PlanetMath: generalized continuum hypothesis

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generalized continuum hypothesis

(Axiom)

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.

"generalized continuum hypothesis" is owned by yark. [ full author list (2) | owner history (1) ]

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Other names:

generalised continuum hypothesis, GCH

Keywords:

cardinality, cardinal

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Cross-references: ZFC, axioms, continuum hypothesis, ordinal, cardinal, infinite
There are 4 references to this entry.

This is version 11 of generalized continuum hypothesis, born on 2002-01-03, modified 2004-04-02.
Object id is 1184, canonical name is GeneralizedContinuumHypothesis.
Accessed 6384 times total.

Classification:

AMS MSC:

03E50 (Mathematical logic and foundations :: Set theory :: Continuum hypothesis and Martin's axiom)

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