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Continuum, cardinality of the

From Encyclopedia of Mathematics
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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 $2\leq\alpha\leq\mathfrak c$,

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

In particular,

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

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=33006
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article