uncountable

Definition A set is uncountable if it is not countable. In other words, a set $S$ is uncountable, if there is no subset of $\mathbb{N}$ (the set of natural numbers) with the same cardinality as $S$.

1. All uncountable sets are infinite. However, the converse is not true, as $\mathbb{N}$ is both infinite and countable.

2. The real numbers form an uncountable set. The famous proof of this result is based on Cantor’s diagonal argument.

Title	uncountable
Canonical name	Uncountable
Date of creation	2013-03-22 11:59:08
Last modified on	2013-03-22 11:59:08
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	9
Author	yark (2760)
Entry type	Definition
Classification	msc 03E10
Synonym	uncountable set
Related topic	CardinalityOfTheContinuum