page_title: Axiom of power set
hierarchical_section_title: Axiom of power set
context_page_description: 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.
context_section_description: 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.
caption: The elements of the power set of the set {x, y, z}  ordered with respect to inclusion.