Axiom of power set

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:

$\forall A \, \exists P \, \forall B \, [B \in P \iff \forall C \, (C \in B \Rightarrow C \in A)]$

where P stands for the power set of A, $\mathcal{P}(A)$ . In English, this says:

Given any set A, there is a set $\mathcal{P}(A)$ such that, given any set B, B is a member of $\mathcal{P}(A)$ if and only if B is a subset of A. (Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.)

By the axiom of extensionality this set is unique. We call the set $\mathcal{P}(A)$ the power set of A. Thus, the essence of the axiom is that every set has a power set.

The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.

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

$x, y \in X \cup Y$

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

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

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:

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

Note that 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.

References

Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

This article incorporates material from Axiom of power set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.