The Generalized Continuum Hypothesis Can Fail Everywhere

In 1874, Cantor [Cl] showed that every set has cardinality strictly smaller than the cardinality of its power set. Cantor asked [C2] whether for infinite sets X there is a set Y of cardinality strictly between cardinality X and cardinality 2X. The special case of X = Z (the integers) was Hilbert's first problem in his famous list [Hi] (the continuum hypothesis). In this paper we show that it is consistent with Zermelo-Frankel set theory with the full Axiom of Choice (modulo large cardinals) that for every set X there is a set Y such that the cardinality of Y lies strictly between the cardinality of X and the cardinality of the power set of X. It was previously shown in GCdel [G] that the generalized continuum hypothesis (G.C.H) was consistent; i.e., for every infinite cardinal X the cardinal successor to X was 2X. In 1963, Cohen [Co] showed that it was consistent that the continuum hypothesis failed. Easton [E] showed that subject to relatively mild restrictions (Kdnig's theorem) essentially arbitrary behavior of the power set operation could occur at regular cardinals. Singular cardinals presented a significantly more difficult matter. The first work on them was done by Prikry and Silver [P] and [Sil] who showed that the G.C.H. can fail at singular strong limit cardinals. Magidor [Ml] showed that it was consistent that it fail at the first singular strong limit and even that one could have the first failure of the G.C.H. at a singular strong limit. After Magidor's work it was generally believed that arbitrary behavior was possible. Silver, however, showed [Si] that if the G.C.H. holds below a singular cardinal K of uncountable cofinality then it holds at K. Galvin and Hajnal [G-H] showed that under more general conditions the behavior of the power set below a singular cardinal K of uncountable cofinality strongly affects the power set at K. These results are all local results. The consistent global behavior of the power set operation was not settled. We prove the following theorem.