 Another way to bring out the radical character of Dedekind’s work is by returning to his treatment of the infinite. It starts with taking what was long seen as a paradoxical property (to be equinumerous with a proper subset) to be a defining characteristic of infinite sets. Dedekind then adds a systematic and wide-ranging analysis of the finite in terms of the infinite, again a rather bold idea. For conceptual and methodological innovations like these to become accepted, they not only had to open up novel realms for mathematics; the resulting approach also had to lead to progress in more traditional realms (as it did: the clarification of continuity and of mathematical induction and recursion, results in algebra, algebraic number theory, the theory of algebraic functions, etc.).