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.