We have proved that sets are finite, but we have not yet rigorously shown that a set is infinite. We also do not have an exact notion of what it means for a finite set “to have n elements.” Our proof of the former and the definition of the latter will depend on a principle known as the pigeonhole principle. The principle says, informally speaking, that if you have more pigeons than holes, you’ll have to put at least two pigeons in the same hole. Obvious, right? The consequences of this seemingly simple principle are anything but. As a consequence of this principle, we’ll finally be able to define the cardinality of a finite set.