finite and infinite

Distinguishing the cardinality of sets:

A finite set $S$ has a bijection $f:S\to \{1,2,3,\dots ,n\}$ for some natural number $n$, and $n$ is the cardinality of $S$.

An infinite set is not finite.

An infinite set is countably infinite if it has a bijection to the set of natural numbers. Such a set has cardinality ${\mathrm{\aleph }}_0$.