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<div>In the mathematical field of [[set theory]], the '''continuum''' means the [[real numbers]], or the corresponding [[cardinal number]], <math>\mathfrak{c}</math>.<br />
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The ''[[cardinality of the continuum]]'' is the [[cardinality|size]] of the set of real numbers. The [[continuum hypothesis]] is sometimes stated by saying that no [[cardinality]] lies between that of the continuum and that of the [[natural numbers]], <math>\aleph_0</math>.<br />
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== Linear continuum == <br />
{{Main|Linear continuum}}<br />
According to [[Raymond Wilder]] (1965) there are four axioms that make a set ''C'' and the relation < into a '''linear continuum''':<br />
* ''C'' is [[Totally ordered set|simply ordered]] with respect to <.<br />
* If [''A,B''] is a cut of ''C'', then either ''A'' has a last element or ''B'' has a first element. (compare [[Dedekind cut]])<br />
* There exists a non-empty, countable subset ''S'' of ''C'' such that, if ''x,y'' &isin; ''C'' such that ''x'' < ''y'', then there exists ''z'' &isin; ''S'' such that ''x'' < ''z'' < ''y''. ([[separable space|separability axiom]])<br />
* ''C'' has no first element and no last element. ([[Bounded set|Unboundedness axiom]])<br />
These axioms characterize the [[order type]] of the [[real number line]].<br />
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== See also ==<br />
* [[Suslin's problem]]<br />
== References ==<br />
* Raymond L. Wilder (1965) ''The Foundations of Mathematics'', 2nd ed., page 150, [[John Wiley & Sons]].<br />
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[[Category:Set theory]]<br />
[[Category:Infinity]]<br />
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