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'generalized continuum hypothesis'
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| Title of object: |
generalized continuum hypothesis |
| Canonical Name: |
GeneralizedContinuumHypothesis |
| Type: |
Axiom |
| Created on: |
2002-01-03 17:04:10 |
| Modified on: |
2004-04-02 07:37:25 |
| Classification: |
msc:03E50 |
| Keywords: |
cardinality, cardinal |
| Synonyms: |
generalized continuum hypothesis=generalised continuum hypothesis
generalized continuum hypothesis=GCH |
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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. |
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