\begin{definition}[Definition:Countable Set/Also defined as]
Some sources define a countable set to be what is defined on {{ProofWiki}} as a countably infinite set.
That is, they use '''countable''' to describe a set which has ''exactly the same'' cardinality as $\N$.
Thus under this criterion $X$ is said to be countable {{iff}} there exists a bijection from $X$ to $\N$, that is, {{iff}} $X$ is equivalent to $\N$.
However, as the very concept of the term '''countable''' implies that a set '''can be counted''', which, plainly, a finite set can be, it is suggested that this interpretation may be counter-intuitive.
Hence, on {{ProofWiki}}, the term countable set will be taken in the sense as to include the concept of finite set, and countably infinite will mean a countable set which is specifically ''not'' finite.
\end{definition}