Axiom of power set

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 every element of B is also an element of A.

Subset is not used in the formal definition because the subset relation is defined axiomatically; axioms must be independent from each other. By the axiom of extensionality this set is unique, which means 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[edit]

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[edit]

  • 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.