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Generalized Continuum Hypothesis

The generalized continuum hypothesis (GCH) states that there is no cardinal number strictly between any infinite cardinal number aleph_alpha and the cardinality of its power set. In terms of alephs, it is the statement

2^(aleph_alpha)=aleph_(alpha+1)

for every ordinal number alpha. Taking alpha=0 gives the continuum hypothesis, so GCH implies CH.

Conway and Guy (1996, p. 282) attribute the generalized version to Hausdorff in 1908.

Like the continuum hypothesis, the generalized continuum hypothesis is undecidable from the standard Zermelo-Fraenkel axioms together with the axiom of choice, provided those axioms are consistent. GCH holds in the constructible universe (Gödel 1940), while forcing can produce models in which it fails (Cohen 1966).

See also

Aleph, Axiom of Choice, Cardinal Number, Continuum Hypothesis, Forcing, Power Set, Zermelo-Fraenkel Axioms

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References

Cohen, P. J. Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 282, 1996.Gödel, K. The Consistency of the Continuum-Hypothesis. Princeton, NJ: Princeton University Press, 1940.Jech, T. J. Set Theory, 2nd ed. Berlin, Germany: Springer-Verlag, 1997.

Cite this as:

Weisstein, Eric W. "Generalized Continuum Hypothesis." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneralizedContinuumHypothesis.html

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