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Martin Sleziak
Martin Sleziak
9:04 AM
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 and, for , that are the only constraints on permissible values for 2κ when κ is a regular cardinal. == Statement of the theoremEdit == 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 is greater than for each α in the domain of G, and is regular for each α in the domain of G, then there is a model of...
This basically says that the only restrictions for powers of cardinals are monotonicity and König's theorem.
It seems rather amazing fact that all that can be proved about $\beth_\alpha$ are facts which are rather simple. (They can be covered in a more advanced course of set theory. Maybe some universities define cofinality and prove Konig's theorem already in the first set theory course.)
IIRC the exponentiation, i.e., $\aleph_\alpha^{\aleph_\beta}$ has more complicated rules. There are some know results (Bukovský-Hechler and similar stuff). I think I vaguely recall something along the lines that here it is also, in some sense, known what can be shown in ZFC and those results cannot be improved. But maybe I do not recall it correctly.
(Something about this was briefly mentioned in a talk by professor Bukovský, which I attended. But it was intended more as a popular talk for general audience rather than talk about set theory.0
 

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