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 Extended Real Number
An extended real number $x \in \overline{\R}$ is defined as finite iff:
 * $\exists n \in \N: -n < x < n$

That is, an extended real number is finite iff there is a natural number which is greater, and its negative smaller, than it. This is the same as saying that $x$ is neither $+\infty$ nor $-\infty$, which in turn is the same as saying that $x$ is a regular real number (because any real number $x \in \R$ is finite in this definition).

Also see

 * Countable
 * Infinite