Definition:Count

Definition
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.

Also see

 * Definition:Uncountable


 * Principle of Counting