Definition:Finite

Finite Set
A set $S$ is defined as finite iff $\exists n \in \N: S \sim \N_n$.

That is, if there exists an element $n$ of the set of natural numbers $\N$ such that the set of all elements of $\N$ less than $n$ is equivalent to $S$.

Finite Real Number
A real number $x \in \R$ is defined as finite iff:
 * $\exists n \in \N: -n < x < n$

That is, a real number is finite iff there is a natural number which is greater, and its negative smaller, than it.

Also see

 * Countable
 * Infinite