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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 known as

The process of counting is also known as enumeration.

Also see