Suppose S is mapped onto the integers from 1 to n, while another bijection maps S onto the integers from 1 to m, where m < n. Compose these bijections to map the first n integers onto the first m integers. This contradicts the pigeonhole principle; Therefore S has a well defined cardinality, the integer that tells us how many members are in S.
Great! Now we can count.
To be rigorous, integers are finite ordinals, and if S has cardinality n, its members are mapped 1-1 onto the ordinals inside n, which we designate 0 through n-1. The ordinal n acts as the cardinal number for the class of sets that contain n elements.