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 P stands for the power set, , of A. In English, this says:
By the axiom of extensionality this set is unique. We call the set 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 the definition of the Cartesian product of two sets X and Y:
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 in Kripke–Platek set theory which does not contain the power set axiom.
References
 Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by SpringerVerlag, New York, 1974. ISBN 0387900926 (SpringerVerlag edition).
 Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3540440852.
 Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0444868399.
This article incorporates material from Axiom of power set on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
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