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
Contents
and, for
are the only constraints on permissible values for 2κ when κ is a regular cardinal.
Statement of the theorem
Easton's theorem states that if G is a class function whose domain consists of ordinals and whose range consists of ordinals such that
- G is non-decreasing,
- the cofinality of
ℵ G ( α ) ℵ α -
ℵ α
then there is a model of ZFC such that
for each
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 König's theorem.
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.
No extension to singular cardinals
Silver (1975) proved that a singular cardinal of uncountable cofinality cannot be the smallest cardinal for which the generalized continuum hypothesis fails. This shows that Easton's theorem cannot be extended to the class of all cardinals. The program of PCF theory gives results on the possible values of