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 2^κ when κ is a regular cardinal are

(where cf(α) is the cofinality of α) and

If ${\displaystyle G}$ is a class function whose domain consists of ordinals and whose range consists of ordinals such that

then (assuming ZFC is consistent) there is a model of ZFC such that

for each ${\displaystyle \alpha }$ in the domain of ${\displaystyle G}$.

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

The first two conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, while condition 2 follows from Kőnig's theorem.

In Easton's model the powersets of singular cardinals have the smallest possible cardinality compatible with the conditions that ${\displaystyle 2^{\kappa }}$ has cofinality greater than ${\displaystyle \kappa }$ and is a non-decreasing function of ${\displaystyle \kappa }$.