View source for Continuum, cardinality of the

{{TEX|done}}
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 $2\leq\alpha\leq\mathfrak c$,

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

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

$$\mathfrak c=\aleph_1.$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,   A. Mostowski,   "Set theory" , North-Holland  (1968)</TD></TR></table>