\begin{definition}[Definition:Count]
To '''count''' a set $S$ is to establish a bijection between $S$ and an initial segment $\N_n$ of the natural numbers $\N$.
If $S \sim \N_n$ (where $\sim$ denotes set equivalence) then we have '''counted''' $S$ and found it has $n$ elements.
If $S \sim \N$ then $S$ is infinite but countable.
\end{definition}