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  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

61. Dedekind Cuts Of Archimedean Complete Ordered Abelian Groups
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http://www.springerlink.com/index/N1YAKHHTLNEJ3XHH.pdf

62. ULTRAFILTER MAPPINGS AND THEIR DEDEKIND CUTS
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http://www.ams.org/journals/tran/1974-188-00/S0002-9947-1974-0351822-6/S0002-994

63. Dedekind Cuts « Not About Apples
Oct 15, 2009 Then a Dedekind cut is any way of dividing the rational numbers into two nonempty groups so that every element in one group (“the small
http://notaboutapples.wordpress.com/2009/10/15/dedekind-cuts/
Not About Apples
Math for Poets / Poetry for Mathematicians
Dedekind cuts
was harder to imagine than the idea of a world where there are multiple lines through a point parallel to another point. It got me thinking, why is that? definition I will assume that you understand the rational number system (fractions, positive, negative, and zero).  There is a natural order on these numbers (an idea of when one number is greater than/less than/equal to another), which I will also assume you understand. Then a Dedekind cut Then in some sense we can define a real number to be a Dedekind cut.  To phrase it more naturally, for each Dedekind cut, create a number to be at least as large as the numbers in the small group and no larger than the numbers in the large group. , which did not previously exist in the rational number line. In summary, a real number is completely described by which fractions are less than and which are greater. This property of the real number line (as opposed to the rational number line, etc.) is called completeness If the reason you believe that right in between This entry was posted on Thursday, October 15th, 2009 at 12:22 and is filed under

64. Construction Of Real Numbers Dedekind Cuts Rational Number
Construction Of Real Numbers Dedekind Cuts Rational Number Economy.
http://www.economicexpert.com/a/Construction:of:real:numbers.html

65. Soare : Cohesive Sets And Recursively Enumerable Dedekind Cuts.
by RI Soare 1969 - Cited by 32 - Related articles
http://projecteuclid.org/euclid.pjm/1102978065
Log in Title Author(s) Abstract Subject Keyword All Fields FullText more options
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    Cohesive sets and recursively enumerable Dedekind cuts.
    Robert I. Soare Source: Pacific J. Math. Volume 31, Number 1 (1969), 215-231. Primary Subjects: Full-text: Open access PDF File (1609 KB) DjVu File (371 KB) Links and Identifiers Permanent link to this document: http://projecteuclid.org/euclid.pjm/1102978065 Zentralblatt MATH identifier: Mathematical Reviews number (MathSciNet): back to Table of Contents
    References
    [1] J. C. E. Dekker and J. Myhill, Retraceable sets, Canad. J. Math. 10 (1958), 357-373. Mathematical Reviews (MathSciNet): Zentralblatt MATH: [2] D. A. Martin, A theorem on hyperhypersimplesets, J. Qf Symbolic Logic 28 (1963), 273-278. Mathematical Reviews (MathSciNet): Zentralblatt MATH: [3] D. A. Martin, Classesof recursively enumerable sets and degrees of unsolvability, Zeitschr. F. Math Logik und Grundl. Math. 12 (1966), 295-310. Mathematical Reviews (MathSciNet): [4] H. G. Rice, Recursive real numbers, Proc. Amer. Math. Soc. 5 (1954), 784-791.

66. 6 Dedekind Cuts We Take The Real Numbers For Granted, But In Fact
File Format PDF/Adobe Acrobat View as HTML
http://homepages.ius.edu/wclang/m413/2010_fall/notes6.pdf

67. CiteSeerX — The Elementary Theory Of Dedekind Cuts
by M Tressl
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.7675

68. Re Rudin And Dedekind Cuts
Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
http://mathforum.org/kb/thread.jspa?forumID=13&threadID=1788511&messageI

69. Advanced Math Answers | Dedekind Cuts Rational LIFESA, Give An Example For A Cut
Advanced Math Answers for Dedekind Cuts Rational LIFESA, Give an example for a cut corresponding to a rational number.
http://www.cramster.com/answers-mar-09/advanced-math/dedekind-cuts-rational-life

70. RE: "Dedekind" Cuts For Representing Substances
To 'SUO WG' standardupper-ontology@listserv.ieee.org Subject RE Dedekind Cuts for Representing Substances; From Chris Lofting chrislofting@ozemail.com.au
http://grouper.ieee.org/groups/suo/email/msg12966.html
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71. Dedekind Cuts
Real analysis problem with dedekind cuts. A pdf document of the problem is attached. Attached file(s) Attachments. real_analysis_prob.pdf View File
http://www.brainmass.com/homework-help/math/real-variables/144466
Mathematics Homework Solutions
Real analysis problem with dedekind cuts
A pdf document of the problem is attached.
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  • real_analysis_prob.pdf View File

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real_analysis_prob.pdf real analysis problem
A Dedekind cut is described as follows in the Real Mathematical Analysis text by Pugh.
In addition, the following properties are part of the definition of a cut:
(a) A B = Q, A = , B = , A B = .
(b) If a A and b B, then a < b .
(c) A contains no largest element.
what C and D if x > ? if x
note: the cuts determine the real numbers so x in the definition is in fact a real number. how do I show uniqueness? would I show that if x · y = 1 and x · y = 1 then y = y ? How do I use these cuts to show this? Please help. Thanks.

72. Real Analysis/Dedekind S Construction - Wikibooks, Collection Of
May 10, 2010 Today when discussing Dedekind cuts one usually only keeps track of one of We now outline how to make the set of Dedekind cuts forms a
http://en.wikibooks.org/wiki/Real_Analysis/Dedekind's_construction

73. Dedekind Cut - Exampleproblems
In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it does have the leastupper-bound property, i.e., its every nonempty subset that has an
http://www.exampleproblems.com/wiki/index.php/Dedekind_cut
Dedekind cut
From Exampleproblems
Jump to: navigation search In mathematics , a Dedekind cut , named after Richard Dedekind , in a totally ordered set S is a partition of it, ( A B ), such that A is closed downwards (meaning that for all a in A x a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. 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
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

74. Dedekind Cuts
Such a pair is called a Dedekind cut (Schnitt in German). You can think of it as defining a real number which is the least upper bound of the Lefthand
http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/Lectures/A3.html
MT2002 Analysis Previous page
(Some Early History of Set Theory) Contents Next page
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Dedekind cuts
The first construction of the Real numbers from the Rationals is due to the German mathematician Richard Dedekind (1831 - 1916). He developed the idea first in 1858 though he did not publish it until 1872. This is what he wrote at the beginning of the article. He defined a real number to be a pair ( L R ) of sets of rationals which have the following properties.
  • Every rational is in exactly one of the sets
  • Every rational in L R
    Such a pair is called a Dedekind cut Schnitt in German). You can think of it as defining a real number which is the least upper bound of the "Left-hand set" L and also the greatest lower bound of the "right-hand set" R . If the cut defines a rational number then this may be in either of the two sets.
    It is rather a rather long (and tedious) task to define the arithmetic operations and order relation on such cuts and to verify that they do then satisfy the axioms for the Reals including even the Completeness Axiom. Richard Dedekind , along with Bernhard Riemann was the last research student of Gauss . His arithmetisation of analysis was his most important contribution to mathematics, but was not enthusiastically received by leading mathematicians of his day, notably

    • 75. Re: "Dedekind" Cuts For Representing Substances
    To John F. Sowa sowa@BESTWEB.NET Subject Re Dedekind Cuts for Representing Substances; From Markus Pilzcker mp.lists@FREE.FR Date Tue, 8 Mar 2005 185949 +0000
    http://suo.ieee.org/email/msg12981.html
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    Re: "Dedekind" Cuts for Representing Substances

    76. ScienceDirect - Fuzzy Sets And Systems : Fuzzy Real Numbers As Dedekind Cuts Wit
    by U Höhle 1987 - Cited by 40 - Related articles
    http://linkinghub.elsevier.com/retrieve/pii/0165011487900273
    window.onresize = resizeWindow; Username: Password: Remember me Not Registered? Forgotten your username or password? Go to Athens / Institution login All fields Author Advanced search Journal/Book title Volume Issue Page Search tips Font Size: Related Articles Real numbers in the topos of sheaves over the category ...
    Journal of Pure and Applied Algebra
    Real numbers in the topos of sheaves over the category of filters     Original Research Article
    Journal of Pure and Applied Algebra Volume 160, Issues 2-3 25 June 2001 Pages 275-284
    Erik Palmgren
    Abstract
    The sheaves over the category of filters, with the precanonical topology, serve as a universe of sets where nonstandard analysis can be developed along constructive principles. In this paper we show that the Dedekind real numbers of this topos can be characterised as the nonstandard hull of the rational numbers. Moreover, it is proved that the axiom of choice holds on standard sets of the topos.
    Purchase PDF (114 K) Probabilistics Turing machines and recursively enumerab...

    77. Re: Rudin And Dedekind Cuts
    Aug 20, 2008 but if you obtain the reals by the unique models of the structures are isomorphic as ordered fields but in some other sense members
    http://sci.tech-archive.net/Archive/sci.math/2008-08/msg02138.html
    Re: Rudin and Dedekind cuts
    • From Date : Wed, 20 Aug 2008 13:55:05 -0700 (PDT)

    But in this case they don`t have all the properties common, for
    example in this case the objects representing reals are infinite
    ordered sets, but if you obtain the reals by the unique models of the
    second order tarski axioms as the unique complete archimedian ordered
    field and that case the reals have no set structure. Both These
    structures are isomorphic as ordered fields but in some other sense
    they may not be isomorphic.
    You are entirely missing the point. Of course, to simply say that the
    system of R is some complete ordered field is cleaner in the sense
    that we don't get any fo the "distracting extra" features about the members of the carrier set R. But then we don't have a SPECIFIC object that is the system R. On the other hand, ANY specific system we choose to be the system R (whether by Dedekind cuts, equivalence classes of Cauchy sequences, etc) of course brings with it the features of the specific objects that are members of the carrier set of the system of R.

    78. Math Refresher: Dedekind Cut
    Mar 6, 2006 The Dedekind Cut is mathematical construction created by Richard Dedekind to provide a definition for the real numbers.
    http://mathrefresher.blogspot.com/2006/03/dedekind-cut.html
    var BL_backlinkURL = "http://www.blogger.com/dyn-js/backlink_count.js";var BL_blogId = "12604614";
    Math Refresher
    Review of fundamental math concepts in a straight-forward, accessible way.
    Monday, March 06, 2006
    Dedekind Cut
    The Dedekind Cut is mathematical construction created by Richard Dedekind to provide a definition for the real numbers.
    The Dedekind Cut itself is defined in terms of the rational numbers.
    Definition 1 - Dedekind Cut
    A Dedekind cut is defined as the subset of the rational integers Q (ratios of integers ) which is less than
    NOTE: Q is used to present the set of all rational numbers; R is used to represent the set of all real numbers, and Z is used to represent the set of all integers.
    Example:
    We could create a Dedekind Cut around . In this case, we could think of , etc. as elements of the Dedekind Cut. On the other hand, , etc. would not be elements of the cut.
    Definition 2 - Set of Real Numbers R
    The set of real numbers is the set of Dedekind cuts that have the following properties:
    (a) is not empty
    (b) contains no greatest element For any element x , there exists y ∈ α such that x is less than y (c) If x,y

    79. Outofprintmath (was: Re: Rudin And Dedekind Cuts) - Application Forum At ObjectM
    Aug 21, 2008 Outofprintmath (was Re Rudin and Dedekind cuts) Theory.
    http://objectmix.com/theory/740934-outofprintmath-re-rudin-dedekind-cuts.html

    80. 1 Dedekind Cuts ( ¯ )
    File Format PDF/Adobe Acrobat Quick View
    http://www.math.tau.ac.il/~aaro/course/calculus1-2010b-revision presentation-1-h

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