Ordering of Cuts
Regard one Dedekind cut (A, B) as less than another Dedekind cut (C, D) if A is a proper subset of C. Equivalently, if D is a proper subset of B, the cut (A, B) is again less than (C, D). In this way, set inclusion can be used to represent the ordering of numbers, and all other relations (greater than, less than or equal to, equal to, and so on) can be similarly created from set relations.
The set of all Dedekind cuts is itself a linearly ordered set (of sets). Moreover, the set of Dedekind cuts has the least-upper-bound property, i.e., every nonempty subset of it that has any upper bound has a least upper bound. Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set S, which might not have had the least-upper-bound property, within a (usually larger) linearly ordered set that does have this useful property.
Read more about this topic: Dedekind Cut
Famous quotes containing the words ordering of, ordering and/or cuts:
“Make gracious attempts at sanctifying Jenny,
Supply cosmetics for the ordering of her frame,
Think of her as Leda, as a goddess,
Emptying a smile on Redkey, Indiana.”
—Allen Tate (1899–1979)
“The national anthem belongs to the eighteenth century. In it you find us ordering God about to do our political dirty work.”
—George Bernard Shaw (1856–1950)
“You don’t want a general houseworker, do you? Or a traveling companion, quiet, refined, speaks fluent French entirely in the present tense? Or an assistant billiard-maker? Or a private librarian? Or a lady car-washer? Because if you do, I should appreciate your giving me a trial at the job. Any minute now, I am going to become one of the Great Unemployed. I am about to leave literature flat on its face. I don’t want to review books any more. It cuts in too much on my reading.”
—Dorothy Parker (1893–1967)