 The sort of infinities that are most familiar to philosophers are infinities used for counting. In the early 1820s, Bolzano arrived at the idea that an infinite set is one for which there is a bijection between the set and a proper subset of it. (Recall Galileo’s paradox.) Dedekind (1884) gave this as a definition of being infinite. It is easy to prove that a Dedekind-infinite set must contain a set that is just as large as the natural numbers. See the