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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''. 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 == 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「axiom of power set」の詳細全文を読む スポンサード リンク
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