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:

   \forall A, \exists B, \forall C, C \in B \Leftrightarrow (\forall D, D \in C \Rightarrow D \in A)

Or, if we've already defined the subset operation:

   \forall A, \exists \mathcal{P}(A), \forall C, C \in \mathcal{P}(A) \Leftrightarrow C \subseteq 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 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.

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