Difference between revisions of "Continuum"
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(There is no such thing as a continuous set. Link given is to continuous function.) |
(The statement removed is not true. One could add a number at either end of the real line, define the order in the obvious way and one would still have a continuum.) |
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Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>2^{\aleph_0}</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[Linear_order|linear order]]. Generally, when mathematicians say "the continuum", they are referring to one of the first two possibilities (context clarifies which one). The [[Continuum_hypothesis|Continuum Hypothesis]] conjectures that there is no set of cardinality bigger than that of the [[natural number]]s <math>\aleph_0</math>, but smaller than the cardinality of the set of the real numbers, <math>2^{\aleph_0}</math>. | Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>2^{\aleph_0}</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[Linear_order|linear order]]. Generally, when mathematicians say "the continuum", they are referring to one of the first two possibilities (context clarifies which one). The [[Continuum_hypothesis|Continuum Hypothesis]] conjectures that there is no set of cardinality bigger than that of the [[natural number]]s <math>\aleph_0</math>, but smaller than the cardinality of the set of the real numbers, <math>2^{\aleph_0}</math>. | ||
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[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 20:02, November 27, 2008
Mathematically, continuum can refer to the real line, its cardinality , or any continuous connected dense linear order. Generally, when mathematicians say "the continuum", they are referring to one of the first two possibilities (context clarifies which one). The Continuum Hypothesis conjectures that there is no set of cardinality bigger than that of the natural numbers , but smaller than the cardinality of the set of the real numbers, .
