axiom of power set

The axiom of power set appears in most axiomatizations of set theory.

In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory.

Within the framework of Zermelo-Fraenkel set theory, the axiom of power set guarantees the existence of the power set of any given set.

(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.)

In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.

Both these methods produce systems which satisfy the axioms of second-order arithmetic, since the axiom of power set allows us to quantify over the power set of , as in second-order logic.

Even if GST did afford a countably infinite set, GST could not prove the existence of a set whose cardinality is , because GST lacks the axiom of power set.

It can be shown that a class is a proper class if and only if it is equinumerous to V, but von Neumann's axiom does not capture all of the "limitation of size doctrine", because the axiom of power set is not a consequence of it.