uncountable

Definition A set is uncountable if it is not countable.In other words, a set $S$ is uncountable, ifthere 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 nottrue, as $\mathbb{N}$ is both infinite and countable.

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