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The '''Continuum hypothesis''' is a conjecture by [[Georg Cantor]] which states that there is no set with [[cardinality]] greater than all the [[natural number]]s but less than the cardinality of the [[real number]]s (Continuum). The cardinality of such a set would be denoted by the Hebrew letter <math>\aleph</math>. Cantor died without knowing the answer to his conjecture. [[Kurt Godel]] and [[Paul Cohen]] have since shown that the Continuum hypothesis is [[undecidable]] in [[Zermelo-Fraenkel]] Set Theory.

The '''Continuum hypothesis''' is a conjecture by [[Georg Cantor]] which states that there is no set with [[cardinality]] greater than all the [[natural number]]s but less than the cardinality of the [[real number]]s (Continuum). Stated another way, this hypothesis holds that there is no set which as a cardinality between that of the natural numbers and that of the real numbers.

 

The ''Generalized'' Continuum hypothesis states that there is no set whose cardinality lies between the cardinality of a given set and the cardinality of the set of all subsets of that given set.

 

In standard mathematical nomenclature, the cardinality of a set is denoted by the Hebrew letter <math>\aleph</math>.  

 

Cantor died without knowing the answer to his conjecture of the Continuum hypothesis. [[Kurt Godel]] and [[Paul Cohen]] have since shown that the Continuum hypothesis is [[undecidable]] in [[Zermelo-Fraenkel]] Set Theory.

[[category:set theory]]

[[category:set theory]]