# 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. 1.

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

2. 2.

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

Title uncountable Uncountable 2013-03-22 11:59:08 2013-03-22 11:59:08 yark (2760) yark (2760) 9 yark (2760) Definition msc 03E10 uncountable set CardinalityOfTheContinuum