\begin{definition}[Definition: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 $S$.
Equivalently, a '''finite set''' is a set with a count.
\end{definition}