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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:
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\forall A, \exists B, \forall C, C \in B \Leftrightarrow (\forall D, D \in C \Rightarrow D \in A)
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Or, if we've already defined the subset operation:
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\forall A, \exists \mathcal{P}(A), \forall C, C \in \mathcal{P}(A) \Leftrightarrow C \subseteq A
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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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