Axiom of power set

In mathematics, the axiom of power set[1] is one of the Zermelo–Fraenkel axioms of axiomatic set theory.

It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.

is defined in terms of set membership,

Given this, in the formal language of the Zermelo–Fraenkel axioms, the axiom of power set reads: where y is the power set of x, z is any element of y, w is any member of z.

In English, this says: The power set axiom allows a simple definition of the Cartesian product of two sets

: 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: 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.

[2] Not all conceivable subsets are guaranteed to exist.

This article incorporates material from Axiom of power set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.