theory|Kuratowski ordered pair]],

:<math>(x, y) = \{ \{ x \}, \{ x, y \} \} \in \mathcal{P}(\mathcal{P}(X \cup Y)) </math>

and thus the Cartesian product is a set since

:<math> X \times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y)). </math>

One may define the Cartesian product of any [[finite set|finite]] [[class (set theory)|collection]] of sets recursively:

:<math> X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n. </math>

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]].