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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

1. Dedekind Cuts Class Notes/problem Set
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http://www.math.uiuc.edu/~dzaharo2/teaching/mathcampsets/dedekind_cuts.pdf

2. Dedekind Cuts
A selection of articles related to Dedekind Cuts Dedekind cuts Encyclopedia Mathematical analysis. Analysis is the generic name given to any branch of mathematics which depends upon
http://www.experiencefestival.com/dedekind_cuts

3. Dedekind_cuts | Define Dedekind_cuts At Dictionary.com
–noun Mathematics . two nonempty subsets of an ordered field, as the rational numbers , such that one subset is the collection of upper bounds of the second and the second
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4. Dedekind_cuts Synonyms, Dedekind_cuts Antonyms | Thesaurus.com
No results found for dedekind_cuts Please try spelling the word differently, searching another resource, or typing a new word. Search another word or see dedekind_cuts on
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5. Dedekind Cut - Handling Dedekind Cuts
A selection of articles related to dedekind cut handling dedekind cuts
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6. Dedekind Cuts
Dedekind Cuts from WN Network. WorldNews delivers latest Breaking news including World News, US, politics, business, entertainment, science,
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7. Dedekind Cut - Wikipedia, The Free Encyclopedia
In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two nonempty parts A and B, such that all
http://en.wikipedia.org/wiki/Dedekind_cut
Dedekind cut
From Wikipedia, the free encyclopedia Jump to: navigation search Dedekind used his cut to construct the irrational real numbers In mathematics , a Dedekind cut , named after Richard Dedekind , is a partition of the rational numbers into two non-empty parts A and B , such that all elements of A are less than all elements of B and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B . In other words, A contains every rational number less than the cut, and B contains every rational number greater than the cut. The cut itself is in neither set. More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts, ( 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. See also completeness (order theory) The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number , an irrational number (which is also a real number ) is created by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity.

8. PlanetMath: Dedekind Cuts
May 16, 2002 The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind s motivation behind this project
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talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Dedekind cuts (Definition) The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind's motivation behind this project is to notice that a real number , intuitively, is completely determined by the rationals strictly smaller than and those strictly larger than . Concerning the completeness or continuity of the real line , Dedekind notes in [ ] that 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 division of all points into two classes, this severing of the straight line into two portions. Dedekind defines a point to produce the division of the real line if this point is either the least or greatest element of either one of the classes mentioned above. He further notes that the completeness

9. Dedekind Cuts
A dedekind cut comprises two nonempty sets of rationals, L and R, such that each rational appears in exactly one of the two sets, and all the rationals in L
http://www.mathreference.com/top-ms,dcuts.html
Metric Spaces, Dedekind Cuts
Dedekind Cuts
Like a cauchy sequence, a dedekind cut defines a real number. We will show that these definitions are equivalent. However, the cauchy sequence is more general, because it can be applied to arbitrary mettric spaces. Also, most people find cauchy sequences more intuitive, so dedekind cuts are primarily of historical interest. You can skip this page if you like. A dedekind cut comprises two nonempty sets of rationals, L and R, such that each rational appears in exactly one of the two sets, and all the rationals in L (left) are less than all the rationals in R (right). We have cut the line in two, and the cut point becomes the real number. If b is an upper bound for L and c is a lower bound for R, c cannot be less than b, and there can't be any gap between, hence b = c. Since each point is suppose to be in just one set, decide arbitrarily that b belongs to L. In this case the real number is the rational b. One cut is less than another if R contains points not in R . Show this is a partial ordering; in fact it is a linear ordering.

10. Dedekind Cut -- From Wolfram MathWorld
Oct 11, 2010 Dedekind Cuts. §2.2.6 in What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England Oxford University
http://mathworld.wolfram.com/DedekindCut.html
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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 SEE ALSO: 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.
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11. Dedekind Cuts
The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind s motivation behind this project is to notice
http://myyn.org/m/article/dedekind-cuts/

12. Dedekind Cuts | Homework Help
A detailed tutorial on how to determine Dedekind cuts. Step by step tutorial including several examples of Dedekind cuts for reference.
http://homeworkhowto.com/dedekind-cuts/
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    How to Determine Dedekind Cuts
    Description A detailed tutorial on how to determine Dedekind cuts. Step by step tutorial including several examples of Dedekind cuts for reference. Overview A Dedekind cut is a partition of rational numbers into two non-empty sets A and B, such that all elements of A are less than elements of B, and A has no greatest element. The cut itself is a gap that is located between A and B, which is normally found by creating a new, irrational number, and setting it in the gap. What irrational number you use depends on what numbers you have partitioned into the two sets. It is like the number line of advanced algebra, that has both rational and irrational numbers on it instead of just integers. The Dedekind cut was named after Richard Dedekind. Bookmark It
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13. Math Forum - Ask Dr. Math
Sep 30, 2002 Date 10/23/96 at 193015 From mat stern Subject Dedekind cut I have already figured out Dedekind s theory of the rings and number notation.
http://mathforum.org/library/drmath/view/52511.html

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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
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14. Dedekind Cuts Of Rational Numbers -- Math Fun Facts
One way to do this was proposed by Dedekind in 1872, who suggested looking at cuts . A cut C is a proper subset of rational numbers that is nonempty,
http://www.math.hmc.edu/funfacts/ffiles/30005.3.shtml
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Dedekind Cuts of Rational Numbers
Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the length m/n can be obtained by dividing a length m line segment into n equal parts (if you like, this can be done by straightedge and compass). A very natural question you might ask is whether all lengths on the line are rational length? The Greeks knew that this was not the case; the square root of two is in fact irrational and can be obtained as the hypotenuse of a right triangle with side lengths 1 and 1. And there are other lengths (like Pi) which are irrational, but cannot be constructed by straightedge and compass? These numbers (representing lengths) have an ordering, thus can be associated with points along a line. What the above remarks show is that the set rational numbers in this line has "gaps". How does one "fill in the gaps" between the rational numbers One way to do this was proposed by Dedekind in 1872, who suggested looking at "cuts". A

15. Dedekind Cut: Definition From Answers.com
Dedekind cut ( d d kint k t ) ( mathematics ) A set of rational numbers satisfying certain properties, with which a unique real number may be.
http://www.answers.com/topic/dedekind-cut

16. Dedekind Cut In NLab
Idea. Dedekind cuts are a way to make precise the idea that a real number is that which can be approximated (in the absolute value? metric) by rational numbers.
http://ncatlab.org/nlab/show/Dedekind cut
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Dedekind cuts
Idea
Dedekind cuts are a way to make precise the idea that a real number is that which can be approximated (in the absolute value metric) by rational numbers In 1872, Richard Dedekind published Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), in which he pointed out that a real number may be uniquely determined its order relationships with rational numbers. That is, the real number x is determined by its lower set L x and its upper set U x L x a a x U x b x b Dedekind had found great success understanding the ‘ideal numbers’ of ring theory as certain sets, which are what we now understand as ideals . So he defined a ‘real number’ as a pair of sets of rational numbers, the lower and upper sets shown above (actually a slight variation). Such a pair of sets of rational numbers he called a ‘Schnitt’ (English ‘cut’), nowadays called a ‘Dedekind cut’.
Definitions
Exactly which pairs of sets of rational numbers can appear this way? We will define a

17. Dedekind Cut - Wikivisual
In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it has the leastupper-bound property, i.e., its every nonempty subset that has an upper
http://en.wikivisual.com/index.php/Dedekind_cut
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Dedekind cut
From Wikipedia, the free encyclopedia
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 . Dedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom). The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number , an irrational number (which is also a real number ) is created by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity.
Contents
edit Handling Dedekind cuts
It is more symmetrical to use the ( A B ) notation for Dedekind cuts, but each of

18. Arithmetic Of Dedekind Cuts Of Ordered Abelian Groups - Microsoft Academic Searc
Authors Antongiulio Fornasiero, Marcello Mamino. Citations 0 We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a
http://academic.research.microsoft.com/Paper/4753965.aspx
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Arithmetic of Dedekind cuts of ordered Abelian groups Edit Arithmetic of Dedekind cuts of ordered Abelian groups Antongiulio Fornasiero Marcello Mamino We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a suitable representation theorem, the universal part of the theory of such structures. Journal: Annals of Pure and Applied Logic - APAL View or Download The following links allow you to view and download full papers. These links are maintained by other sources not affiliated with Microsoft Academic Search. Reference ... W. Baur Published in 1976. Partially Ordered Algebraic Systems Citations: 126 L. Fuchs

19. Construction Of The Real Numbers - Wikipedia, The Free Encyclopedia
We form the set of real numbers as the set of all Dedekind cuts A of , and define a total ordering on the real numbers as follows We embed the rational numbers into the reals by
http://en.wikipedia.org/wiki/Construction_of_the_real_numbers
Construction of the real numbers
From Wikipedia, the free encyclopedia Jump to: navigation search In mathematics , there are several ways of defining the real number system as an ordered field . The synthetic approach gives a list of axioms for the real numbers as a complete ordered field . Under the usual axioms of set theory , one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic . Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field.
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20. Real Number In NLab
Two successful approaches were developed in 1872, Richard Dedekind? 's definition of real numbers as certain sets of rational numbers (called Dedekind cuts) and Georg Cantor? 's
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