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         Dedekind Cuts:     more detail
  1. Order Theory: Zorn's Lemma, Well-Order, Total Order, Interval, Supremum, Ordered Pair, Dedekind Cut, Infimum, Ultrafilter, Monotonic Function
  2. Real Number: Square Root of 2, Equivalence Class, Decimal Representation, Cauchy Sequence, Dedekind Cut, Archimedean Property, Complete Metric Space

41. Math Forum Discussions
If we accept the definition of real numbers as dedekind cuts, between No, Ithink the Dedekind cut definition of the reals makes it an
http://mathforum.org/kb/thread.jspa?threadID=1177385&messageID=3871060

42. Math Forum Discussions
in your lines to obey the property of dedekind cuts that you quoted I thought you were the one who wanted to use dedekind cuts to prove something.
http://mathforum.org/kb/thread.jspa?threadID=1177385&messageID=3872599

43. Program Files\Netscape\Communicator\Program\dedexxx
and integral calculus when the thought of a Dedekind cut came to him. was his redefinition of irrational numbers in terms of dedekind cuts which,
http://www.andrews.edu/~calkins/math/biograph/199900/biodedek.htm
    Julius Wihelm Dedekind
      6 Oct 1831 in Braunschweig, duchy of Braunschweig (Germany) 12 Feb 1916 in Braunschweig, duchy of Braunschweig (Germany)
        Early Life
      Richard Dedekind was born into the life of research and experimental theory. His father was a professor at the Collegium Carolinum in Brunswick. His mother was the daughter of a professor, also working at the Collegium Carolinum in Brunswick. He was not the first into the family of professors. He was the youngest of four children and never marred. Although he never married he still had the company of the opposite sex. He lived with one of his sisters, who also never married, for most of his adult life. He attended school throughout his child hood at Brunswick. At the age of seven Richard's interest was not in the field of Mathematics. Soon he attended the school of Martino-Catherineum which was a good school for the study of physics and chemistry. However Richard's soon took a disliking to the fields of physics because of the imprecise logical structure and soon he returned to the field of Mathematics. Between high school and entering a university Dedekind entered the Collegium Carolinumin 1848 at the age of 16. There he received a good understanding of basic Mathematics. The studies of differential and integral calculus, analytic geometry and the foundation of analysis. He entered the University of Guttingen in the spring of 1850 with a solid grounding in mathematics.

44. \documentclass{article} \usepackage{latexsym} \newtheorem
We have to decide what that means; either $r$ is a Dedekind cut or it is an This is where we should start worrying about dedekind cuts and Cauchy
http://www.bath.ac.uk/~masgcs/book1/amplifications/ch8q8_2.txt

45. Surreal Numbers
dedekind cuts are fairly concrete, or at least understandable without too much All the real numbers have been created previously with Dedekind s cuts.
http://www.usna.navy.mil/MathDept/wdj/surreal_numbers.html
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.

46. MathematicalArgument.html
We could, in fact, generalize the definition of dedekind cuts by simply replacingMATH with Be an arbitrary set of dedekind cuts that is bounded above,
http://www.umsl.edu/~siegel/SetTheoryandTopology/MathematicalArgument.html

47. Re: Isomorphism Between Mereology And Boolean Algebra Without Least Elem
dedekind cuts, however, are *never* used by working mathematicians. Nobody usesa Dedekind cut to calculate a square root, a cosine, or a logarithm or
http://suo.ieee.org/email/msg13184.html
Thread Links Date Links Thread Prev Thread Next Thread Index Date Prev ... Date Index
Re: Isomorphism between Mereology and Boolean algebra without least element
http://www.cs.nyu.edu/mailman/listinfo/fom "The FOM list is intended to provide a venue for discussing the provocative, sometimes controversial, ideas which drive contemporary research in foundations of mathematics..." Jay > > John

48. Re: Isomorphism Between Mereology And Boolean Algebra Without Least Elem
dedekind cuts for defining real numbers, however, are different. So Peano saxioms are part of objectlevel math, but dedekind cuts are part of
http://suo.ieee.org/email/msg13181.html
Thread Links Date Links Thread Prev Thread Next Thread Index Date Prev ... Date Index
Re: Isomorphism between Mereology and Boolean algebra without least element

49. \documentclass[a4paper]{article} \usepackage{amssymb} \pagestyle
our printers \begin{document} \begin{center}{\huge On dedekind cuts inPolynomially Bounded,}\end{center} \begin{center}{\huge $o$Minimal Expansions of
http://www.amsta.leeds.ac.uk/events/logic97/abstracts/tressl.txt

50. Some Number Theory
Now we define objects (called dedekind cuts) that consist of two sets of We call the set of dedekind cuts defined above the positive real numbers.
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

51. Sensei's Library: Toms Technical Introduction To CGT/ Discussion
However, the relation between Conway numbers and dedekind cuts is nice. A pageon Nim, and impartial games generally would be good.
http://senseis.xmp.net/?TomsTechnicalIntroductionToCGT/Discussion

52. EEVL | Full Record
construction of the real numbers through dedekind cuts, introduction to decimal approximations, qary, isomorphism, dedekind cuts, infinities,
http://www.eevl.ac.uk/show_full.htm?rec=1002096535-22422

53. AW: Fw: [Fis] Is The Distinction Between Internalism And Externalism Morethan Ap
As Koichiro remarks, Dedekind s approach is externalist. get from there tothe reals, unless we assume that the reals are nothing but dedekind cuts.
http://webmail.unizar.es/pipermail/fis/2005-February/000920.html
AW: Fw: [Fis] Is the distinction between internalism and externalism morethan aparticular philosophy of mathematics?
Karl Javorszky kj04@chello.at
Thu, 10 Feb 2005 13:53:15 +0100 fis-bounces@listas.unizar.es [mailto: fis-bounces@listas.unizar.es ]Im Auftrag von John Collier Gesendet: Donnerstag, 10. Februar 2005 13:02 An: Koichiro Matsuno; fis@listas.unizar.es is the BASIS of the distinction between the internalist and the externalist positions something more than a different conceptualization of number and relation? collierj@ukzn.ac.za Philosophy and Ethics, University of KwaZulu-Natal, Durban 4041 South Africa T: +27 (31) 260 3248 / 260 2292 F: +27 (31) 260 3031 multipart/alternative attachment An HTML attachment was scrubbed... URL: http://webmail.unizar.es/pipermail/fis/attachments/62cac1e8/attachment.htm

54. Fw: [Fis] Is The Distinction Between Internalism And Externalismmorethan Apartic
As Koichiro remarks, Dedekind s approach is externalist. get from there tothe reals, unless we = assume that the reals are nothing but dedekind cuts.
http://webmail.unizar.es/pipermail/fis/2005-February/000921.html
Fw: [Fis] Is the distinction between internalism and externalismmorethan aparticular philosophy of mathematics?
Gordana Dodig Crnkovic gordana.dodig-crnkovic@mdh.se
Thu, 10 Feb 2005 14:58:25 +0100

55. On Gödel's Philosophy Of Mathematics, Chapter I
Granted, the idea of dedekind cuts is heuristically related to a Gödel isparticularly concerned with dedekind cuts because of his desire to derive all
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

56. 2.4. The Real Number System
Construction of R via dedekind’s cuts; Construction of R classes via equivalenceof Cauchy sequences . Right now, however, it will be more important to
http://www.shu.edu/projects/reals/infinity/reals.html
2.4. The Real Number System
IRA In the previous chapter we have defined the integers and rational numbers based on the natural numbers and equivalence relations. We have also used the real numbers as our prime example of an uncountable set. In this section we will actually define - mathematically correct - the 'real numbers' and establish their most important properties. There are actually several convenient ways to define R . Two possible methods of construction are:
  • Construction of R via Dedekind’s cuts
  • Construction of R classes via equivalence of Cauchy sequences .
Right now, however, it will be more important to describe those properties of R that we will need for the remainder of this class. The first question is: why do we need the real numbers ? Aren’t the rationals good enough ? Theorem 2.4.1: No Square Roots in Q There is no rational number x such that x = x * x = 2 Proof Thus, we see that even simple equations have no solution if all we knew were rational numbers. We therefore need to expand our number system to contain numbers which do provide a solution to equations such as the above. There is another reason for preferring real over rational numbers: Informally speaking, while the rational numbers are all 'over the place', they contain plenty of holes (namely the irrationals). The real numbers, on the other hand, contain no holes. A little bit more formal, we could say that the rational numbers are not closed under the limit operations, while the real numbers are. More formally speaking, we need some definitions.

57. Dedekind, Richard
study of CONTINUITY and definition of the real numbers in terms of dedekind cuts ,the nature of number and mathematical induction, definition of finite
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

58. Dedekind Cut: Information From Answers.com
dedekind cut In mathematics , a dedekind cut in a totally ordered set S is apartition of it, ( A , B ), such that A is closed downwards (meaning.
http://www.answers.com/topic/dedekind-cut
showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Dedekind cut Wikipedia 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 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
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

59. Dedekind Cut - Definition Of Dedekind Cut In Encyclopedia
In mathematics, a dedekind cut in a totally ordered set S is a partition ofit, (A, B), such that A is closed downwards (meaning that for any element x in S
http://encyclopedia.laborlawtalk.com/Dedekind_cut
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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

60. Dedekind Cut@Everything2.com
So let a dedekind cut be made at some rational point q1. This constructivelydefines some QL1. Then take q2, q3, to create QL2, QL3, with the proviso
http://www.everything2.com/index.pl?node=Dedekind Cut

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