41. Dedekind Cut - Definition In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S http://www.wordiq.com/definition/Dedekind_cut

Dedekind cut - Definition In mathematics , a Dedekind cut in a totally ordered set S is a partition of it, ( A B ), such that A is closed downwards (meaning that for any element x in S , if a is in A and x a , then x is in A as well) and B is closed upwards. 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 Contents showTocToggle("show","hide") 1 Handling Dedekind cuts 2 Ordering Dedekind cuts 3 The cut construction of the real numbers 4 Additional structure on the cuts ... 7 See also 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 either a pair with A the interval a ), in which case B must be [ a A the interval a ], in which case B must be ( a Ordering Dedekind cuts If a is a member of S then the set a ); by identifying

42. Construction Of The Reals Via Dedekind Cuts We discuss the construction of the set of real numbers as Dedekind cuts of rational numbers. http://web.mat.bham.ac.uk/R.W.Kaye/seqser/constrreals3

43. Dedekind Cut (mathematics) -- Britannica Online Encyclopedia Dedekind cut (mathematics), in mathematics, concept advanced in 1872 by the German mathematician Richard Dedekind that combines an arithmetic formulation of http://www.britannica.com/EBchecked/topic/155424/Dedekind-cut

document.write(''); Search Site: Advanced Search With all of these words With the exact phrase With any of these words Without these words Home SHOP BROWSE BLOG ... HELP Username: Password: Remember me Forgot your password? Help Email " is the e-mail address you used when you registered. Password " is case sensitive. If you need additional assistance, please contact customer support Enter the e-mail address you used when enrolling for Britannica Premium Service and we will e-mail your password to you. Dedekind-cut My Britannica CREATE MY Dedekind cut NEW ARTICLE ... SAVE Dedekind cut Table of Contents: Dedekind cut Article Article Related Articles Related Articles Citations Article Related Articles Citations ARTICLE from the Dedekind cut in mathematics , concept advanced in 1872 by the German mathematician Richard Dedekind that combines an arithmetic formulation of the idea of continuity with a rigorous distinction between rational and irrational numbers . Dedekind reasoned that the real numbers form an ordered continuum, so that any two numbers x and y must satisfy one and only one of the conditions x y x y , or x y . He postulated a cut that separates the continuum into two subsets, say

44. Dedekind Cuts RECIPROCALS OF INTERVALS, because I think discussion of this detail may Even in the context of interval analysis, I would prefer to put the emphasis This result is the http://sci.tech-archive.net/Archive/sci.math.research/2008-03/msg00030.html

Dedekind cuts From Date : Sat, 8 Mar 2008 16:28:03 +0000 David Cantrell queried the definition of the reciprocal of an interval that I gave in my posting. I have replied to this under a new heading, RECIPROCALS OF INTERVALS, because I think discussion of this detail may distract attention from the other mathematical issues that I raised about Dedekind cuts. Even in the context of interval analysis, I would prefer to put the emphasis on the FUNDAMENTAL THEOREM OF INTERVAL ANALYSIS, pointing out that I meant to apply this to totally defined continuous functions: I would like to know where this has been stated and proved in the interval literature. The "semantic" proof of this uses the Scott topology on the interval domain, and is given in Section 2 of our paper. However, I also have a purely syntactic proof, which is the main result in my draft paper called "Interval analysis without intervals". All three of these papers are available from www.PaulTaylor.EU/ASD/analysis.php I am also keen to know who first - showed that Dedekind complete implies Archimedean, classically, and

45. Arithmetic Of Dedekind Cuts Of Ordered Abelian Groups File Format PDF/Adobe Acrobat Quick View http://www.dm.unipi.it/~fornasiero/articles/scrolls.pdf

46. Dedekind Cuts: Real Numbers As Partitions Of The Ordered Field Of Rational Numbe appletmagic.com Thayer Watkins Silicon Valley Tornado Alley USA http://www.sjsu.edu/faculty/watkins/dedekind.htm

applet-magic.com Thayer Watkins Silicon Valley USA Dedekind Cuts: Real Numbers as Partitions of the Ordered Field of Rational Numbers Real In the 19th century mathematicians began to find ways to derive the real numbers. One such formulation was due to the French mathematician Augustin-Louis Cauchy (1789-1857) and identified real numbers with convergent sequences. See Cauchy . Another was from the German mathematician Richard Dedekind (1831-1916) and his formulation is the topic of the material which follows. or 1/a. In addition to the previous properties the order relation must be compatible with the operations. This means Dedekind Cuts (Schnitten) Dedekind's formulation is now called Dedekind cuts This notion of (R, S) as an ordered pair of complementry sets is redundant. The partition can be exactly identified by either R or S. Later only the lower set R will be used in the analysid but for now the notion of a partition will be used briefly. Change of Terminology Instead of a partitioning pair of sets (R, S) the presentation will be in terms of sets of rational numbers which are closed downward and which have no maximum elements. Such sets will ultimately be identified as real numbers, but temporarily it is appropriate to call them dedekind sets. Note that there are some simple lemmas that hold for dedekind sets. For example:

47. 2.15.1 Dedekind Cuts 2.15.1 Dedekind Cuts. A real number tex2html_wrap_inline33691 is represented by a cut tex2html_wrap_inline33693 , tex2html_wrap_inline33695 . http://www.dgp.toronto.edu/~mooncake/thesis/node61.html

Next: 2.15.2 Cauchy Sequences Up: 2.15 Real Representations Previous: 2.15 Real Representations 2.15.1 Dedekind Cuts A real number is represented by a cut . Every cut has the property that for all As presented, the cut represents . Disallowing this special cut gives a representation for all non-negative real numbers. In general, Most numbers have a representation that cannot be written out directly since the representation is an infinite set. Operations on reals are inherited from the corresponding operations on rationals. For example, a binary operation on two real numbers, represented by cuts X and Y , is given by: Difficulties are encountered when generalizing this to negative real numbers. If a cut is simply redefined to be a subset of , then the product of two cuts is not a cut if the multiplicands correspond to negative numbers. See [ ] for further details concerning this representation and associated methods. Next: 2.15.2 Cauchy Sequences Up: 2.15 Real Representations Previous: 2.15 Real Representations Jeff Tupper March 1996

48. Dedekind Cuts Many of the accounts of Dedekind cuts that I have seen, including the original one, wave their hands in the most blatant way, especially with regard to http://www.paultaylor.eu/ASD/dedras/classical

Dedekind cuts Paul Taylor Many of the accounts of Dedekind cuts that I have seen, including the original one, wave their hands in the most blatant way, especially with regard to multiplication. Nevertheless, several deep and powerful ideas have been incorporated into this theory in the course of 150 years. This posting is a summary of them, and was written in the hope that somebody might tell me who first discovered each of them. However, whilst I am always glad to receive mathematical, historical and philosophical comments from my colleagues, I would respectfully ask, on this occasion in particular, that they first check them against the actual papers that they cite, and against the bibliography of my paper with Andrej Bauer The Dedekind Reals in Abstract Stone Duality www.PaulTaylor.EU/ASD/dedras This paper originally appeared in the proceedings of Computability and Complexity in Analysis , held in Kyoto in August 2005. It has now been accepted for a journal, although we are still tightening the proofs, narrative and bibliography. Abstract Stone Duality categories mailing list on 18 August 2007.

49. Dedekind-completion - Definition Another generalization surreal numbers. A construction similar to Dedekind cuts is used for the construction of surreal numbers. See also. Cauchy sequence http://www.wordiq.com/definition/Dedekind-completion

Dedekind-completion - Definition In mathematics , a Dedekind cut in a totally ordered set S is a partition of it, ( A B ), such that A is closed downwards (meaning that for any element x in S , if a is in A and x a , then x is in A as well) and B is closed upwards. 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 Contents showTocToggle("show","hide") 1 Handling Dedekind cuts 2 Ordering Dedekind cuts 3 The cut construction of the real numbers 4 Additional structure on the cuts ... 7 See also 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 either a pair with A the interval a ), in which case B must be [ a A the interval a ], in which case B must be ( a Ordering Dedekind cuts If a is a member of S then the set a ); by identifying

50. Dedekind Cut 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 http://translate.roseville.ca.us/ma/enwiki/en/Dedekind_cut

Mul tili ngual Ar chi ve Po wer ed by Wor ldLi ngo Dedekind cut Home Multilingual Archive Index Ch oo se your la ngua ge: English Italiano Deutsch Nederlands ... Svenska var addthis_pub="anacolta"; Dedekind cut 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 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.

51. Dedekind Cuts Synonyms, Dedekind Cuts Antonyms | Thesaurus.com No results found for Dedekind cuts Please try spelling the word differently, searching another resource, or typing a new word. Search another word or see Dedekind cuts on http://thesaurus.com/browse/Dedekind cuts

52. The Real Numbers Dedekind Cuts DEFINITION. A (Dedekind) Cut Is File Format PDF/Adobe Acrobat Quick View http://www.math.sunysb.edu/~mde/142S_03/gw4.pdf

53. Re: "Dedekind" Cuts For Representing Substances To Markus Pilzcker mp.lists@FREE.FR , John F. Sowa sowa@BESTWEB.NET Subject Re Dedekind Cuts for Representing Substances; From Jay Halcomb jhalcomb8 http://suo.ieee.org/email/msg12982.html

Thread Links Date Links Thread Prev Thread Next Thread Index Date Prev ... Date Index Re: "Dedekind" Cuts for Representing Substances To : "Markus Pilzcker" < mp.lists@FREE.FR >, "John F. Sowa" < sowa@BESTWEB.NET Subject : Re: "Dedekind" Cuts for Representing Substances From : "Jay Halcomb" < jhalcomb8@comcast.net Date : Tue, 8 Mar 2005 13:09:20 -0800 Cc cg@CS.UAH.EDU john.a.thompson@BOEING.COM jcabral@CYC.COM >, "'SUO WG'" < standard-upper-ontology@listserv.ieee.org References 16941.63013.842494.519603@calefacens.low-entropy.hn.org Sender owner-standard-upper-ontology@ieee.org References Re: "Dedekind" Cuts for Representing Substances From: Prev by Date: Re: How about Self-Inconsistency Next by Date: Re: "Dedekind" Cuts for Representing Substances Prev by thread: Re: "Dedekind" Cuts for Representing Substances Next by thread: Feature Engineeriing! Index(es): Date Thread

54. Dedekind Cut Articles And Information A Dedekind cut in information totally ordered set S is information partition consistent with it, (A, B), such that A example closed downwards (meaning that http://neohumanism.org/d/de/dedekind_cut.html

Home Archaeology Astronomy Biology ... Contact Us Advert: Buy Gold Current Article Dedekind cut A Dedekind cut in a totally ordered set S is a partition of it, ( A B ), such that A is closed downwards (meaning that whenever a is in A and x a , then x is in A as well), B is closed upwards. If a is a member of S x in S x x in S x a a , so that the linearly ordered set S may be regarded as embedded within the set of all Dedekind cuts of S . If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts is strictly bigger than S A B less than C D A is a proper subset of C , or, equivalently D is a proper subset of B . In this way, the set of all Dedkind cuts is itself a linearly ordered set, and, moreover, it does have 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 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 A a in Q a < 2 or a B b in Q b b Generalization: Dedekind completions in posets More generally, in a

55. Dedekind Cuts Of Partial Orderings Dedekind cuts of partial orderings Dedekind cuts are a clever trick for defining the reals given the rationals. Such a cut considers a set C of rationals such that if x is in C and http://www.cap-lore.com/MathPhys/Cuts.html

Dedekind cuts of partial orderings We may take any partial ordering and consider such cuts. The result is always a lattice. There is another different unique (within isomorphism) lattice associated with any partial ordering. There is for any partial ordering some unique smallest lattice in which it is embedded. The lattice may contain new elements but the new ordering, restricted to the old PO will contain no new orderings. This construction is also found in security considerations. The orange book provides a theory of security classifications that implicitly defines a lattice. In a particular computer system it is likely that some of the lattice values will be unused. This may cause some confusion. It should not any more than noting that the boolean or command of the CPU need not in an application produce all possible values in order to make the set of all possible values a useful concept with which to reason. It is the same with the lattice of security classifications. When the partial ordering is finite and total the cuts add nothing of interest.

56. PDF - Project Gutenberg S Essays On The Theory Of Numbers, By File Format PDF/Adobe Acrobat Quick View http://www.gutenberg.org/files/21016/21016-pdf.pdf

57. Fuzzy Real Numbers As Dedekind Cuts With Respect To A Multiple-valued Logic by U Höhle 1987 - Cited by 40 - Related articles http://portal.acm.org/citation.cfm?id=36913

58. Re: Dedekind Cuts, Fundamental Sequences: Why? sequences or Dedekind cuts are useful in defining completeness. The constructions mentioned do not succeed in producing a complete view of what sort of reasoning about http://sci.tech-archive.net/Archive/sci.math/2007-06/msg01066.html

Re: Dedekind Cuts, Fundamental Sequences: why? From Date : Wed, 06 Jun 2007 00:08:20 -0400 Dave Seaman wrote: On Tue, 05 Jun 2007 09:57:58 -0400, Hatto von Aquitanien wrote: Aatu Koskensilta wrote: On 2007-06-05, in sci.math, Hatto von Aquitanien wrote: That tells me how they are useful, not why. But what I really meant by my previous comment is that he was begging the question as to why fundamental sequences or Dedekind cuts are useful in defining completeness. And I still do not know if there was a clearly defined objective which can subsequently be verified as being accomplished. They provide a concrete way of specifying a complete ordered field with a countable dense subset, reducing talk about reals to talk about naturals and sets of naturals. That is where Weyl rejected the development. He basically argues that using sets of rational numbers as a means of defining the real numbers is not supported by any form of proof that it is even meaningful. It is also fairly clear that he doubts such a proof could be given. I am certainly no expert on Weyl, but your description implies that he

59. Dedekind Cut : Definitions + Information Dedekind cut Definitions Information images, discuss, define, news. http://www.blurbwire.com/topics/Dedekind_cut

60. Is 0.999... = 1? Dedekind Cuts Dedekind cuts are usually defined in the ring of rational numbers, but if we are interested in decimal numbers, we will want to work with a different ring. http://math.fau.edu/richman/HTML/999.htm

Is 0.999... = 1? Fred Richman Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives. F. Faltin, N. Metropolis, B. Ross and G.-C. Rota, The real numbers as a wreath product A serious question Arguing whether 0.999... is equal to 1 is a popular sport on the newsgroup sci.math-a thread that will not die. It seems to me that people are often too quick to dismiss the idea that these two numbers might be different. The issues here are closely related to Zeno's paradox, and to the notion of potential infinity versus actual infinity. Also at stake is the sanctity of the current party line regarding the nature of real numbers. Many believers in the equality think that we may no longer discuss how best to capture the intuitive notion of a real number by formal properties. They dismiss any idea at variance with the currently fashionable views. They claim that skeptics who question whether the real numbers form a complete ordered field are simply ignorant of what the real numbers are, or are talking about a different number system. One argument for the equality goes like this. Set

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