\begin{definition}[Definition:Countable Set/Countably Infinite/Definition 1]
Let $S$ be a set.
$S$ is '''countably infinite''' {{iff}} there exists a bijection:
:$f: S \to \N$
where $\N$ is the set of natural numbers.
That is, it is an infinite set of the form:
:$\set {s_0, s_1, \ldots, s_n, \ldots}$
where $n$ runs over all the natural numbers.
An infinite set is '''countably infinite''' if it is countable, and is uncountable otherwise.
\end{definition}