uncountableUncountable2001-11-16 00:49:452006-07-12 11:53:44Definition\usepackage{amssymb}
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{\bf Definition} A set is \emph{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 $S$.
\begin{enumerate}
\item All uncountable sets are infinite. However, the converse is not
true, as $\N$ is both infinite and countable.
\item The real numbers form an uncountable set. The famous proof
of this result is based on Cantor's diagonal argument.
\end{enumerate}