# generalized continuum hypothesis

The 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.

 Title generalized continuum hypothesis Canonical name GeneralizedContinuumHypothesis Date of creation 2013-03-22 12:05:31 Last modified on 2013-03-22 12:05:31 Owner yark (2760) Last modified by yark (2760) Numerical id 15 Author yark (2760) Entry type Axiom Classification msc 03E50 Synonym generalised continuum hypothesis Synonym GCH Related topic AlephNumbers Related topic BethNumbers Related topic ContinuumHypothesis Related topic Cardinality Related topic CardinalExponentiationUnderGCH Related topic ZermeloFraenkelAxioms