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continuum hypothesis (Axiom)

The continuum hypothesis states that there is no cardinal number $ \kappa$ such that $ \aleph_0<\kappa <2^{\aleph_0}$.

An equivalent statement is that $ \aleph_1 =2^{\aleph_0}$.

It is known to be independent of the axioms of ZFC.

The continuum hypothesis can also be stated as: there is no subset of the real numbers which has cardinality strictly between that of the reals and that of the integers. It is from this that the name comes, since the set of real numbers is also known as the continuum.



"continuum hypothesis" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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See Also: axiom of choice, Zermelo-Fraenkel axioms, generalized continuum hypothesis

Other names:  CH
Also defines:  continuum
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Cross-references: integers, strictly, cardinality, real numbers, subset, ZFC, axioms, equivalent, cardinal number
There are 14 references to this entry.

This is version 10 of continuum hypothesis, born on 2002-01-03, modified 2006-11-02.
Object id is 1183, canonical name is ContinuumHypothesis.
Accessed 8579 times total.

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