generalized continuum hypothesisGeneralizedContinuumHypothesis2002-01-03 17:04:102004-04-02 07:39:44Axiomcardinalitycardinal\usepackage{amssymb}
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The \emph{generalized continuum hypothesis} states that for any infinite cardinal $\lambda$ there is no cardinal $\kappa$ such that $\lambda <\kappa <2^{\lambda}$.
An equivalent condition is that $\aleph_{\alpha+1}=2^{\aleph_\alpha}$ for every ordinal $\alpha$.
Another equivalent condition is that $\aleph_\alpha=\beth_\alpha$ for every ordinal $\alpha$.
Like the continuum hypothesis, the generalized continuum hypothesis is known to be independent of the axioms of ZFC.