Axiom of power set

From Wikipedia, the free encyclopedia
  (Redirected from Axiom of the power set)
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:

where y is the Power set of x, . In English, this says:

Given any set x, there is a set such that, given any set z, z is a member of if and only if every element of z is also an element of x.

More succinctly: for every set , there is a set consisting precisely of the subsets of .

Note the subset relation is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership, . By the axiom of extensionality, the set is unique.

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 and :

Notice that

and thus the Cartesian product is a set since

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

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.