Axiom of power set

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Template:No footnotes 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:

Given any set A, there is a set such that, given any set B, B is a member of 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

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

  • 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.

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