Axiom of power set

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In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory.

In the formal language of the Zermelo-Fraenkel axioms, the axiom reads:

<math>\forall A, \exists\; {\mathcal{P}(A)}, \forall B: B \in {\mathcal{P}(A)} \iff (\forall C: C \in B \implies C \in A)</math>

Or in words:

Given any set A, there is a set <math>\mathcal{P}(A)</math> such that, given any set B, B is a member of <math>\mathcal{P}(A)</math> if and only if B is a subset of A.

By the axiom of extensionality this set is unique. We call the set <math>\mathcal{P}(A)</math> the power set of A. Thus the essence of the axiom is:

Every set has a power set.

The axiom of power set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory.

Consequences

The Power Set Axiom allows the 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>

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 collection of sets recursively:

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

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