Axiom of power set

Consequences

The power set axiom allows a simple definition of the Cartesian product of two sets ${\displaystyle X}$ and ${\displaystyle Y}$:

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

Notice that

${\displaystyle x,y\in X\cup Y}$

${\displaystyle \{x\},\{x,y\}\in \mathcal{P}(X\cup Y)}$

and, for example, considering a model using the Kuratowski ordered pair,

${\displaystyle (x,y)=\{\{x\},\{x,y\}\}\in \mathcal{P}(\mathcal{P}(X\cup Y))}$

and thus the Cartesian product is a set since

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

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

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

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.

Limitations

The power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do. Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is the constructible universe. But in other models of ZF, the universe could contain sets that are not constructible.