21. Dedekind Cuts - Pedagogical dedekind cuts. Pedagogical Reasoning. At the definitions. Here I willemphasize the reasoning behind how I present dedekind cuts. One http://www-math.bgsu.edu/~cbennet/math417/Portfolio/Picture 8/Pedreas.htm

Dedekind Cuts Pedagogical Reasoning At the end of the introduction, I give my reasons for choosing the Dedekind cut definition of the real line over the other two classical definitions. Here I will emphasize the reasoning behind how I present Dedekind cuts. One of the things that I have noticed in teaching students is that they sometimes disbelieve the statement that .999...=1, which is not particularly surprising when one considers that many studies on high school students show that this equivalence is very hard to get people to accept and understand. The other idea that I try to get across in the section on Dedekind cuts is the importance of approximating real numbers and how such approximations can change answers. In particular, I discuss the case of the faulty Pentium computer chip in the 1990s, and I also discuss how these things affect calculators (as you can see on the homework). At least in my classes, a lack of understanding of the difficulties that approximations cause is something I have seen consistently, although curiously, at Bowling Green, I find that students are too ready to believe that their calculators are correct, while at MSU, I think they are too quick to assume that round off error is a problem.

22. Dedekind's Cuts Dedekind's cuts. post a message on this topic post a message on a new topic 29 Sep1998 Dedekind's cuts, by Alan Hill 3 Oct 1998 dedekind cuts, by todd trimble http://mathforum.org/epigone/alt.math.undergrad/dunyobe

a topic from alt.math.undergrad Dedekind's cuts post a message on this topic post a message on a new topic 29 Sep 1998 Dedekind's cuts , by Alan Hill 3 Oct 1998 dedekind cuts , by todd trimble The Math Forum

23. Math Forum - Ask Dr. Math dedekind cuts. I hope that I've helped you understand a little about Dedekindcuts. Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/library/drmath/view/52511.html

Associated Topics Dr. Math Home Search Dr. Math Dedekind Cuts Date: 10/23/96 at 19:30:15 From: mat stern Subject: Dedekind cut I have already figured out Dedekind's theory of the rings and number notation. I still cannot figure out what his theory of the Dedekind cut is. Could you please send me a couple of sentences of what his cut is all aboutso that a 7th grader can understand? I have contacted several mathamatic instructors and they know but cannot tell me in language that I can understand. http://mathforum.org/dr.math/ http://mathforum.org/dr.math/ Associated Topics High School History/Biography High School Number Theory Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

24. Dedekind Cut -- From MathWorld member. Real numbers can be defined using either dedekind cuts or Cauchysequences. CantorDedekind Axiom, Cauchy Sequence. References. http://mathworld.wolfram.com/DedekindCut.html

Foundations of Mathematics Set Theory General Set Theory Dedekind Cut A set partition of the rational numbers into two nonempty subsets and such that all members of are less than those of and such that has no greatest member. Real numbers can be defined using either Dedekind cuts or Cauchy sequences Cantor-Dedekind Axiom Cauchy Sequence References Courant, R. and Robbins, H. "Alternative Methods of Defining Irrational Numbers. Dedekind Cuts." §2.2.6 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 71-72, 1996. Jeffreys, H. and Jeffreys, B. S. "Nests of Intervals: Dedekind Section." §1.031 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 6-8, 1988. Author: Eric W. Weisstein Wolfram Research, Inc.

25. Real Numbers And Order Completeness We will look briefly at one of them, the one identifying real numbers with Dedekindcuts Proposition 129 The sum of dedekind cuts is again a Dedekind cut. http://www.iwu.edu/~lstout/NewTheoremlist/node22.html

Next: Finite and Infinite Up: Numbers Previous: Rationals Real numbers and order completeness The real numbers correspond to all possible measurements of lengths of a line. They differ from the rationals in that the order is complete. The completeness of the order on the reals is used in proving such results in calculus as the Intermediate Value Theorem, the Theorem on the Existence of Maxima and Minima, Convergence of Monotone Bounded Sequences. There are at least two major constructions of the real numbers from the rationals. We will look briefly at one of them, the one identifying real numbers with Dedekind cuts: Definition 59 A Dedekind cut on the rationals consists of two subsets and of such that both and are nonempty if and only if there is a rational with is an open interval going to if and only if there is a rational with is an open interval going to If and then . (In particular note that For any there are rationals and with and are arbitrarily close together.) Example: Any rational number gives a Dedekind cut: given we let and . Such a rational cut will have . An irrational cut will have An example of an irrational cut is and It is reasonably easy to define addition of Dedekind cuts and an order relation: Definition 60 The sum of the Dedekind cuts is the cut with lower set and upper set given by Proposition 129 The sum of Dedekind cuts is again a Dedekind cut.

26. Dedekind Richard Dedekind's major contribution was a redefinition of irrationalnumbers in terms of dedekind cuts. He introduced the notion http://members.tripod.com/sfabel/mathematik/database/Dedekind.html

Julius Wihelm Richard Dedekind Born: 6 Oct 1831 in Braunschweig, duchy of Braunschweig (now Germany) Died: 12 Feb 1916 in Braunschweig, duchy of Braunschweig (now Germany) Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page Richard Dedekind 's major contribution was a redefinition of irrational numbers in terms of Dedekind cuts. He introduced the notion of an ideal which is fundamental to ring theory. Gauss His major contribution was a major redefinition of irrational numbers in terms of Dedekind cuts. He published this in Stetigkeit und Irrationale Zahlen in 1872. His analysis of the nature of number and mathematical induction, including the definition of finite and infinite sets and his work in number theory, particularly in algebraic number fields, is of major importance. In 1874 he met Cantor while on holiday in Interlaken and was sympathetic to his set theory. 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.

27. CST LECTURES: Lecture 3 See Lecture 2. Lecture 3, first part More on the constructive theoryof dedekind cuts, based on Rudin(1964). 1. (1.15) continued. http://www.cs.man.ac.uk/~petera/Padua_Lectures/lect3.html

Lectures on Constructive Set Theory (Padua, Spring 1998) Lecture 3: 8th May (16.30 - 18.00) See Lecture 2 Lecture 3, first part: More on the constructive theory of Dedekind cuts, based on Rudin(1964). 1. (1.15) continued. The proposition (1.15) expresses that the cut A can be aproximated arbitrarily closely by a rational number, a property that is surely an essentail property of real numbers. We have seen that to prove 1.15 constructively for a cut A we can assume that A satisfies II'; i.e. that A is a cut'. In fact this is a necessary as well as sufficient condition. So we have the following proposition for a cut A. Prop: The cut A satisfies (1.15) iff A is a cut'. Def: Note that any decidable cut is a cut'. Also note that decidable cuts can be irrational. For example the irrational cut 2. Addition of cuts. As in Rudin(1964), if A, B are cuts then we can define and show that A+B is a cut. Moreover it is a cut' if both A and B are. We can prove the basic properties of +; e.g. it is commutative and associative, with identity 0*. Also r*+s* = (r+s)*. 3. Subtraction of cuts.

28. CST LECTURES: Lecture 2 The constructive approach to dedekind cuts. We follow chapter 1 of Rudin(1964).Rudin(1964) Principles of Mathematical Analysis, McGrawHill, 2nd edition. http://www.cs.man.ac.uk/~petera/Padua_Lectures/lect2.html

Lectures on Constructive Set Theory (Padua, Spring 1998) Lecture 2: 6th May (16.30 - 18.00): Introduction, continued See Lecture 1 We continue with our list of possible principles for constructive mathematics. 3. Constructions as Algorithms Principle A construction of an object must provide an algorithm for producing the object. This principle links constructive mathematics with computability. The Russian school use Church's Thesis so that they identify constructions with programs in a suitable programming language. The following supplementary principle should perhaps have its own heading. 3a. Determinateness of Constructions A construction must be determinate. 4. Vicious Circle Principle Impredicative definitions should not be used; i.e. an object d should not be defined in terms of a set D which has among its elements the object d. In particular a set A of natural numbers should not be defined using quantifiers, (for all sets X of natural numbers) or (for some set X of natural numbers), which quantify a variable X that ranges over all sets of natural numbers. (Here d is the set A and D is the `set' of all sets of natural numbers.) Such definitions have been thought to give rise to semantic paradoxes. The concept of impredicativity was first considered in the early parts of this century by Poincare, Russell, Weyl and others. In the 1960s a precise characterisation of `classical predicative mathematics' was formulated by Feferman and Schutte.

29. Www.amsta.leeds.ac.uk/events/logic97/abstracts/tressl.txt hoffset=40pt \voffset=-20pt \textwidth 15.3cm \textheight 22cm % to fit our printers\begin{document} \begin{center}{\huge On dedekind cuts in Polynomially http://www.amsta.leeds.ac.uk/events/logic97/abstracts/tressl.txt

30. On Gödel's Philosophy Of Mathematics, Chapter I are blocked.7 If one however wishes to derive totally his mathematics from hislogic, it is found that the process of dedekind cuts, the fundamental method http://www.friesian.com/goedel/chap-1.htm

Chapter I It is well-known that many programs, devised in order to insure the clarity of mathematical concepts, as well as to secure the foundations of mathematics, concepts of classical mthematics are indeed understood and are "sufficiently clear for us to be able to recognize their soundness...."[ I pass now to the most important of Russell's investigations in the field of the analysis of the concepts of formal logic, namely those concerning the logical paradoxes and their solution. By analyzing the paradoxes to which Cantor's set theory had led, he freed them from all mathematical technicalities, thus bringing to light the amazing fact that our logical intuitions (i.e., intuitions concerning such notions as: truth, concept, being, class, etc.) are self-contradictory. He then investigated where and how these common-sense assumptions of logic are to be corrected....[ It might seem at first that the set-theoretical paradoxes would doom to failure such an undertaking, but closer examination shows that they cause no trouble at all. They are a very serious problem, not for mathematics, however, but rather for logic and epistemology. As far as sets occur in mathematics (at least in the mathematics of today, including all of Cantor's set theory), they are sets of integers, or of rational numbers, (i.e., of pairs of integers), or of real numbers (i.e., sets of rational numbers), or of functions of real numbers (i.e., of sets of pairs of real numbers), etc. When theorems about all sets

31. 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://www.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

32. Poster Of Dedekind Richard Dedekind. lived from 1831 to 1916. Dedekind's major contributionwas a redefinition of irrational numbers in terms of dedekind cuts. http://www-gap.dcs.st-and.ac.uk/~history/Posters2/Dedekind.html

33. Quotations By Dedekind foundation for arithmetic. Opening of the paper in which dedekind cutswere introduced. Numbers are the free creation of the human mind. http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Dedekind.html

34. Is 0.999... = 1? dedekind cuts. Let cut D denote the set of all dedekind cuts in D. Define thesum of two cuts in the usual way. u + v = {x + y x is in u and y is in v}. http://www.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

35. Dedekind One remarkable piece of work was his redefinition of irrational numbers in termsof dedekind cuts which first came to him as he was thinking about how to teach http://www.wactc.wo.k12.ri.us/csstudents02/heathers/math/dedekind.html

Richard Dedekind, bron Oct. 6, 1831, died 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. Richard Dedekind(1831-1916) was born on October 6, 1831, in Brunswick, Germany, the birthplace of Gauss. He was the youngest of four children. Dedekind was the first university teacher to lecture on Galois theory. He introduced the concept of a field, replaced the concept of a permutation group by the abstract group concept, and, in 1858, introduced a purely arithmetic definition of continuity. Dedekind is most remembered for his concept of "Dedekind cut", which he introduced in 1872. He was criticized on this theory by mathematicians such a Kronecker, Weierstrass, and Russell.

36. Logikseminarier Våren 2002 Och Hösten 2003 the dedekind cuts in dense unbounded linear orders. 18 september. Jonas EliassonSheaves and Ultrasheaves. For arbitrary dense orders these are dedekind cuts. http://www.matematik.su.se/matematik/forskning/logik/Logiksemvt02.html

Matematiska institutionen Avd matematik Hemsida Matematik Matematisk statistik Biblioteket Logikseminarier hösten 2002 och våren 2003 Seminarieledare jesper@matematik.su.se Vårterminen 2003 19 mars kl 10.00 Johan Granström: Imperative lambda calculus Onsdag 19 mars, kl. 10.00 i sal 16, hus 5, Kräftriket. Abstract: Imperative Lambda Calculus (ILC) is a bridge between functional and mainstream programming languages. Johan Granström will discuss methods of interpreting ILC in dependent type theory. 5 mars kl 13.15 Benno van den Berg (Utrecht): Inductive types, ex/reg-completions, and realizability Onsdag 5 mars, kl. 13.15 (OBS Tid!) Matematiska institutionens seminarierum 3513 (plan 5, hus 3) MIC, Polacksbacken, Uppsala 19 februari kl 13.00 Erik Palmgren: Constructive completions of ordered sets, groups and fields II Matematiska institutionens seminarierum 3513 (plan 5, hus 3) MIC, Polacksbacken, Uppsala. 5 februari kl 13.00 Erik Palmgren: Constructive completions of ordered sets, groups and fields kl. 13.00, Matematiska institutionens seminarierum 3513 (plan 5, hus 3) MIC, Polacksbacken, Uppsala. OBS Ny tid och lokal! Abstract: The constructive real numbers are known to verify only a weakened form of the axioms for total order. We examine two kinds of completions of such orders. For arbitrary dense orders these are Dedekind cuts. For (nonarchimedean) ordered groups and fields, we consider so-called Cauchy cuts. As always, there is the problem how to represent the collection of cuts as a set in type theory. We show how this can be done using provable choice principles, in particular a generalisation of dependent choice.

37. Practical Foundations Of Mathematics Show how to add dedekind cuts and multiply them by rationals, justifyingthe case analysis of the latter into positive, zero and negative. http://www.dcs.qmul.ac.uk/~pt/Practical_Foundations/html/s2e.html

Practical Foundations of Mathematics Paul Taylor Exercises II Give a construction of the integers ( Z ) from the natural numbers such that z m n m n z Show how to add and multiply complex numbers as pairs of reals, verifying the commutative, associative and distributive laws and the restriction of the operations to the reals. The volume-flow (in m s ) down a pipe of radius r of a liquid under pressure p is c h n r m p k for some dimensionless c , where h is the dynamic viscosity , in units of kg m s . Find n m and k Show how to add Dedekind cuts and multiply them by rationals , justifying the case analysis of the latter into positive, zero and negative. What do your definitions say when the cuts represent rationals? Verify the associative, commutative and distributive laws. Express 3 and 6 as Dedekind cuts, and hence show that Let x L U ) and y M V ) be Dedekind cuts of Q define a Dedekind cut of R . Calling it x y , verify the usual laws for multiplication, without using case analysis [ n m ) which satisfies n m n m m n Show how to add Cauchy sequences and to multiply them by rational numbers.

38. Practical Foundations Of Mathematics In Ded72 he used these dedekind cuts of the set of rational numbers to definereal numbers, and went on to develop their arithmetic and analysis. http://www.dcs.qmul.ac.uk/~pt/Practical_Foundations/html/s21.html

Practical Foundations of Mathematics Paul Taylor Constructing the Number Systems The growth of algebra from the sixteenth to the nineteenth century made the idea of number more and more general, apparently demanding ever greater acts of faith in the existence and meaningfulness of irrational, negative and complex quantities. Then in 1833 William Rowan Hamilton showed how complex numbers ( C ) could be defined as pairs of real numbers, and the arithmetic operations by formulae involving these pairs. Ten years later he discovered a similar system of rules with four real components, the quaternions. The rationals ( Q ) may also be represented in the familiar way as pairs of integers ( Z ), although now there are many pairs representing each rational (Example ), and the positive and negative integers may be obtained from the natural numbers ( N ) in a similar way. This leaves the construction of the reals ( R ) from the rationals. The real numbers The course of the foundations of mathematics in the twentieth century was set on 24 November 1858, when Richard Dedekind first had to teach the elements of the differential calculus, and felt more keenly than before the lack of a really scientific foundation for analysis. In discussing the approach of a variable magnitude to a fixed limiting value, he had to resort to geometric evidences. Observing how a point divides a line into two parts, he was led to what he saw as the essence of continuity: R EMARK 2.1.1 If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this severing of the straight line into two portions.

39. 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

40. Citation Processing Letters archive Volume 19 , Issue 4 (November 1984) toc ProbabilisticTuring machines and recursively enumerable dedekind cuts Authors M Chrobak http://portal.acm.org/citation.cfm?id=2353&coll=portal&dl=GUIDE&CFID=11111111&CF

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