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The [[Cardinal number|cardinal number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257801.png" />, i.e. the cardinality of the set of all subsets of the natural numbers. The following sets have the cardinality of the continuum: 1) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257802.png" /> of all real numbers; 2) the set of all points in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257803.png" />; 3) the set of all irrational numbers in this interval; 4) the set of all points of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257805.png" /> is a positive integer; 5) the set of all transcendental numbers; and 6) the set of all continuous functions of a real variable. The cardinality of the continuum cannot be represented as a countable sum of smaller cardinal numbers. For any cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257806.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257807.png" />,
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The [[Cardinal number|cardinal number]] $\mathfrak c=2^{\aleph_0}$, i.e. the cardinality of the set of all subsets of the natural numbers. The following sets have the cardinality of the continuum: 1) the set $\mathbf R$ of all real numbers; 2) the set of all points in the interval $(0,1)$; 3) the set of all irrational numbers in this interval; 4) the set of all points of the space $\mathbf R^n$, where $n$ is a positive integer; 5) the set of all transcendental numbers; and 6) the set of all continuous functions of a real variable. The cardinality of the continuum cannot be represented as a countable sum of smaller cardinal numbers. For any cardinal number $\alpha$ such that $2\leq\alpha\leq\mathfrak c$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257808.png" /></td> </tr></table>
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$$\alpha^{\aleph_0}=\mathfrak c.$$
  
 
In particular,
 
In particular,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257809.png" /></td> </tr></table>
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$$2^{\aleph_0}=3^{\aleph_0}=\ldots=\aleph_0^{\aleph_0}=\aleph_1^{\aleph_0}=\mathfrak c^{\aleph_0}=\mathfrak c.$$
  
 
The [[Continuum hypothesis|continuum hypothesis]] states that the cardinality of the continuum is the first uncountable cardinal number, that is,
 
The [[Continuum hypothesis|continuum hypothesis]] states that the cardinality of the continuum is the first uncountable cardinal number, that is,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c02578010.png" /></td> </tr></table>
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$$\mathfrak c=\aleph_1.$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,  A. Mostowski,  "Set theory" , North-Holland  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,  A. Mostowski,  "Set theory" , North-Holland  (1968)</TD></TR></table>

Revision as of 19:13, 17 August 2014

The cardinal number $\mathfrak c=2^{\aleph_0}$, i.e. the cardinality of the set of all subsets of the natural numbers. The following sets have the cardinality of the continuum: 1) the set $\mathbf R$ of all real numbers; 2) the set of all points in the interval $(0,1)$; 3) the set of all irrational numbers in this interval; 4) the set of all points of the space $\mathbf R^n$, where $n$ is a positive integer; 5) the set of all transcendental numbers; and 6) the set of all continuous functions of a real variable. The cardinality of the continuum cannot be represented as a countable sum of smaller cardinal numbers. For any cardinal number $\alpha$ such that $2\leq\alpha\leq\mathfrak c$,

$$\alpha^{\aleph_0}=\mathfrak c.$$

In particular,

$$2^{\aleph_0}=3^{\aleph_0}=\ldots=\aleph_0^{\aleph_0}=\aleph_1^{\aleph_0}=\mathfrak c^{\aleph_0}=\mathfrak c.$$

The continuum hypothesis states that the cardinality of the continuum is the first uncountable cardinal number, that is,

$$\mathfrak c=\aleph_1.$$

References

[1] K. Kuratowski, A. Mostowski, "Set theory" , North-Holland (1968)
How to Cite This Entry:
Continuum, cardinality of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuum,_cardinality_of_the&oldid=32983
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article