Definition:Countable Set/Countably Infinite/Definition 1

Definition
Let $S$ be a set.

$S$ is countably infinite 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.

Also see

 * Equivalence of Definitions of Countably Infinite Set