# Axiom of power set

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:

- ∀
*A*, ∃ *B*, ∀ *C*, *C* ∈ *B* ↔ (∀ *D*, *D* ∈ *C* → *D* ∈ *A*);

or in words:

- Given any set
*A*, there is a set *B* such that, given any set *C*, *C* is a member of *B* if and only if, given any set *D*, if *D* is a member of *C*, then *D* is a member of *A*.

To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that

*C* is a

subset of

*A*.
Thus, what the axiom is really saying is that, given a set

*A*, we can find a set

*B* whose members are precisely the subsets of

*A*.
We can use the

axiom of extensionality to show that this set

*B* is unique.
We call the set

*B* the

*power set* of

*A*, and denote it

**P***A*.
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.