Easton's Theorem, proved by William Easton in 1970 while extending an earlier result of Robert Solovay's, shows that for regular cardinals the generalized continuum hypothesis can fail in essentially any pattern consistent with two basic constraints: that the continuum function is non-decreasing, and that its value at a cardinal has cofinality greater than that cardinal. Using the technique of forcing, Easton showed that any function meeting these two conditions can be realized as the actual value of two to the kappa for every regular cardinal kappa in some model of set theory, a freedom that Silver's theorem later showed does not extend to singular cardinals.
Facts
StatementFor every regular cardinal kappa, the only constraints on the value of two raised to kappa are that kappa is less than the cofinality of two raised to kappa, and that the function is non-decreasing, so kappa less than lambda implies two raised to kappa is less than or equal to two raised to lambda; subject only to those two conditions, forcing can realize any pattern of values for two raised to kappa across the regular cardinals simultaneously. 1 Classification
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Source Easton's theorem, Wikipedia
Sources
1. Easton's theorem, Wikipedia
Lead section
Easton (1970) (extending a result of Robert M.
References section, Easton 1970 citation
Easton, W. (1970), Powers of regular cardinals, Ann. Math. Logic, 1 (2): 139-178.
In Branch: Set Theory, Lead sentence
In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets.
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