Definition:Countable/Also defined as

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Some sources define countable to be what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as countably infinite.

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

Thus under this criterion $X$ is said to be countable if and only if there exists a bijection from $X$ to $\N$, that is, if and only if $X$ is equivalent to $\N$.

However, as the very concept of the term countable implies that a collection can be counted, which, plainly, a finite can be, it is suggested that this interpretation may be counter-intuitive.

Hence, on $\mathsf{Pr} \infty \mathsf{fWiki}$, the term countable will be taken in the sense as to include the concept of finite, and countably infinite will mean a countable collection which is specifically not finite.