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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 $2\leq\alpha\leq\mathfrak c$,
 
  
$$\alpha^{\aleph_0}=\mathfrak c.$$
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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 $2\leq\alpha\leq\mathfrak c$,
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$$\alpha^{\aleph_0}=\mathfrak c \ .$$
  
 
In particular,
 
In particular,
  
$$2^{\aleph_0}=3^{\aleph_0}=\dots=\aleph_0^{\aleph_0}=\aleph_1^{\aleph_0}=\mathfrak c^{\aleph_0}=\mathfrak c.$$
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$$2^{\aleph_0}=3^{\aleph_0}=\dots=\aleph_0^{\aleph_0}=\aleph_1^{\aleph_0}=\mathfrak c^{\aleph_0}=\mathfrak c \ .$$
  
The [[Continuum hypothesis]] states that the cardinality of the continuum is the first uncountable cardinal number, that is,
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By [[Cantor theorem|Cantor's theorem]] the cardinal $2^{\aleph_0}$ is strictly greater than $\aleph_0$: that is, $\mathfrak c$ is uncountable.  The [[Continuum hypothesis]] states that the cardinality of the continuum is the first uncountable cardinal number, that is,
  
$$\mathfrak c=\aleph_1.$$
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$$\mathfrak c=\aleph_1 \ .$$
  
 
====References====
 
====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,  A. Mostowski,  "Set theory" , North-Holland  (1968)</TD></TR>
 
<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,  A. Mostowski,  "Set theory" , North-Holland  (1968)</TD></TR>
 
</table>
 
</table>
 
[[Category:Set theory]]
 

Latest revision as of 20:11, 10 January 2015

2020 Mathematics Subject Classification: Primary: 03E10 Secondary: 03E50 [MSN][ZBL]

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,

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

By Cantor's theorem the cardinal $2^{\aleph_0}$ is strictly greater than $\aleph_0$: that is, $\mathfrak c$ is uncountable. 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=33744
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article