Namespaces
Variants
Actions

Continuum, cardinality of the

From Encyclopedia of Mathematics
Revision as of 17:16, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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)
How to Cite This Entry:
Continuum, cardinality of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuum,_cardinality_of_the&oldid=32983
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article