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

1. All uncountable sets are infinite. However, the converse is not true, as $\N$ 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.