finite

A set $S$ is finite if there exists a natural number $n$ and a bijection from $S$ to $n$. Note that we are using the set theoretic definition of natural number, under which the natural number $n$ equals the set $\{0,1,2,\mathrm{\dots },n-1\}$. If there exists such an $n$, then it is unique, and we call $n$ the cardinality of $S$.

Equivalently, a set $S$ is finite if and only if there is no bijection between $S$ and any proper subset of $S$.