|
( Uncountable set)
In mathematics, an uncountable set is an infinite set which is too big to be countable. The uncountability of a set is closely related to its cardinal number a set is uncountable if its cardinal number is larger than that of the natural numbers. The related term nondenumerable set is used by some authors as a synonym for "uncountable set" while other authors define a set to be nondenumerable if it is not an infinite countable set. There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions holds The first three of these characterizations can be proved equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles. The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (and even the set of all infinite sequences consisting only of zeros and ones) and the set of all subsets of the set of natural numbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or  , or  (beth-one).
|
Uncountable set Subcategories
Uncountable set Articles
|
| 
