Kerodon

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.