Dedekind cut
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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).
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. Template:Quotation
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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' for example the lower part a. For example it is shown that the typical Dedekind cut in the real numbers is a pair with A the interval ( −∞, a), and B the interval [b, +∞).
Ordering Dedekind cuts
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., its every nonempty subset 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.
The cut construction of the real numbers
The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers. A typical Dedekind cut of the rational numbers is given by
- <math>A = \{ a\in\mathbb{Q} : a^2 < 2 \lor a\le 0 \},</math>
- <math>B = \{ b\in\mathbb{Q} : b^2 \ge 2 \land b > 0 \}.</math>
This cut represents the real number<math>\sqrt{2}</math> in Dedekind's construction. Note that the equality <math>b^2 = 2</math> cannot hold since that would imply that<math>\sqrt{2}</math> is rational.
Additional structure on the cuts
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.
Another completion is obtained by the following construction: For each subset A of S, let Au denote the set of upper bounds of A, and let Al 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 (Au)l = A; it is ordered by inclusion. The Dedekind-Macneille completion is generally a sublattice of the lattice of order ideals; S is embedded in it in the same way.
Another generalization: surreal numbers
A construction similar to Dedekind cuts is used for the construction of surreal numbers.
References
- Dedekind, Richard, Essays on the Theory of Numbers, "Continuity and Irrational Numbers," Dover: New York, ISBN 0-486-21010-3
See also
es:Cortaduras de Dedekind fr:Coupure de Dedekind he:חתכי דדקינד nl:Dedekindsnede fi:Dedekindin leikkaus