Difference between revisions of "Continuum hypothesis"
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| − | The '''Continuum hypothesis''' is a conjecture by [[Georg Cantor]] which states that there is no set with cardinality | + | 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>. |
| − | [[category: | + | 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. |
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| + | [[Category:Set Theory]] | ||
| + | [[category:infinity]] | ||
Latest revision as of 00:18, March 31, 2019
The Continuum hypothesis is a conjecture by Georg Cantor which states that there is no set with cardinality greater than all the natural numbers but less than the cardinality of the real numbers (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 .
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.