Difference between revisions of "Continuum"

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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]]. 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>.
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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>.
  
The continuum is called so because it was the first (and most prominent) [[Continuous|continuous]] set studied by mathematicians.  
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The continuum is called so because it was the first (and most prominent) [[Continuous|continuous]] set studied by mathematicians. No additional numbers may be added to the continuum (real line) without losing its dense linear order without endpoints. Since the [[Complex_number|complex numbers]] add the [[Imaginary_number|imaginary number]], i, to the real line, this is one reason they have no natural linear order. Any linear order of the complex numbers needs the [[Axiom of Choice]] to be constructed.
  
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]

Revision as of 03:57, 5 February 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. No additional numbers may be added to the continuum (real line) without losing its dense linear order without endpoints. Since the complex numbers add the imaginary number, i, to the real line, this is one reason they have no natural linear order. Any linear order of the complex numbers needs the Axiom of Choice to be constructed.