Definition:Finite

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

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 is defined as finite iff it is a real number.

Also see

 * Countable
 * Infinite