Difference between revisions of "Continuum"
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(Removing incorrect statement. The Long line without endpoints is a counterexample to the claim.) |
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Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>\mathfrak{c}</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[Linear_order|linear order]] without [[endpoint]]s. 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>\mathfrak{c}</math>. | Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>\mathfrak{c}</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[Linear_order|linear order]] without [[endpoint]]s. 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>\mathfrak{c}</math>. | ||
| − | The continuum is called so because it was the first (and most prominent) [[Continuous|continuous]] set studied by mathematicians | + | The continuum is called so because it was the first (and most prominent) [[Continuous|continuous]] set studied by mathematicians. Any linear order of the complex numbers needs the [[Axiom of Choice]] to be constructed. |
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 19:09, February 5, 2009
Mathematically, continuum can refer to the real line, its cardinality , or any continuous connected dense linear order without endpoints. 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 .
The continuum is called so because it was the first (and most prominent) continuous set studied by mathematicians. Any linear order of the complex numbers needs the Axiom of Choice to be constructed.