Definition:Countably Infinite

Definition
Let $S$ be a set.

Then $S$ is countably infinite iff there is a bijection $f: S \to \N$, where $\N$ is the set of natural numbers.

That is, it is an infinite set of the form:
 * $\left\{{s_0, s_1, \ldots, s_n, \ldots}\right\}$

where $n$ runs over all the natural numbers

An infinite set is countably infinite if it is countable, and is uncountable otherwise.

The cardinality of a countably infinite set is denoted by the symbol $\aleph_0$ (aleph null).

Also defined as
Some sources use the term countable to mean specifically countably infinite, that is, not including finite sets as countable.

That is, to use countable to describe a set which has exactly the same cardinality as $\N$.

That is, $X$ is said under this criterion to be countable iff there exists a bijection from $X$ to $\N$, i.e. equivalent to $\N$.

However, this definition also seems to be going out of fashion, as the very concept of the term countable implies that a set can be counted, which, plainly, a finite set can be.

Also see

 * Definition:Countable Set

From Infinite Set has Countably Infinite Subset it is seen that the Axiom of Countable Choice implies that $\aleph_0$ is the smallest possible cardinality of an infinite set.