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In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it into two non-empty parts, (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. 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 is not 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 is 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 is an intersection of a line with another line that crosses it. It is not a gap. When one line crosses another in geometry, it is 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 is 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.

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 greatest 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, +∞). In this case, we say that b is represented by the cut (A,B).

Regard one Dedekind cut { A, B } as less than another Dedekind cut { C, D } if A is a proper subset of C, or, equivalently D is a proper subset of B. In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it has the least-upper-bound property, i.e., every nonempty subset of it that has an upper bound has a least upper bound. Embedding S within a larger linearly ordered set that does have the least-upper-bound property is the purpose.

A typical Dedekind cut of the rational numbers is given by

- $A\; =\; \{\; ainmathbb\{Q\}\; :\; a^2\; <\; 2\; lor\; ale\; 0\; \},$

- $B\; =\; \{\; binmathbb\{Q\}\; :\; b^2\; ge\; 2\; land\; b\; >\; 0\; \}.$

This cut represents the irrational number $sqrt\{2\}$ in Dedekind's construction. Note that the equality $b^2\; =\; 2$ cannot hold since that would imply that$sqrt\{2\}$ is rational.

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.

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

- (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.

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

- Dedekind, Richard, Essays on the Theory of Numbers, "Continuity and Irrational Numbers," Dover: New York, ISBN 0-486-21010-3. Also available at Project Gutenberg.

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Last updated on Sunday October 05, 2008 at 18:41:45 PDT (GMT -0700)

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Last updated on Sunday October 05, 2008 at 18:41:45 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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