- Axiom of power set
mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory.
formal languageof the Zermelo-Fraenkel axioms, the axiom reads:
where "P" stands for the power set, , of "A". In English, this says:
Given anyset "A", there is a set such that, given any set "B", "B" is a member of if and only if"B" is a subsetof "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.)
axiom of extensionalitythis 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 is generally considered uncontroversial and it, or an equivalent axiom, appears in most alternative
axiomatizations of set theory.
The Power Set Axiom allows the definition of the
Cartesian productof two sets and :
and thus the Cartesian product is a set since
Note that the existence of the Cartesian product can be proved in
Kripke–Platek set theorywhich does not contain the power set axiom.
*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.
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