Axiom of power set

The subset relation \subseteq is not a primitive notion in formal set theory and is not used in the formal language of the Zermelo–Fraenkel axioms. Rather, the subset relation \subseteq is defined in terms of set membership, \in. Given this, in the formal language of the Zermelo–Fraenkel axioms, the axiom of power set reads: :\forall x \, \exists y \, \forall z \, [z \in y \iff \forall w \, (w \in z \Rightarrow w \in x)] where y is the power set of x, z is any element of y, w is any member of z. In English, this says: :Given any set x, there is a set y such that, given any set z, this set z is a member of y if and only if every element of z is also an element of x. == Consequences ==

The power set axiom allows a simple definition of the Cartesian product of two sets X and Y: : X \times Y = \{ (x, y) : x \in X \land y \in Y \}. Notice that :x, y \in X \cup Y :\{ x \}, \{ x, y \} \in \mathcal{P}(X \cup Y) and, for example, considering a model using the Kuratowski ordered pair, :(x, y) = \{ \{ x \}, \{ x, y \} \} \in \mathcal{P}(\mathcal{P}(X \cup Y)) and thus the Cartesian product is a set since : X \times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y)). One 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. 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. == References ==