Axiom of power set
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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:
- <math>\forall A, \exists\; {\mathcal{P}A}, \forall B: B \in {\mathcal{P}A} \iff (\forall C: C \in B \implies C \in A)<math>
Or in words:
- Given any set A, there is a set PA such that, given any set B, B is a member of PA if and only if B is a subset of A.
By the axiom of extensionality this set is unique. We call the set B the power set of A, and denote it PA. Thus the essence of the axiom is:
- Every set has a power set.
The axiom of power set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory.fr:Axiome de l'ensemble des parties