Difference between revisions of "Continuum"
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| − | Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>2^\aleph</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[ | + | Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>2^\aleph</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[Linear_order|linear order]]. Generally, when mathematicians say "the continuum", they are refering to one of the first two possibilities (context clarifies which one). The [[Continuum_hypothesis|Continuum Hypothesis]] conjectures that the cardinality of the set is <math>\aleph</math>. |
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. 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. | 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. 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. | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 18:13, January 12, 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 refering to one of the first two possibilities (context clarifies which one). The Continuum Hypothesis conjectures that the cardinality of the set is .
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. Since the complex numbers add the imaginary number, i, to the real line, this is one reason they have no natural linear order.