Continuum, cardinality of the

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)