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:
“Our goal as a parent is to give life to our children’s learning—to instruct, to teach, to help them develop self-discipline—an ordering of the self from the inside, not imposition from the outside. Any technique that does not give life to a child’s learning and leave a child’s dignity intact cannot be called discipline—it is punishment, no matter what language it is clothed in.”
—Barbara Coloroso (20th century)
“Our goal as a parent is to give life to our children’s learning—to instruct, to teach, to help them develop self-discipline—an ordering of the self from the inside, not imposition from the outside. Any technique that does not give life to a child’s learning and leave a child’s dignity intact cannot be called discipline—it is punishment, no matter what language it is clothed in.”
—Barbara Coloroso (20th century)
“No man, said Birkin, cuts another man’s throat unless he wants to cut it, and unless the other man wants it cutting. This is a complete truth. It takes two people to make a murder: a murderer and a murderee.... And a man who is murderable is a man who has in a profound if hidden lust desires to be murdered.”
—D.H. (David Herbert)