词条 | Axiom of power set |
释义 |
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, this set 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. ConsequencesThe Power Set Axiom allows a simple definition of the Cartesian product of two sets and : Notice that and, for example, considering a model using the Kuratowski ordered pair, 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
1 : Axioms of set theory |
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