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:
where P stands for the power set of A, . In English, this says:
- Given any set A, there is a set such that, given any set B, B is a member of if and only if B is a subset of A. (Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.)
By the axiom of extensionality this set is unique. We call the set the power set of A. Thus, the essence of the axiom is that every set has a power set.
The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.
Read more about Axiom Of Power Set: Consequences
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