41. Abstract 18/12/98 K. Grue 18/12/98 Klaus Grue dedekind cuts as a means for constructing kappaScott domains. Thetalk focuses on the Dedekind cut construction and its properties. http://www.logique.jussieu.fr/semlam/98_99/981218grue.html

18/12/98: Klaus Grue Dedekind cuts as a means for constructing kappa-Scott domains abstract The talk focuses on the Dedekind cut construction and its properties.

42. Dedekind Cut The Dedekind cut is named after Richard Dedekind, who invented this constructionin order to represent the real numbers as dedekind cuts of the rational numbers http://www.fastload.org/de/Dedekind_cut.html

Dedekind cut Home Up About 'Dedekind cut' Advertisement A Dedekind cut in an ordered field 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 and A has no maximum. 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 B b in Q b this is a stub article See also: Cauchy sequence This article is licensed under the GNU Free Documentation License You may copy and modify it as long as the entire work (including additions) remains under this license. You must provide a link to http://www.gnu.org/copyleft/fdl.html To view or edit this article at Wikipedia, follow this link

43. I Didn T Have Time To Write A Short Letter, So I Wrote A Long One as { q in Q q 0 or q^2 2 }. For dedekind cuts A and B, we have multiplicationdefined as AB = { rs r is in A; r 0; s is in B; and s 0 }. Take A = B http://math.berkeley.edu/~laurent/104/remarks-3

44. Some Number Theory Now we define objects (called dedekind cuts) that consist of two sets of integers(L,U). Here every element of the set of positive rationals is either element http://homepages.cwi.nl/~dik/english/mathematics/numa.html

This (and subsequent) pages will not only be about pure number theory, but also some additional mathematics from other fields, but I will group it together under number theory. If you are not confident with the concepts of ring and field , you might first want to look at the page describing these concepts . Also you might to want to look at the page describing equivalence and ordering relations before you continue. I will first talk about some classes of numbers and how they are constructed. The sets are conventionally noted by some script or other form of a single letter, which I will give in bold below. N The natural numbers. Just the counting numbers as we know them, starting at either or 1 and continuing through 2, 3, etc. (Note, some mathematicians include in this set, some do not. I will use here the anglosaxon method to include 0.) (Second note: there are ways to properly define this set using the Peano axioms. I will not follow this and assume the set as given.) Z The integers. The set above with the usual operations of addition and multiplication does not form a ring. What is lacking is the additive inverse. We can augment that set by the objects (-0), (-1), (-2), ..., and use the ring axioms to get some properties (e.g. (-a).(-b) = ab.) The result is a commutative ring. It is indeed more, it is an integral domain (there are no zero divisors and there is a multiplicative unit). The letter Z probably comes from the German Zahlen , just meaning numbers. Note that when saying a number is positive, anglosaxon mathematics implies that the number is 1 or larger, French mathematics uses the term also for 0. The anglosaxon term

45. Richard Dedekind 1872, published paper on dedekind cuts to define real numbers. 1874, metCantor. 1879, published paper on purely arithmetic definition of continuity. http://dbeveridge.web.wesleyan.edu/wescourses/2001f/chem160/01/Who's Who/richard

Home Science Humanities Cantor ... Mendel Richard Dedekind Biography Photo Gallery Links to Outside Sources German mathematician who developed a major redefinition of irrational numbers in terms of arithmetic concepts. Although not fully recognized in his lifetime, his treatment of the ideas of the infinite and of what constitutes a real number continues to influence modern mathematics. Biography born October 6 entered University of Gottingen with solid math background from Collegium Carolinum in Brunswick received doctorate, last pupil of Gauss awarded habilitation degrees, began teaching at Gottingen began friendship with Dirichlet appointed to Polytechnikum in Zurich and began teaching appointed to Brunswick Polytechnikum (Callegium Carolinum upgraded), elected to the Gottingen Academy supplemented Dirichlet's lectures and introduced notion of an "ideal," a term he coined published paper on "Dedekind cuts" to define real numbers met Cantor published paper on purely arithmetic definition of continuity elected to Berlin Academy published joint paper with Heinrich Weber which applies his theory of ideals to the theory of Reimann Surfaces retired from Brunswick Polytechnikum elected to Academy of Rome, the Leopoldino-Carolina Naturae Curiosorum Academia, and the Academie des Sciences in Paris

46. Weierstrass, Dedekind And Cantor (d) dedekind cuts. Dedekind went on to call such cuts irrational numbers,and the set of all cuts he called the real numbers.. (e) Completeness. http://www.maths.uwa.edu.au/~schultz/3M3/L28Weierstr,Dede,Cantor.html

Weierstrass, Dedekind and Cantor The function concept The precursors of the notion of function in mathematics are the trigonometric tables of Ptolemy's Almagest (about 150). The Babylonians had produced tables of the positions of heavenly bodies throughout the year, but Ptolemy also showed how to interpolate, and thus treated these motions as continuous functions of time. We have seen how Oresme, and the other mediaeval scholastics defined and studied the intensity of magnitudes and even drew graphs. However, the originators of Calculus, Galileo, Fermat, Descartes, Barrow, Leibniz and Newton thought in terms of curves rather than functions; in other words they considered those functions which were defined in terms of known functions such as polynomials, trigonometric and logarithmic functions, conic sections, paths of rolling balls etc. Probably Euler was the first to recognize that a function was something that needed a definition. He said: a function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities. By this, he probably had in mind to include power series and indefinite integrals among the objects which should be considered as functions.

47. ExpectNothing! - April Fools Is For Fags! 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 http://www.expectnothing.com/?page=story&post=4082

48. Xml Version= 1.0 Encoding= Utf-8 ? !DOCTYPE Omdoc SYSTEM assertion id= gwff.THM568 type= conjecture CMP The idea of this theorem isthat if a collection of real numbers (given by dedekind cuts) is bounded from http://www.mathweb.org/~mbase/content/tps/tps.REALS-THMS.omdoc

The TPS Project kohlhase application/omdoc+xml Dataset The TPS library: http://gtps.math.cmu.edu/tps.html The formalization can be freely distributed, maintaining reference to the TPS source. Theorem suggested by Figure 3 in Parnas' HUG93 paper on theorems to prove TRANSITIVE-LAW AND IRREFLEXIVE-LAW IMPLIES FORALL x(R).~.less(ORR) x 0(R) AND x = OR less x AND less x OR less x AND x = Part of theorem suggested by Figure 3 in Parnas' HUG93 paper on theorems to prove IRREFLEXIVE-LAW IMPLIES FORALL x(R).~.less(ORR) x 0(R) AND x = Part of theorem suggested by Figure 3 in Parnas' HUG93 paper on theorems to prove TRANSITIVE-LAW AND IRREFLEXIVE-LAW IMPLIES FORALL x(R).~.less(ORR) x 0(R) AND less x The idea of this theorem is that if a collection of real numbers (given by Dedekind Cuts) is bounded from below, then the inf of the collection is a real number. HOWEVER(!!!): I accidentally defined DEDEKIND-CUT wrong. See DDEDEKIND-CUT for the corrected definition. SEE ALSO: THM569, THM570, THM571, THM572, DEDEKIND-CUT, DDEDEKIND-CUT. S(O(OB)) SUBSET [DEDEKIND-CUT(O(OB)(OBB)) LESSEQ] AND EXISTS x(OB) S x AND EXISTS c(OB) [[DEDEKIND-CUT LESSEQ] c AND FORALL x.S x IMPLIES c SUBSET x] IMPLIES [DEDEKIND-CUT LESSEQ].SETINTERSECT S Similar to THM568, but also proves SETINTERSECT S is glb of S (wrt SUBSET). HOWEVER(!!!): I accidentally defined DEDEKIND-CUT wrong. See DDEDEKIND-CUT and THM571 for the corrected thm. S(O(OB)) SUBSET [DEDEKIND-CUT(O(OB)(OBB)) LESSEQ] AND EXISTS x(OB) S x AND EXISTS c(OB) [[DEDEKIND-CUT LESSEQ] c AND FORALL x.S x IMPLIES c SUBSET x] IMPLIES GLB(O(OB)(O(OB))(O(OB)(OB))(O(OB))) [DEDEKIND-CUT LESSEQ] [LAMBDA x LAMBDA y(OB).x SUBSET y] S.SETINTERSECT S

49. Math History - Age Of Liberalism 1858, Dedekind discovers a rigorous method to define irrational numberswith dedekind cuts . The idea comes to him while he is thinking http://lahabra.seniorhigh.net/pages/teachers/pages/math/timeline/mLiberalism.htm

Math History Timeline Age of Liberalism 1848-1914 A.D. Math History Prehistory and Ancient Times Middle Ages Renaissance Reformation ... 20th Century ... non-Math History Prehistory and Ancient Times Middle Ages Renaissance Reformation ... External Resources Chebyshev publishes On Primary Numbers In his paper On a New Class of Theorems Sylvester first uses the word "matrix". Bolzano's book Paradoxien des Undendlichen (Paradoxes of the Infinite) is published three years after his death. It introduces his ideas about infinite sets. Liouville publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville numbers". In particular he gave the example 0.1100010000000000000000010000... where there is a 1 in place n! and elsewhere. Riemann's doctoral thesis contains ideas of exceptional importance, for example "Riemann surfaces" and their properties. Francis Guthrie poses the Four Colour Conjecture to De Morgan.

50. D Alembert Jean Le Rond (1717-1783) France Dedekind s construction of the real numbers using `dedekind cuts was part of theeffort of Dedekind, Cantor, and Weierstrass, and others to bring a rigor to http://www.mlahanas.de/Stamps/Data/Mathematician/D.htm

D D'Alembert Jean Le Rond (1717-1783) France Biography Davidoglou Anton (1876-1958) Romania Romanian mathematician, author, educator. He studied partial differential equations. Romania 2613 Dedekind Julius Wilhelm Richard Biography Was sind und was sollen die Zahlen? (What are the numbers and what do they mean?) . These contributions, called part of the arithmetization of analysis, illustrate Dedekind's arithmetic and algebraic viewpoint. In algebraic number theory Dedekind introduced his theory of ideals to restore unique factorization; today integral domains in which every ideal is a unique product of prime ideals are called Dedekind domains. The GDR stamp presents an ideal written as a product of prime ideals. Democritus of Abdera Born: about 460 BC in Abdera, Thrace, Greece Died: about 370 BC Biography Descartes Rene Biography

51. Archimedes Plutonium Autobiography Axiom of Choice Reals can be arranged (ordered) so that every subset underthis same ordering has a first element is equivalent to dedekind cuts I am http://www.archimedesplutonium.com/File1995_07.html

AUTOBIO: 1995 Jul by Archimedes Plutonium this is a return to my website location http://www.archimedesplutonium.com/

52. Forschungsbericht - Prof. Dr. Manfred Knebusch Translate this page in polynomial beschränkten, teilweise in allen o-minimalen Strukturen erreicht(siehe, M. Tressl, “The elementary theory of dedekind cuts in polynomially http://www.uni-regensburg.de/Universitaet/Forschungsbericht/aktuell/nat1/prof6.h

Titelseite Inhaltsverzeichnis Weiter Prof. Dr. Manfred Knebusch Leitung: Prof. Dr. M. Knebusch Mitarbeiter/in: Dr. D. Zhang Manisbewertungen und Prüfererweiterungen Manisbewertungen spielen in der Geometrie des Raumes V(k) der rationalen Punkte einer algebraischen Varietät V über einem nicht algebraisch abgeschlossenen Körper k eine natürliche und wichtige Rolle, insbesondere in den Fällen k = R (reelle Geometrie) und k = Q p (p-adische Geometrie). Die kommutativen Ringerweiterungen A B, die sich mit Hilfe der Manisbewertungen voll beschreiben lassen und umgekehrt zum Verständnis der Manisbewertungen vorrangig Bedeutung haben, sind die sogenannten Prüfererweiterungen Diese wurden bislang in der Literatur nur selten untersucht, abgesehen von dem Spezialfall, das B der totale Quotientenring von A ist, über den es eine ausufernde Literatur gibt. Leider ist dieser Spezialfall für die Geometrie von V(k) fast immer irrelevant. Man braucht statt dessen Kenntnis über „Holomorphieringe“ verschiedener Art. Wir konnten im Berichtszeitraum Band 1 (Kapitel 1 – 3) einer Monographie zu diesem Thema abschließen und veröffentlichen, sodann Kapitel 4 und 5 in Angriff nehmen. Dabei standen Zusammenhänge zwischen Prüfererweiterungen und reeller Algebra im Zentrum des Interesses.

53. Mathematical Masterpieces: Teaching With Original Sources In an attempt to explain calculus better to his students, Dedekind constructed thereal numbers through what is now known as dedekind cuts, from which their http://math.nmsu.edu/~history/masterpieces/masterpieces.html

Next: References Mathematical Masterpieces: Teaching with Original Sources Mathematics, New Mexico State University, Las Cruces, NM 88003 Vita Mathematica: Historical Research and Integration with Teaching R. Calinger (ed.), MAA, Washington, 1996, pp. 257260] Our upper-level university honors course, entitled Great Theorems: The Art of Mathematics To achieve our aims we have selected mathematical masterpieces meeting the following criteria. First, sources must be original in the sense that new mathematics is captured in the words and notation of the inventor. Thus we assemble original works or English translations. When English translations are not available, we and our students read certain works in their original French, German, or Latin. In the case of ancient sources, we must often depend upon restored originals and probe the process of restoration. Texts selected also encompass a breadth of mathematical subjects from antiquity to the twentieth century, and include the work of men and women and of Western and non-Western mathematicians. Finally, our selection provides a broad view of mathematics building upon our students' background, and aims, in some cases, to reveal the development over time of strands of mathematical thought. At present the masterpieces are selected from the following. ARCHIMEDES: The Greek method of exhaustion for computing areas and volumes, pioneered by Eudoxus, reached its pinnacle in the work of Archimedes during the third century BC. A beautiful illustration of this method is Archimedes's determination of the area inside a spiral. [

54. Categories: Re: Real Interval Halving irrationals along with infinity (thinking of the real line projectively) are thenobtained as the empty rays, all of which make distinct dedekind cuts in the http://north.ecc.edu/alsani/ct99-00(8-12)/msg00054.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index categories: Re: Real interval halving To categories@mta.ca Subject : categories: Re: Real interval halving From pratt@cs.stanford.edu Date : Tue, 04 Jan 2000 11:44:07 -0800 In-reply-to Pine.LNX.4.10.10001021406570.4933-100000@triples.math.mcgill.ca Sender cat-dist@mta.ca References categories: Re: Real interval halving From: Prev by Date: categories: Re: Real interval halving Next by Date: categories: Re: Tate reals Prev by thread: categories: Re: Real interval halving Next by thread: categories: Re: banach operations Index(es): Date Thread

55. The Turing Closure Of An Archimedean Field - Boldi, Vigna Rose Saint John (1994) (Correct) Related documents from cocitation More All5 Recursion theory and dedekind cuts (context) - Soare - 1969 5 Degrees of http://citeseer.ist.psu.edu/boldi97turing.html

56. More Real Number Paradoxes — Mathematics — PAIAS The are many other paradoxes, some that are fundamentally variants of the VanishingRemainders Paradoxes, concern Cauchy sequences, dedekind cuts, and related http://paias.org/Mathematics/Paradoxes/morerealnumberparadoxes.htm

Palo Alto Institute for Advanced Study , worth looking at, but still in early stages SECTIONS More Real Number Theory Paradoxes Yet More Real Number Theory Paradoxes More Real Number Theory Paradoxes The Vanishing Remainder Paradoxes are not the only paradoxes in Real Number Theory. The are many other paradoxes, some that are fundamentally variants of the Vanishing Remainders Paradoxes, concern Cauchy sequences, Dedekind cuts, and related topics. They can be difficult to classify as to whether they are properly paradoxes in Real Number Theory, Set Theory, Measure Theory, or some other theory. There are further paradoxes that have been standardly overlooked. The “Countable Reals Paradox” and the “Non-Denumerable Rationals Paradox” (neither, again, widely recognized nor having official names) straddle set theory and real number theory. If we look at the “limit” of the closed interval [0,1/ n ] “as n goes to T (infinity)”, by standard theory we get [0], an interval/set of “measure zero”. Standard real number theory and standard measure theory have both ignored the “vanishing remainder”.

57. EEVL | Full Record There are many exercises and optional topics (isomorphism of complete ordered fields,construction of the real numbers through dedekind cuts, introduction to http://www.eevl.ac.uk/show_full.htm?rec=1002096535-22422

58. Forelesninger I MA 370 (Mat 301) Våren 1998 Forelesninger i MA 370 (Mat 301) våren 1998. Dato 24.02.98, Tema Dedekindog Cantor. § 16.2.1, F, dedekind cuts. Ø, Katz side 686 23, 24, 26. http://home.hia.no/~aasvaldl/kurs/ma370_3.html

Dato Tema : Dedekind og Cantor. F Dedekind cuts. Katz side 686: 23, 24, 26 F Katz side 686: 27, 28. F Uendelige mengder. Tellbarhet. Tellbarhet av algebraiske tall, men ikke av reelle tall. Eksistens av transendentale tall. Katz side 687: 30, 31*). F Mengde teori. F Dedekind og aksiomatisering av de naturlige tall. F *) Et tall x kalles algebraisk hvis det finnes et polynom, p, med heltallige koeffisienter slik at p(x)=0. Anta p(t)=a + a t + a t + ... + a n t n med alle a i heltall og a n ulik 0. n

59. HOL/Complex/README PReal The positive reals constructed using dedekind cuts; Rational Therational numbers constructed as equivalence classes of integers; http://www.cl.cam.ac.uk/Research/HVG/Isabelle/library/HOL/HOL-Complex/README.htm

Complex: The Complex Numbers This directory defines the type complex of the complex numbers, with numeric constants and some complex analysis. The development includes nonstandard analysis for the complex numbers. Note that the image HOL-Complex includes theories from the directories HOL/Real and HOL/Hyperreal . They define other types including real (the real numbers) and hypreal (the hyperreal or non-standard reals). CLim Limits, continuous functions, and derivatives for the complex numbers CSeries Finite summation and infinite series for the complex numbers CStar Star-transforms for the complex numbers, to form non-standard extensions of sets and functions Complex The complex numbers NSCA Nonstandard complex analysis NSComplex Ultrapower construction of the nonstandard complex numbers Real: Dedekind Cut Construction of the Real Line Lubs Definition of upper bounds, lubs and so on, to support completeness proofs. PReal The positive reals constructed using Dedekind cuts Rational The rational numbers constructed as equivalence classes of integers RComplete The reals are complete: they satisfy the supremum property. They also have the Archimedean property.

60. Richard Julius Wilhelm Dedekind notion of the dedekind cut is an early example of a formal procedure that can beused to partition a set (with the understanding that certain cuts , ie those http://www.engr.iupui.edu/~orr/webpages/cpt120/mathbios/rdedek.htm

RICHARD JULIUS WILHELM DEDEKIND Richard Dedekind was a German mathematician who was born in 1831 in Brunswick. His father was a professor of law. Dedekind studied at Gottingen where he later taught. He also taught at the Zurich polytechnic for a few years. He then became the professor of mathematics in the technical school of Brunswick where he taught for half a century. He was a bachelor and he lived with his unmarried sister, Julie, until her death in 1914. Dedekind made many original and important contributions to the theory of algebraic numbers. He died at the age of 85 in 1916. In 1872, he published a book, Continuity and Irrational Numbers , in which he attempted to remove all ambiguities and doubts as to how irrational numbers fitted into the domain of arithmetic. Some items to be considered in this work are as follows (all numbers are shown in base ten arithmetic): A rational number can be expressed in the form of a fraction a/b where a and b are integers. A number which cannot be expressed as a rational fraction is an irrational number. For example, . The class of real numbers is made up of rational and irrational numbers. A rational number can be expressed in decimal notation and where the decimal does not terminate (end in zeroes), it repeats itself periodically. For example, 10/13 =.769230.769230.769230 and 14/11 = 1.27.27.27. An irrational number when expressed as a decimal does not terminate or exhibit the periods. It is impossible to exactly express numbers such as

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