dans sa coupure de Dedekind. Nous montrons Cgalement que la somme de deux reels dont le dfc est calculable en temps polynomial peut Ctre un reel dont le. and Repetition Deleuze defines ‘limit’ as a ‘genuine cut [coupure]’ ‘in the sense of Dedekind’ (DR /). Dedekind, ‘Continuity and Irrational Numbers’, p. C’est à elle qu’il doit l’idée de la «coupure», dont l’usage doit permettre selon Dedekind de construire des espaces n-dimensionnels par-delà la forme intuitive .

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Dedekind cut – Wikipedia

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Dedekind cut

This page was last edited on 28 Novemberat A related completion that preserves all existing sups and infs of S is obtained by the following derekind Unsourced material may be challenged and removed. Retrieved from ” https: Dedekind cut sqrt 2.

In this case, we say that b is represented by the cut AB. This article needs additional citations for verification. Order theory Rational numbers.

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The important purpose of the Dedekind cut is to work with number sets that are not complete. June Learn how and when to remove this template message. In this way, set inclusion can be used to represent the ordering of numbers, and all other relations greater thanless than or equal toequal toand so on can be dedekijd created from set relations. The Dedekind-MacNeille completion is the smallest complete lattice with S embedded in it.


The cut can represent a number beven though the dedekkind contained in the two sets A and B do not actually include the number b that their dedskind represents.

If B has a smallest element among the rationals, the cut corresponds to that rational.

The cut itself can represent a number not in the original collection of numbers most often rational numbers. It is more symmetrical to use the AB notation for Dedekind cuts, but each coupute A and B does determine the other. Every real number, rational or not, is equated to one and only one cut of rationals. In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.

Retrieved from ” https: The set of all Dedekind cuts is itself a linearly ordered set of sets.

One completion of S is the set of its downwardly closed subsets, ordered by inclusion. Views View Edit History.

Contains information outside the scope of the article Please help improve this article if you can. Articles needing additional references from March All articles needing additional references Articles needing cleanup from June All pages needing cleanup Cleanup tagged articles with a reason field from June Wikipedia pages needing cleanup from June A similar construction to that used by Dedekind cuts was used in Euclid’s Elements book V, definition 5 to define proportional segments.

March Learn how and when to remove this template message. Moreover, the set of Dedekind cuts has the least-upper-bound propertyi. The timestamp is only as accurate as the clock in the camera, and it may be completely wrong.


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”. To establish this truly, one must show that this really is a cut and that it is the square root of two. A Dedekind cut is a partition of the rational numbers into two non-empty sets A and Bsuch that all elements of A are less than all elements of Band A contains no greatest element.

File:Dedekind cut- square root of two.png

Please help improve this article by adding citations to reliable sources. Views Read Edit View history. Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set Swhich might not have had the least-upper-bound property, within a usually larger linearly ordered set that does have this useful property.

However, neither claim is immediate. Summary [ edit ] Description Dedekind cut- square root of two. From Wikipedia, the free encyclopedia. Similarly, every cut of reals is identical to the cut produced by a specific real number which can be identified as the smallest element of the B set.