generalized continuum hypothesis

The generalized continuum hypothesis states that for any infinite cardinal $\lambda$ there is no cardinal $\kappa$ such that .

An equivalent condition is that ${\mathrm{\aleph }}_{\alpha +1}=2^{{\mathrm{\aleph }}_{\alpha }}$ for every ordinal $\alpha$. Another equivalent condition is that ${\mathrm{\aleph }}_{\alpha }={\mathrm{\beth }}_{\alpha }$ 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