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21. 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.

22. Instructor Hentzel Office Phone 515-294-8141 E-mail Hentzel
Cauchy sequences and dedekind cuts. We start with Cauchy sequences. The same for dedekind cuts.
http://www.math.iastate.edu/hentzel/class.166.03/Apr.26
< epsilon. Cauchy sequences are just the convergent sequences, but the big thing is that Cauchy did not have to already know the limit before he knew the sequence converged. Now to Cauchy, the real numbers are just the Cauchy sequences. He show that the term by term sum, difference, product, quotient of Cauchy sequences is still a Cauchy sequence. Thus he had a model of the real numbers. Dedekind used another strategy. He plugged the holes in the rational number line. If he could divide the rationals into two halfs, where each number in one half was less than each number in the other half, he had a Dedekind cut. Thus to represent Sqrt[2] the two haves were: Negative and those positive rationals whose squares were less than 2. Those positive rationals whose squares were greater than 2. Dedekind showed that the sum, difference, product, quotient, of two Dedekind cuts was again a Dedekind cut, and to him, the real numbers were these Dedekind cuts of rationals. Again, lurking behind all of this is the original question, If one has a Cauchy sequence, presumably one should be able to express the numbers in the sequence. This requires some sort of rule. The "Set" of all Cauchy Sequences sweeps it under the rug because we presumably have all of them. But is "all" just those which can be expressed with a formula in words? Or will we allow other ones. - The same for Dedekind cuts. Usually one states them as the set of all rationals with some property. Again, we are limited to words.

23. 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
Quotations by Richard Dedekind
Opening of the paper in which Dedekind cuts were introduced. Numbers are the free creation of the human mind. Main index Quotations index Biographies Index
History Topics ... Anniversaries for the year
JOC/EFR February 2003 The URL of this page is:
School of Mathematics and Statistics
University of St Andrews, Scotland http://www-history.mcs.st-andrews.ac.uk/Quotations/Dedekind.html

24. Reals Via Dedekind Cuts
Theorem Real Numbers as dedekind cuts. The proof isnot done, sorry. To Theory Glossary Map (bgw).
http://pirate.shu.edu/projects/reals/infinity/proofs/r_dedek.html
Theorem: Real Numbers as Dedekind Cuts
The proof is not done, sorry. Map (bgw)

25. Dedekind Cut
the set is a Dedekind cut that gets identified with a, so that the linearly orderedset S may be regarded as embedded within the set of all dedekind cuts of S
http://www.sciencedaily.com/encyclopedia/dedekind_cut
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Dedekind cut
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 whenever a is in A and x a , then x is in A as well) and B is closed upwards. If a is a member of S then the set is a Dedekind cut that gets identified with 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

26. [FOM] Real Numbers
When I say that something, like the real numbers are dedekind cuts , canbe true, I mean that it becomes true under such an interpretation.
http://www.cs.nyu.edu/pipermail/fom/2003-June/006874.html
[FOM] real numbers
Neil Tennant neilt at mercutio.cohums.ohio-state.edu
Fri Jun 20 15:04:22 EDT 2003 ... I can finally explain "can". When I say that something, like "the real numbers are Dedekind cuts", can be true, I mean that it becomes true under such an interpretation. More information about the FOM mailing list

27. [FOM] 211:Coding In Reverse Mathematics 2
In the left Dedekind cut coding, equality two left dedekind cuts are consideredequal if and only if they have the same nonmaximum elements.
http://www.cs.nyu.edu/pipermail/fom/2004-February/007905.html
[FOM] 211:Coding in Reverse Mathematics 2
Harvey Friedman friedman at math.ohio-state.edu
Wed Feb 4 10:52:44 EST 2004 http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with proofs. Type Harvey Friedman in the window. This is the 211th in a series of self contained numbered postings to FOM covering a wide range of topics in f.o.m. The list of previous numbered postings #1-149 can be found at http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html More information about the FOM mailing list

28. 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

29. Between Cardinal 0 And Cadinal 1 - Physics Help And Math Help - Physics Forums
My advanced calc text opts to define completeness in terms of dedekind cuts asfollows A Dedekind cut of a field is a pair of nonempty sets A and B whose
http://www.physicsforums.com/showthread.php?t=6896&page=2

30. Julius Wihelm Richard 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.stetson.edu/~efriedma/periodictable/html/Db.html
Julius Wihelm Richard Dedekind
Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today. One remarkable piece of work was his redefinition of irrational numbers in terms of Dedekind cuts which first came to him as he was thinking about how to teach calculus. His work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is also of major importance. Among Dedekind's other notable contributions to mathematics were his editions of the collected works of Dirichlet, Gauss, and Riemann. His study of Dirichlet's work did, in fact, to lead to his own study of algebraic number fields, as well as to his introduction of ideals. In a joint paper with Heinrich Weber published in 1882, he applies his theory of ideals to the theory of Riemann surfaces. This gave powerful results such as a purely algebraic proof of the Riemann-Roch theorem. Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas. Dedekind's notion of an ideal was taken up and extended by Hilbert and then later by Emmy Noether. This led to the unique factorization of integers into powers of primes to be generalised to ideals in other rings.

31. Georg Ferdinand Ludwig Philipp Cantor
Dedekind. Cantor published a paper on trigonometric series in 1872in which he defined what are now known as dedekind cuts . In
http://www.stetson.edu/~efriedma/periodictable/html/Ca.html
Georg Ferdinand Ludwig Philipp Cantor
After early education at home from a private tutor, Georg Cantor attended primary school in St. Petersburg, then in 1856 when he was 11 years old his family moved to Germany. His teachers gave him outstanding reports, which mentioned in particular his exceptional skills in mathematics, in particular trigonometry. He entered the Polytechnic of Zurich in 1862. His studies at Zurich, however, were cut short by the death of his father in June 1863. Cantor moved to the University of Berlin where he became friends with Herman Schwarz, a fellow student. Cantor attended lectures by Weierstrass, Kummer and Kronecker. While at Berlin, Cantor became much involved with the Mathematical Society being president of the Society during 1864-65. He was also part of a small group of young mathematicians who met weekly in a wine house. After receiving his doctorate in 1867 in number theory, Cantor taught at a girl's school in Berlin. Then, in 1868, he joined the Schellbach Seminar for mathematics teachers in Halle. There the direction of Cantor's research turned away from number theory and towards analysis. This was due to Heine, one of his senior colleagues, who challenged Cantor to prove the open problem on the uniqueness of representation of a function as a trigonometric series. Cantor solved the problem in 1870. He published further papers between 1870 and 1872 dealing with trigonometric series. Cantor was promoted to Extraordinary Professor in 1872, and in that year he began a friendship with Dedekind. Cantor published a paper on trigonometric series in 1872 in which he defined what are now known as "Dedekind cuts". In 1873, Cantor proved the rational numbers and the algebraic numbers are countable and that the real numbers were not countable. He published this in a paper in 1874, and it is in this paper that the idea of a one-to-one correspondence appears for the first time, albeit indirectly.

32. PETER ACZEL S TOPICS
followers. Both Brouwer and Bishop construct the reals using Cauchy sequences.But dedekind cuts can also be used if one is careful. With
http://www.cs.man.ac.uk/~petera/Math-MSc-projects/mar04.html
PETER ACZEL'S TOPICS
    1) Impredicativity
    This topic originated at the turn of the century with Russell's attempt to resolve his paradox through his `vicious Circle Principle' that `no totality can contain members defined in terms of itself'. The project could trace the history of this idea through the debate with Poincare, Russell and Whitehead's Principia Mathematica with its Theory of Types and the Axiom of Reducibility and the work of Weyl in the first part of the century, through the more recent work of Schutte and Feferman on a precise concept of Predicative Mathematics, to the present day issues in the Foundations of Constructive Mathematics. A more narrowly conceived project could focus on one or other technical aspect of this topic.
    2) A survey of the variety of semantics for intuitionistic propositional logic.
    Intuitionistic logic is the subsystem of classical logic that was first axiomatised by Heyting as the correct logic for the intuitionistic approach to mathematics initiated by Brouwer. Over the years many styles of complete semantics for intuitionistic logic have been developed; e.g. algebraic, topological, Beth trees, Kripke structures, realisability, categorical, type theoretical, etc ... . The aim of the project is to survey and relate these semantics for intuitionistic propositional logic
    3) The constructive real numbers
    There are several standard approaches to the construction of the complete ordered field of real numbers, perhaps the two main ones being Cantor's Cauchy sequence approach and the other being the Dedekind cut approach. As the theory of complete ordered fields is categorical (i.e. not only is there a model, but any two models are isomorphic) all approaches are equivalent.

33. 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.

34. Richard Dedekind --  Encyclopædia Britannica
of St Andrews Biography of this 19thcentury German mathematician noted forhis redefinition of irrational numbers in terms of dedekind cuts and his
http://www.britannica.com/eb/article?eu=30201

35. From Ags@seaman.cc.purdue.edu (Dave Seaman) Subject Re Zeno
Some examples of dedekind cuts are given by a) L = { x in Q x 1 } and R = Q\L.b) R = { x in Q x^2 = 2 } and L = Q\R. c) L = { x in Q x sum_(i=0 to
http://www.math.niu.edu/~rusin/known-math/00_incoming/reals
From: ags@seaman.cc.purdue.edu (Dave Seaman) Subject: Re: Zeno gets naked :-) Date: 14 Jul 2000 10:13:35 -0500 Newsgroups: sci.math Summary: [missing] In article , Pax wrote: >"Dave Seaman"

36. Archimedes Plutonium
My conjecture which I have not yet proved is that AC = Dedekind cut = Reals. Thatis the same as taking the padics and doing dedekind cuts on the p-adics.
http://www.iw.net/~a_plutonium/File107.html
Understanding the Axiom of Choice (AC)
by Archimedes Plutonium this is a return to website location www.iw.net/~a_plutonium

37. 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

38. Documentclass[a4paper]{article} \usepackage{amssymb} \pagestyle
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

39. 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.qmw.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.

    • 40. 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.qmw.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.

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