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Continuum (view source)

Revision as of 19:34, February 16, 2009

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Mathematically, '''continuum''' can refer to the [[real line]], its [[cardinality]] <math>\mathfrak{c}</math>, or any [[continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[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]] 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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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 number]]s 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.

[[Category:Mathematics]]

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