A set $X$ is countable if there is an injection from $X$ to $\N$.
$X$ is countably infinite if there is a bijection from $X$ to $\N$.
$X$ is uncountable if it is not countable.
A set $X$ is countable if there is an injection from $X$ to $\N$.
$X$ is countably infinite if there is a bijection from $X$ to $\N$.
$X$ is uncountable if it is not countable.