Principle of Counting
Let $T$ be a set such that $T \sim \N_n$.
- $\forall m \in \N: n \ne m \implies T \nsim \N_m$
That is, we want to know how many objects there are in our set.
What we are doing is trying to match up our set with one of the sets $\N_n$.
That's what we (effectively) do when we count. We pick a set $\N_n$ and compare it in size to see whether it matches the one we're counting. If it doesn't, we pick another one and try again.
In fact, what we do is test them in order. We give each $n \in \N$ a name, and we learn what order they go in and which name matches which $n$. Then we separate the set we are counting into two subsets: the ones counted and the ones still to be counted. As we move each object one by one from the latter into the former, we recite the name of the element of $n$ which follows the previous one we have just counted.
When there are no more elements in the ones still to be counted subset, the $n$ we have reached is the name of the $\N_n$ which is equivalent to the ones counted subset.
This is (confusingly) not to be confused with the Fundamental Principle of Counting, otherwise known as the counting principle.