Axiom of power set
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set the existence of a set, the power set of, consisting precisely of the subsets of. 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.
Formal statement
The subset relation is not a primitive notion in formal set theory and is not used in the formal language of the Zermelo–Fraenkel axioms. Rather, the subset relation 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:
Consequences
The 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:
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.