Easton's theorem

In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets.

Easton (1970) (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2κ when κ is a regular cardinal are (where cf(α) is the cofinality of α) and If G is a class function whose domain consists of ordinals and whose range consists of ordinals such that then there is a model of ZFC such that for each

in the domain of G. The proof of Easton's theorem uses forcing with a proper class of forcing conditions over a model satisfying the generalized continuum hypothesis.

In Easton's model the powersets of singular cardinals have the smallest possible cardinality compatible with the conditions that 2κ has cofinality greater than κ and is a non-decreasing function of κ.

This shows that Easton's theorem cannot be extended to the class of all cardinals.