 Set-theoretic assumptions and procedures already inform Dedekind’s Stetigkeit und irrationale Zahlen, as we saw. In particular, the system of rational numbers is assumed to be an infinite set; the collection of arbitrary cuts of rational numbers is treated as another infinite set; and when supplied with an order relation and arithmetic operations on its elements, the latter gives rise to a new complete ordered field. Parallel moves can be found in the sketches, from Dedekind’s Nachlass, on how to introduce the integers and the rational numbers. Once more we start with an infinite system, here that of all the natural numbers, and new number systems are constructed out of it set-theoretically (although the full power set is not needed in those cases). Finally, Dedekind uses set-theoretic techniques in his other mathematical work as well (e.g. in his treatment of modular arithmetic and his construction of ideals as infinite sets, to be discussed further below). It should be emphasized that the application of such techniques was novel and bold at the time. While a few mathematicians, such as Cantor, used them as well, many others, like Kronecker, rejected them. In fact, by working seriously with actual infinities Dedekind took a stance incompatible with that of his teacher Gauss, who had allowed for the infinite only as a “manner of speaking” (cf. Ferreirós 1999, ch. 7, also Edwards 1983).