In the formal language of the ZermeloFraenkel axioms, the axiom reads:
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 extension 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:
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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