Dedekind cut Totally Explained

Everything about Dedekind Complete totally explained

In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B. The original and most important cases are Dedekind cuts for rational numbers and real numbers. Dedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom). See also completeness (order theory). The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it isn't on a real rational number, an irrational number (which is also a real number) is created by the mathematician. Through the use of this device, there's considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity.
Dedekind used the ambiguous word cut (Schnitt) in the geometric sense. That is, it's an intersection of a line with another line that crosses it. It isn't a gap. When one line crosses another in geometry, it's said to cut that line. In this case, one of the lines is the number line. Both lines have one point in common. At that one point on the number line, if there's no rational number, the mathematician posits or arbitrarily places an irrational number. This results in the positioning of a real number at every point on the continuum.

Handling Dedekind cuts

It is more symmetrical to use the (A,B) notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one 'half' — say, the lower one — and call any downward closed set A without last element a "Dedekind cut".
If the ordered set S is complete, then every set B in a Dedekind cut (A, B) must have a minimal element b, hence we must have that A is the interval (−∞, b), and B the interval [b, +∞).

Ordering Dedekind cuts

Regard one Dedekind cut is rational.

Additional structure on the cuts

ť See: Construction of real numbers

Generalization: Dedekind completions in posets

More generally, if S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding of S into L. The notion of complete lattice generalizes the least-upper-bound property of the reals.
One completion of S is the set of its downwardly closed subsets (also called order ideals), ordered by inclusion. S is embedded in this lattice by sending each element x to the ideal it generates.

Dedekind-MacNeille completion

A related completion that preserves all existing sups and infs of S is obtained by the following construction: For each subset A of S, let A^u denote the set of upper bounds of A, and let A^l denote the set of lower bounds of A. (These operators form a Galois connection.) Then the Dedekind-MacNeille completion of S consists of all subsets A for which ť (A^u)^l = A;

it is ordered by inclusion. The Dedekind-MacNeille completion is generally a smaller lattice than the lattice of order ideals; S is embedded in it in the same way. It is the smallest lattice with S embedded in it.
The Dedekind-MacNeille completion of a Boolean algebra is a complete Boolean algebra.

Another generalization: surreal numbers

A construction similar to Dedekind cuts is used for the construction of surreal numbers.

Allusions

In his chapter on Henri Bergson, the author C.E.M. Joad employed imagery that was similar to Dedekind's concept of the cut. Joad was trying to explain how Bergson saw the mind as an instrument that projected permanent objects onto the experience of constant change. "The intellect, then, is a purely practical faculty, which has evolved for the purposes of action. What it does is to take the ceaseless, living flow of which the universe is composed and to make cuts across it, inserting artificial stops or gaps in what is really a continuous and indivisible process. The effect of these stops or gaps is to produce the impression of a world of apparently solid objects. These have no existence as separate objects in reality; they are, as it were, the design or pattern which our intellects have impressed on reality to serve our purposes." This is reminiscent of Dedekind's creation of a new irrational number at every gap in the continuous number line at which there's no existing real number.

Bibliography

Dedekind, Richard, Essays on the Theory of Numbers, "Continuity and Irrational Numbers," Dover: New York, ISBN 0-486-21010-3

Further Information

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