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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, \mathcal{P}(A). In English, this says:
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.
The Power Set Axiom allows a simple 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 without using the power set axiom, as in the case of the Kripke–Platek set theory.
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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