Dedekind's Real Numbers

Richard Dedekind's characterization of the real numbers as the system of cuts of rational numbers is by now the standard in almost every mathematical book on analysis or number theory. In the philosophy of mathematics Dedekind is given credit for this achievement, but his more general views are discussed very rarely and only superrcially. For example, Leo Corry, who dedicates a whole chapter of his Modern Algebra and the Rise of Mathematical Structures (1996) writes: \Dedekind deened the system of the real numbers as the collection of all cuts of rationals" (Corry, 1996 ] , p. 73). In this paper I will present Dedekind's own views of his \deenition" and \cre-ation" of the real numbers, and elucidate what he meant by saying that the real numbers \correspond" to the cuts. The upshot of this discussion will be that Corry's statement will be revealed as an obvious, but not uncommon (cf. Landau, 1917 ] , p. 55: \And every such cut, that corresponds to no rational number, deenes an irrational number" (my translation); Weyl, 1919 ] , p. 111: \if we, along with Dedekind, conceive of a real number as a (specially constituted) set of rational numbers"; Maddy, 1992 ] , p. 81: \by identifying real numbers with certain sets (called`Dedekind-cuts'), Dedekind. .. ") misinterpretation.