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Remark 4.7.2.14 (The Generalized Continuum Hypothesis). Let \kappa be a cardinal and let S be a set of cardinality \kappa . We let 2^{\kappa } denote the cardinality of the collection P(S) of subsets of S. Proposition 4.7.2.8 then supplies an inequality \kappa ^{+} \leq 2^{\kappa }. The continuum hypothesis asserts that equality holds in the case \kappa = \aleph _0. The generalized continuum hypothesis is the stronger assertion that \kappa ^{+} = 2^{\kappa } for every infinite cardinal \kappa . It is also known to be independent of the axioms of Zermelo-Fraenkel set theory.