 Dedekind’s published this account of the real numbers only in 1872, fourteen years after developing the basic ideas on which it relies. It was not the only account proposed at the time; indeed, various mathematicians addressed this issue, including: Weierstrass, Thomae, Méray, Heine, Hankel, Cantor, and, somewhat later, Frege (Dieudonné 1985, ch. 6, Boyer &amp; Merzbach 1991, ch. 25, Jahnke 2003, ch. 10). Most familiar among their alternative approaches is probably Cantor’s, also published in 1872. Instead of using “Dedekind cuts”, Cantor works with (equivalence classes of) Cauchy sequences of rational numbers. The system of such (classes of) sequences can also be shown to have the desired properties, including continuity. Like Dedekind, Cantor starts with the infinite set of rational numbers; Cantor’s construction again relies essentially on the full power set of the rational numbers, here in the form of arbitrary Cauchy sequences; and the two resulting systems are isomorphic, as is well known. In these set-theoretic respects the two treatments are thus equivalent. What sets apart Dedekind’s treatment of the real numbers, from Cantor’s and all the others, is the clarity he achieves with respect to the central notion of continuity. His treatment is also more elegantly and maturely structuralist, in a sense to be spelled out further below.