 Dedekind does not just assume, or simply postulate, the existence of infinite sets; he tries to prove it. For that purpose, he considers “the totality of all things that can be objects of my thought” and argues that this “set” is infinite (Dedekind 1888a, section v). He also does not just presuppose the concept of infinity; he defines it (in terms of his three basic notions of logic, as well as the definable notions of subset, union, intersection, etc.). The definition is as follows: A set of objects is infinite—“Dedekind-infinite”, as we now say—if it can be mapped one-to-one onto a proper subset of itself. (A set can then be defined to be finite if it is not infinite in this sense.) Moving a step closer to arithmetic, this leads to the notion of a “simple infinity” (or “inductive set”). A rigorous introduction of that notion involves Dedekind’s innovative idea of a “chain”. As one would say in contemporary terminology, a chain is the minimal closure of a set A in a set B containing A under a function f on B (where being “minimal” is captured of in terms of the general notion of intersection).