 It is not the cuts themselves with which Dedekind wants to work in the end, however. Instead, for each cut—those corresponding to rational numbers, but especially those corresponding to irrational quantities—he “creates” a new object, a “real number” determined by the cut (cf. Dedekind 1876b, 1888b). Those objects, together with an order relation and arithmetic operations defined on them (in terms of the corresponding cuts), form the crucial system for him. Next, two properties of the new system are established: The rational numbers can be embedded into it, in a way that respects the order and the arithmetic operations (a corresponding field homomorphism exists); and the system is continuous, or line-complete, with respect to its order. What we get overall is the long missing, unified criterion of identity for rational and irrational numbers, both of which are now treated as elements in an encompassing number system (isomorphic to but distinct from the system of cuts). Finally Dedekind indicates how explicit, rigorous, and straightforward proofs of various facts about the real numbers can be given along such lines, including ones that had been accepted without proof so far. These include: basic rules of operation with square roots; and the theorem that every increasing bounded sequence of real numbers has a limit value (a result equivalent to, among others, the more well-known intermediate value theorem.)