Continuum, cardinality of the

The cardinal number , 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 of all real numbers; 2) the set of all points in the interval ; 3) the set of all irrational numbers in this interval; 4) the set of all points of the space , where 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 such that ,

In particular,

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

References

[1]	K. Kuratowski, A. Mostowski, "Set theory" , North-Holland (1968)