41. ACM Guide Reviewer Index 2, Probabilistic Turing machines and recursively enumerable dedekind cuts M Chrobak, B S. Chlebus Information Processing Letters November 1984 Volume 19 Issue http://portal.acm.org/reviewers.cfm?part=author&row=W&idx=1&idx2=20&query=P26782

42. What Are The 'real Numbers,' Really? In particular, the decimal expansions, the dedekind cuts, and the equivalence classesof Cauchy sequences, though they appear to be entirely different, all http://www.cartage.org.lb/en/themes/Sciences/Mathematics/calculus/realnumbers/co

Themes Science Mathematics Calculus ... Real Number We have used the real numbers in some of our preceding discussions. For instance, the complex numbers are ordered pairs of real numbers, and our example of infinitesimals involved rational functions with real coefficients. In effect, we "borrowed" the real numbers we used the reals in examples, even though we hadn't formally defined them yet; we just relied on the informal and intuitive understanding that students already have, based on the geometric line. Trust me, there is no circular reasoning here I won't use the "borrowed" concepts when I finally get around to defining the real numbers. You'll see that if you actually work through all the details. (I'm not claiming that this web page is more than an outline.) The definition of the reals depends on two more theorems, both of which are difficult to prove. Theorem 1. There exists a Dedekind-complete ordered field. The literature contains many different proofs of this theorem. I think three are simple enough to deserve mention here: Proof using decimal expansions.

43. What Are The 'real Numbers,' Really? Wouldn't it be easier to simply define the real numbers to be the dedekind cuts,or define the real numbers to be the decimal expansions, or something like that http://www.cartage.org.lb/en/themes/Sciences/Mathematics/calculus/realnumbers/fi

Themes Science Mathematics Calculus ... Real Number Definition. The real number system is that unique algebraic structure represented by all Dedekind-complete ordered field. You might wonder why mathematicians want to use such a complicated definition. Wouldn't it be easier to simply define the real numbers to be the Dedekind cuts, or define the real numbers to be the decimal expansions, or something like that? That is the approach taken in some elementary textbooks, but ultimately it is less productive. When we actually use the real number system in proofs, the properties that we need are not specifically the properties of (for instance) Dedekind cuts or of decimal expansions. Rather, the properties we need are the axioms of a Dedekind complete ordered field. It is much simpler to think in terms of those axioms. To think of "numbers" as being cuts or expansions would just encumber us with extra baggage. The cuts or expansions are models they are useful for the job proving Theorem 1, but they are useful for little else. Once they've done that job, we can discard and forget them. If you wish, you can now think of the points on a line as

44. Program Files\Netscape\Communicator\Program\dedexxx One remarkable piece of work was his redefinition of irrational numbers in termsof dedekind cuts which, as we mentioned above, first came to him as early as http://www.andrews.edu/~calkins/math/biograph/biodedek.htm

Julius Wihelm Richard Dedekind Birth: October 6, 1831 in Braunschweig, duchy of Braunschweig (Germany) Death: February 12, 1916 in Braunschweig, duchy of Braunschweig (Germany) Early Life William Dedekind was born on October 6, 1831. He was born in what is now Germany. He was the last of four children to be born to his parents. He was not the first in his family tree to be a professor,his father and grandmother were both professors. His entire life was surrounded around research and theories. He attended many schools in his early life. One being Martino-Catherineum which gave him a good background in the sciences. And his basic mathematical background came from Collegium Carolinumin in 1848. He studied integral calculus, analytic geometry and the foundation of anlysis. His math and science background prepared him for the University of Guttingen in 1850. Schooling and mid-life Though Guttigen was a college that many famous mathematiciansattended at the time Dedekind did it was a dismal place to study math. Even though Gauss taught courses his were not at the level of the sciences. The math Dedekind did

45. 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.

46. Surreal Numbers dedekind cuts are fairly concrete, or at least understandable without too much difficulty.So where can we find these dreamlike, fantastical, surreal numbers? http://www.usna.edu/MathDept/wdj/surreal1.htm

Surreal Numbers [Introduction] [Definitions] [Games] [Examples with Numbers] ... [Links] I: Introduction I once heard this joke. "Q: How many surrealists does it take to screw in a light bulb? A: A fish." Subtract one from infinity. Now take the square root of that. Did you get 42? Surreal. It seems unlikely to use a word like this to describe a set of numbers. How unreal or hallucinatory have numbers ever seemed to you? Cantor built the infinite ordinal numbers and Dedekind constructed the real numbers from the rationals using "cuts". ( Dedekind's construction has similarities to the construction of surreal numbers.) The infinity of ordinal numbers is not too tricky for most people to understand. Dedekind cuts are fairly concrete, or at least understandable without too much difficulty. So where can we find these dreamlike, fantastical, surreal numbers? One might wonder how you can do calculations with infinity. Enter surreal numbers. John Conway created (or you might say discovered) this new construction of numbers, which include the real numbers plus much more. The purpose of this paper is to give the reader a general understanding of what surreal numbers are and their significance. Starting at the very beginning, a definition would probably be most appropriate. Unfortunately we can't call on Webster, so Donald Knuth will have to do the job instead.

47. Real Numbers The Dedekind Cut. dedekind cuts define all real numbers. ie both rational andirrational numbers. Definition in terms of positive and negative rationals. http://room.anu.edu.au/DoM/firstyear/poetry/RealNumbers.html

Rational Numbers Rational numbers introduced via ratios. Ratio of 2 lengths; eg 5 to 4. Then one can find a length that can serve as a unit so that there are 4 and 5 units, respectively. To begin with it is assumed that any 2 lengths can be compared. From ratios one obtains the concept of rational number by adopting a criterion of identity The ratio m to n identifies the same number as the ratio m to n iff m x n = m x n The arithmetical operations can be defined similarly in terms of arithmetical operations on natural numbers. Representation of the rationals on a line (with a beginning, the origin , but no end). Needed is the specification of a unit, beginning at the origin, by identifying on the line the rational number 1. Incommensurability There are pairs of line segments that cannot be compared as to their length. No length can be found that could serve as a unit for both. Example: The diagonal of a square But lengths can be compared with one another. It is our concepts that are inadequate. Hence the introduction of irrational numbers. More points are needed on the line.

48. Archimedes Plutonium dedekind cuts on the rationals produce all the reals, including the rationals and integers. Unioning them is not necessary. Examples. http://www.newphys.se/elektromagnum/physics/LudwigPlutonium/File009.html

Lowest things of mathematics; foundation of math by Archimedes Plutonium this is a return to website location http://www.newphys.se/elektromagnum/physics/LudwigPlutonium/ - From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) Newsgroups: talk.philosophy.misc,sci.math Subject: Lowest things of mathematics; what math starts with Date: 6 Feb 1999 07:40:47 GMT Organization: lowest things of math Lines: 33 Distribution: world Message-ID: On 1998/12/13 I wrote in References: In article Estraven References: In article Estraven writes: > I do not understand the content of your argument. Please answer a few > questions to clarify things for me: Sorry, I do not have the time. But it is quite simple to confirm that I am correct when I say there cannot be a construction of Reals from lower entities. Simply look at Euclidean Geometry. IF the Reals can be constructed from lower parts, then Euclidean Geometry can be constructed from lower geometries. From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) Newsgroups: sci.math,talk.philosophy.misc Subject: Re: Lowest things of mathematics; what math starts with Date: 8 Feb 1999 07:43:33 GMT Organization: Pu Lines: 25 Distribution: world Message-ID: References: In article roconnor@undergrad.math.uwaterloo.ca (Russell Steven Shawn O'Connor) writes: > This is not true. The construction of the rationals from the integers > produces all rationals including the integers. Rationals are equivalence > classes of integers [

49. 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/

50. Tatra Mountains In this paper, we present a Boolean, pointfree characterization of fuzzy observables,using Boolean-valued dedekind cuts and the theory of Boolean powers. http://tatra.mat.savba.sk/paper.php?id_paper=59

51. Introduction: A Brief History Of Axiomatic Set Theory convergent sequences of rationals. Shortly after Dedekind published hisfamous characterization of the real numbers as dedekind cuts . http://www.jboden.demon.co.uk/SetTheory/history.html

Introduction: A brief History of Axiomatic Set Theory Defining the Real Numbers By the middle of the 19 th Century it had become apparent that mathematics lacked rigour. As a result a trend towards Analysis began, because it was recognized that mathematics were not built on solid foundations. In particular, infinity, partly in the guise of infinitesimals (in calculus) and also as in infinite sequences had allowed mathematicians to define infinite series and so onto functions without necessarily paying proper attention to the prerequisites for convergence. Indeed, although it was simple to view rationals as being simply fractions or just a pair of integers it was not properly appreciated that the situation was not so simple for real numbers. Since mathematicians were defining infinite sequences and series of rational terms which had an irrational limit, some means needed to be found to properly define the real numbers. Georg Cantor - in brief Between 1862 and 1867 Cantor completed his dissertation on number theory. During this period he attended lectures given by Weierstrass Kummer and Kronecker . In 1870 he was introduced by Heine to the difficult problem of proving that if two functions have the same Fourier series then they are identical, with the possible exception of a finite number of points. Despite the failure of mathematicians such as

52. Abstract 18/12/98 K. Grue Retour à la page du séminaire 18/12/98 Klaus Grue dedekind cuts as ameans for constructing kappaScott domains. abstract In Berline Grue 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.

53. Mathematics List Part 3 Complete text. ( Said to be the original paper where Dedekind defined irrationalnumbers in terms of dedekind cuts. ) MATH13261 $150.00. Dedekind, Richard. http://www.significantbooks.com/mthl3.htm

Go back to Specialized Book Lists Index To inquire about any of these items contact us by e-mail at: inquiry@significantbooks.com Start of Math page 3. Go To Math page Mathematics List Page 3. Authors, Ch. thru Et. Cline, Randall E. Elements of the Theory of Generalized Inverses for Matrices. 86 pp. Math Dept. Univ. Tennessee. (1979) Softcover. Very good condition. (PART OF: Modules and Monographs in Undergraduate Mathematics and its Applications Project. ) MATH13140 $10.00 Coble, Arthur B. Algebraic Geometry and Theta Functions. 282 pp. American Mathematical Society 1929 (Hardback) Good condition, some edge wear. ( American Mathematical Society Colloquium Publications Vol. X. ) MATH13403 $35.00 Coburn, Nathaniel. Vector and Tensor Analysis. 341 pg. Macmillan (1955) GD MATH10167 $25.00 Coddington,. Earl A. Cogan, E. J., R. L. Davis, J. G. Kemeny, et. al. Modern Mathematical Methods and Models. 2 volumes. 328, 313 pp. Mathematical Association of America. (1958, 1959). Hardcover. Good condition. (Vol. 1; Multicomponent methods. Vol. 2; Mathematical models) (A Book of Experimental Text Materials. Produced under the direction of the Committee on the Undergraduate Program.) MATH11994 $30.00 Cogan, E. J., R. L. Davis, J. G. Kemeny, et. al.

54. Re: Symbolic Math_Quantum Mechanics_Continued Dedekind published his definition of the real numbers by dedekind cuts also in1872 and in this paper Dedekind refers to Cantor's 1872 paper which Cantor had http://www.mdlug.org/archives/mdlug-pro/msg00036.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Symbolic Math_Quantum Mechanics_Continued To Subject : Re: Symbolic Math_Quantum Mechanics_Continued From : "Tom Canu" < Date : Sat, 24 Nov 2001 16:34:05 +0000 Bcc Cc lapiet@xxxxxxxx Delivered-to : mailing list mdlug-pro@radiusnet.net Mailing-list : contact mdlug-pro-help@radiusnet.net; run by ezmlm http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Cantor.html http://www.shu.edu/projects/reals http://explorer.msn.com/intl.asp Prev by Date: OT: Hunters on Lists Next by Date: Re: OT: Hunters on Lists Previous by thread: Symbolic Math_Quantum Mechanics_Continued Next by thread: Re: Symbolic Math_Quantum Mechanics_Continued Index(es): Date Thread

55. Study Group illusion, fourth dimensional space, the laws of probability and other advanced conceptsin mathematics (eg space/time continuums, dedekind cuts, Poincaré cuts http://www.artscienceresearchlab.org/events/lectures.htm

"Riding Between the Lions" Art Science Research Laboratory Weekly Seminar Thursdays, 7-8:30 P.M. (soon via streaming video) "Reading between the lines Riding between the lions" -Marcel Duchamp (from Marcel Duchamp, Notes , Paul Matisse, ed. Paris: Centre Georges Pompidou, 1980.) The Art Science Research Laboratory announces its study and discussion group "Riding Between the Lions," a weekly gathering dedicated to the study of science and art in the late nineteenth and early twentieth centuries. Conducted by Stephen Jay Gould and Rhonda Roland Shearer, the education-oriented seminar invites participation from graduate students, professors and independent scholars interested in the disciplines of art, art history, computer technology, mathematics, and science. The seminar aims to promote a general understanding of science and develop key tools that will enhance research in art/science education. "Riding Between the Lions" is first exploring the role of science in the work of Marcel Duchamp. Scientific concepts of the late nineteenth and early twentieth centuries greatly informed Duchamp's art. Throughout his career, several scientific realmsmathematics, optics, and perceptual theoryplayed an essential role in Duchamp's creative process. His interest and invention can be found in his particular use of shadow projections, optical illusion, fourth dimensional space, the laws of probability and other advanced concepts in mathematics (e.g. space/time continuums, Dedekind cuts, Poincaré cuts). His expertise in mathematics and science is unique not only in early twentieth-century art but in all Western art history.

56. Browse Probabilistic Turing machines and recursively enumerable dedekind cuts, Article,01/01/87. Information transfer under different sets of protocols, Article, 06/01/86. http://www.reviews.com/Browse/Browse_reviewer2.cfm?reviewer_id=107831

57. 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

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. Dedekind's Real Numbers set of rational numbers''; Maddy1992, p. 81 ``by identifying real numbers withcertain sets (called `dedekindcuts'), dedekind '' misinterpretation. http://www.phil.cmu.edu/dschlimm/texts/reals.html

Dedekind's Real Numbers Dirk Schlimm Department of Philosophy Carnegie Mellon University Pittsburgh, PA 15213 email: dschlimm@andrew.cmu.edu February 22, 1999 Abstract Richard Dedekind's characterization of the real numbers as the system of cuts of rational numbers is by now the standard in almost every mathematical book on analysis or number theory. In the philosophy of mathematics Dedekind is given credit for this achievement, but his more general views are discussed very rarely and only superficially. For example, Leo Corry, who dedicates a whole chapter of his Modern Algebra and the Rise of Mathematical Structures (1996) writes: ``Dedekind defined the system of the real numbers as the collection of all cuts of rationals'' ([ ], p. 73). In this paper I will present Dedekind's own views of his ``definition'' and ``creation'' of the real numbers, and elucidate what he meant by saying that the real numbers ``correspond'' to the cuts. The upshot of this discussion will be that Corry's statement will be revealed as an obvious, but not uncommon (cf. [ ], p. 55: ``And every such cut, that corresponds to no rational number, defines an irrational number'' (my translation); [

60. Dedekind, Richard study of CONTINUITY and definition of the real numbers in terms of dedekind "cuts", the Category Science Math History People...... 12, 1916, was a German mathematician known for his study of CONTINUITY and definitionof the real numbers in terms of dedekind cuts ; his analysis of the http://euler.ciens.ucv.ve/English/mathematics/dedekind.html

Dedekind, Richard Richard Dedekind, b. Oct. 6, 1831, d. Feb. 12, 1916, was a German mathematician known for his study of CONTINUITY and definition of the real numbers in terms of Dedekind "cuts"; his analysis of the nature of number and mathematical induction, including the definition of finite and infinite sets; and his influential work in NUMBER THEORY, particularly in algebraic number fields. Among his most notable contributions to mathematics were his editions of the collected works of Peter DIRICHLET, Carl GAUSS, and Georg Riemann. Dedekind's study of Dirichlet's work led to his own study of algebraic number fields, as well as his introduction of ideals. He developed this concept into a theory of ideals that is of fundamental importance in modern algebra. Dedekind also introduced such fundamental concepts as RINGS. Author: J. W. Dauben Homepage e-mail: webmaster@euler.ciens.ucv.ve © 2000 Mathematics School Science Faculty, Central University of Venezuela

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