 While the general idea of reducing analysis to arithmetic, as opposed to geometry, was not new at the time—Dedekind shared it with and adopted it from his teachers Gauss and Dirichlet (Ferreirós 1999, ch. 4, Merzbach 2018)—the particular manner in which he proceeded was quite original. The crucial issue, or the linchpin, for him was the notion of continuity. To get clearer about that notion, he compared the system of rational numbers to the points on a geometric line. Once a point of origin, a unit length, and a direction have been picked for the latter, the two systems can be correlated systematically: each rational number corresponds, in a unique and order-preserving way, to a point on the line. But a further question arises: Does each point on the line correspond to a rational number? Crucially, this question can be reformulated in terms of Dedekind’s idea of “cuts”  defined directly on the rational numbers, so that any geometric intuition concerning continuity can be put aside. Namely, if we divide the whole system of rational numbers into two disjoint parts while preserving their order, is each such division determined by a rational number? The answer is no, since some correspond to irrational numbers (e.g., the cut consisting of \(\{x : x^2 \lt 2\}\) and \(\{x : x^2 \gt 2\}\) which corresponds to \(\sqrt{2}\)). In this explicit, purely arithmetic, and precise sense, the system of rational numbers is not continuous, i.e. not line-complete.