continuum hypothesis
The hypothesis proposed by Cantor that there is no set with a cardinality greater than that of the natural numbers but less than the cardinality of the set of all subsets of the set of natural numbers (the power set of that set). The generalized continuum hypothesis is that the cardinality of any power set of an infinite set is the next highest cardinality after that of the set itself. Gödel proved in 1938 that the continuum hypothesis is consistent with classical set theory; the American mathematician Paul Cohen proved in 1963 that its negation is so too, i.e. the hypothesis is independent of the other axioms.

Philosophy dictionary. . 2011.

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  • continuum hypothesis — Math. a conjecture of set theory that the first infinite cardinal number greater than the cardinal number of the set of all positive integers is the cardinal number of the set of all real numbers. [1935 40] * * * ▪ mathematics       statement of… …   Universalium

  • continuum hypothesis — noun The hypothesis which states that any infinite subset of must have the cardinality of either the set of natural numbers or of itself …   Wiktionary

  • continuum hypothesis — Math. a conjecture of set theory that the first infinite cardinal number greater than the cardinal number of the set of all positive integers is the cardinal number of the set of all real numbers. [1935 40] …   Useful english dictionary

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  • generalized continuum hypothesis — noun The hypothesis that, for each ordinal <math> alpha</math>, there is no cardinal number strictly between <math> aleph alpha</math> and <math style= vertical align: 0%; >2^ aleph alpha</math>, i.e. <math… …   Wiktionary

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  • Continuum (set theory) — In the mathematical field of set theory, the continuum means the real numbers, or the corresponding cardinal number, . The cardinality of the continuum is the size of the real numbers. The continuum hypothesis is sometimes stated by saying that… …   Wikipedia

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