DEF-CNT. Countable and Uncountable Set.
$X$ is countable if there is an injection from $X$ to $\N$, i.e.
\[\lvert X\rvert \leq \aleph_0.\]$X$ is countably infinite if there is a bijection from $X$ to $\N$, i.e.
\[\lvert X\rvert = \aleph_0.\]$X$ is uncountable if it is not countable. If $\AC$ is assumed, $X$ is uncountable if and only if
\[\lvert X\rvert > \aleph_0.\]