1. Dedekind Cut -- From MathWorld Dedekind Cut. A set partition of has no greatest member. Real numberscan be defined using either dedekind cuts or Cauchy sequences. http://mathworld.wolfram.com/DedekindCut.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon 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 search 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. Eric W. Weisstein. "Dedekind Cut." From

2. Dedekind Cut - Wikipedia, The Free Encyclopedia is a Dedekind cut that gets identified with a, so that the linearly ordered set Smay be regarded as embedded within the set of all dedekind cuts of S. If the http://en.wikipedia.org/wiki/Dedekind_cut

Dedekind cut From Wikipedia, the free encyclopedia. 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 S A B less than C D A is a proper subset of C , or, equivalently D is a proper subset of B . In this way, the set of all Dedkind cuts is itself a linearly ordered set, and, moreover, it does have the least-upper-bound property, i.e., its every nonempty subset that has an upper bound has a least upper bound. Embedding S within a larger linearly ordered set that does have the least-upper-bound property is the purpose. The Dedekind cut is named after Richard Dedekind , who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers . A typical Dedekind cut of the rational numbers is given by This cut represents the real number in Dedekind's construction.

3. PlanetMath: Dedekind Cuts dedekind cuts, (Definition). The purpose of dedekind cuts is to providea sound logical foundation for the real number system. Dedekind s http://planetmath.org/encyclopedia/DedekindCuts.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List 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

4. PlanetMath Example Of Definable Type See Also example of a universal structure, dedekind cuts Keywordsdense linear order. This object s parent. Crossreferences o http://planetmath.org/encyclopedia/ExampleOfDefinableType.html

5. Dedekind Cut x a } } 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.fact-index.com/d/de/dedekind_cut.html

Main Page See live article Alphabetical index Dedekind cut 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), B is closed upwards. If a is a member of S x in S x x in S x a 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 S A B less than C D A is a proper subset of C , or, equivalently D is a proper subset of B . In this way, the set of all Dedkind cuts is itself a linearly ordered set, and, moreover, it does have the least-upper-bound property, i.e., its every nonempty subset that has an upper bound has a least upper bound. Embedding S within a larger linearly ordered set that does have the least-upper-bound property is the purpose. The Dedekind cut is named after Richard Dedekind , who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers . A typical Dedekind cut of the rational numbers is given by A a in Q a < 2 or a B b in Q b b Generalization: Dedekind completions in posets More generally, in a

6. Construction Of Real Numbers 1. Construction by dedekind cuts. Real numbers can be constructed as Dedekindcuts of rational numbers. Construction by decimal expansions. http://www.fact-index.com/c/co/construction_of_real_numbers.html

Main Page See live article Alphabetical index Construction of real numbers Real numbers can be constructed from rational numbers in various ways: Table of contents 1 Construction from Cauchy sequences 2 Construction by Dedekind cuts 3 Construction by decimal expansions 4 Construction from ultrafilters ... 5 Construction from surreal numbers Construction from Cauchy sequences If we have a space where Cauchy sequences are meaningful (such as a metric space , i.e., a space where distance is defined, or more generally a uniform space ), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completion ). By starting with rational numbers and the metric d x y x y p -adic numbers instead.) Let R be the set of Cauchy sequences of rational numbers. Cauchy sequences ( x n ) and ( y n ) can be added, multiplied and compared as follows: x n y n x n y n x n y n x n y n x n y n N such that x n y n n N Two Cauchy sequences are called equivalent if the sequence ( x n y n ) has limit 0. This does indeed define an equivalence relation , it is compatible with the operations defined above, and the set

7. Dedekind Cuts. Up Contents. Next Real numbers, other definitions. dedekind cuts. Wecall the set of all dedekind cuts, the set of reals, R. http://hemsidor.torget.se/users/m/mauritz/math/num/real.htm

Created 980227. Last change 980728. Previous : Irrational Numbers, Algebraic Numbers . Up : Contents . Next : Real numbers, other definitions Dedekind cuts. If we choose a rational number q, we can use this to split the rationals in two sets, one larger than the number q, and one smaller than q. The number q itself can we choose to be in the upper. The lower set is now defined as Q In general if we have a relation p(x) such that it can be used to divide all rationals in two sets where all numbers in one of the sets are larger than all numbers in the other set we can use this to define a cut of the rationals. One of the two sets is then, Q and the other is the complement set to this set. We then write the cut as, c where the other part of the set is implicitly defined. We can omit the ' Q ' because the cut is per definition over the rationals. Such a cut can now be of three kinds, either, as the first one we looked at, a cut where the upper set has a lowest rational number, or a cut where the lowest set has a highest rational number, and finally, a cut where neither set has a highest or

8. The Reals. As you may recall, one way of defining the reals was by using Dedekind scuts. We do now define the the real numbers to be a dedekind cuts. http://hemsidor.torget.se/users/m/mauritz/math/num/setreal.htm

Created 980623. Last change 980731. Previous : Rational Numbers . Up : Contents . Next : Complex Numbers The Reals. The reals are substantially different from the others. As you may recall, one way of defining the reals was by using Dedekind's cuts . A Dedekind's cut is a set or rational numbers such that, 1 : All numbers in the complement to the cut are larger than any number in the cut. 2 : The cut has no largest element. Q c will thus be a Dedekind's cut. We do now define the the real numbers to be a Dedekind cuts. b, and that a=b if and only if the sets are equal. We can embed the rational numbers in the reals by, c And we can define arithmetic on the reals. We could also define a real using a Cauchy sequence . A Cauchy sequence is a sequence, x ,x ,x ,...such m -x n one and only one number, and that is the number the sequence defines. We will use this method when constructing a set of reals. The set of Reals, R This is a most tricky one. The main problem is that most 'thinkable' reals are not constructable. A constructable, c will be a Dedekind's cut.

9. Theorem: R Via Dedekind Cuts Theorem R via dedekind cuts. The proof is not done, sorry. Context Context.Interactive Real Analysis, ver. 1.9.3 (c) 19942000, Bert G. Wachsmuth. http://www.shu.edu/projects/reals/infinity/proofs/r_dedek.html

Theorem: R via Dedekind Cuts The proof is not done, sorry. Context Interactive Real Analysis , ver. 1.9.3 (c) 1994-2000, Bert G. Wachsmuth

10. Dedekind Cuts Previous page (Some Early History of Set Theory), Contents, Next page (Farey sequences).dedekind cuts. Such a pair is called a Dedekind cut (Schnitt in German). http://www.gap-system.org/~john/analysis/Lectures/A3.html

MT2002 Analysis Previous page (Some Early History of Set Theory) Contents Next page (Farey sequences) 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

11. Some Early History Of Set Theory MT2002 Analysis Previous page (Some definitions of the concept of continuity),Contents, Next page (dedekind cuts). Some Early History of Set Theory. http://www.gap-system.org/~john/analysis/Lectures/A2.html

MT2002 Analysis Previous page (Some definitions of the concept of continuity) Contents Next page (Dedekind cuts) Some Early History of Set Theory Problems with definitions of what a set could be go back very far. Note that since George Boole (1815 - 1864) showed how Set Theory and Logic are really the same thing, these problems can crop up in a logical background. The first to spot a difficulty was the Greek philosopher Epimenedes in the 6th Century BC. He said, All Cretans are liars but since he was himself a Cretan one may deduce that if he was telling the truth, he was lying and vice versa. This attracted a lot of debate and became well-known enough to be referred to by St Paul (in his Epistle to Titus ). It was made into a clearer paradox by the 4th Century BC philosopher Eubulides who said, This statement is false which leads to the same kind of problem. It might seem that these problems of "self-reference" are not too serious, but when in the 19th Century mathematicians tried to define sets of elements by statements like X x x satisfies some property P For example, let

12. Dedekind Cut Analysis about the problems, but between the time in the course (projects were also due soon),the lack of student understanding about the place of dedekind cuts in the http://gallery.carnegiefoundation.org/cbennett/Picture08/anal.htm

Analysis In analyzing what happened with the Dedekind cut section of the course, I find myself taking two separate tracks. First, there is what I wrote at the time that the homework assignments were turned in. Namely, Tuesday December 5, 2000: So what went wrong? I think one key issue was that the problems themselves needed a better framing. Not so as to get them to write what I wanted, but rather, I think I overestimated where the students were on understanding and proof on this concept. I suspect a better way to start problems 2 and 3 would be to ask them to use technology explicitly at first, and then force them to consider mathematically framing the arguments so as to establish generic bounds on errors. All of that said, I think the Dedekind Cut section worked surprisingly well after seeing the final exams and having the exit interview with the students. In particular, three of the students chose to do the Dedekind cut question on the final, which is much higher than usual. Moreover, two of the students actually told me that they found the Dedekind cut section one of the best in the class, because it tied everything together. In particular, John said in a discussion of covering Dedekind cuts: It (Dedekind cuts) was really cool, awesome

13. Dedekind Cuts - Pedagogical dedekind cuts. Pedagogical Reasoning. At the classical definitions. HereI will emphasize the reasoning behind how I present dedekind cuts. http://gallery.carnegiefoundation.org/cbennett/Picture08/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.

14. 2.15.1 Dedekind Cuts 2.15.1 dedekind cuts. A real number is represented by a cut , . Every cuthas the property that for all As presented, the cut represents . http://www.dgp.utoronto.ca/~mooncake/thesis/node61.html

Next: 2.15.2 Cauchy Sequences Up: 2.15 Real Representations Previous: 2.15 Real Representations 2.15.1 Dedekind Cuts A real number is represented by a cut . Every cut has the property that for all As presented, the cut represents . Disallowing this special cut gives a representation for all non-negative real numbers. In general, Most numbers have a representation that cannot be written out directly since the representation is an infinite set. Operations on reals are inherited from the corresponding operations on rationals. For example, a binary operation on two real numbers, represented by cuts X and Y , is given by: Difficulties are encountered when generalizing this to negative real numbers. If a cut is simply redefined to be a subset of , then the product of two cuts is not a cut if the multiplicands correspond to negative numbers. See [ ] for further details concerning this representation and associated methods. Next: 2.15.2 Cauchy Sequences Up: 2.15 Real Representations Previous: 2.15 Real Representations Jeff Tupper March 1996

15. Dedekind Cuts Of Partial Orderings dedekind cuts of partial orderings. dedekind cuts are a clever trickfor defining the reals given the rationals. Such a cut considers http://www.cap-lore.com/MathPhys/Cuts.html

Dedekind cuts of partial orderings We may take any partial ordering and consider such cuts. The result is always a lattice. There is another different unique (within isomorphism) lattice associated with any partial ordering. There is for any partial ordering some unique smallest lattice in which it is embedded. The lattice may contain new elements but the new ordering, restricted to the old PO will contain no new orderings. This construction is also found in security considerations. The orange book provides a theory of security classifications that implicitly defines a lattice. In a particular computer system it is likely that some of the lattice values will be unused. This may cause some confusion. It should not any more than noting that the boolean or command of the CPU need not in an application produce all possible values in order to make the set of all possible values a useful concept with which to reason. It is the same with the lattice of security classifications. When the partial ordering is finite and total the cuts add nothing of interest.

16. 408 Course Notes Mar. 16, Karen, Countability of algebraic integers, Mar. 18, Brian, dedekind cuts,Mar. 30, Cynthia, Continued Fractions, Apr. 1, Dan, Quaternions and rotations,Apr. http://homepage.mac.com/vogtmann/projects.html

Class Project This will consist of a report to the class, in a presentation of 30 minutes to 1 hour, on a topic related to the mathematical foundations or natural mathematical extensions of topics typically taught in a high school. Each presentation should include A non-trivial proof Two or three exercises which will help the rest of the class understand the topic Further mathematical questions which arise naturally from the material presented Historical perspective on the topics presented is also encouraged. Schedule of projects DATE Student Title Mar. 16 Karen Countability of algebraic integers Mar. 18 Brian Dedekind cuts Mar. 30 Cynthia Continued Fractions Apr. 1 Dan Quaternions and rotations Apr. 6 Martin Solving cubic equations Apr. 8 Daanish Ideas for Projects 1. Decimal representations of real numbers. Explain why .99999... is equal to 1 using Dedekind's definition. Give a definition of real numbers in which these decimals represent different numbers. Explore decimal representations of real numbers in bases other than 10. 2. Notions of infinity. Explain countability and prove that the real numbers, as we constructed them, are not countable. Show that the rationals, and the constructible numbers are countable; in fact the algebraic numbers (roots of all polynomials) are countable.

17. 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

18. Re: On Defining Pi By John Conway John Conway conway@math.Princeton.EDU Date Thu, 16 Jan 1997 121808 0500 It strue of course that the problems of specifying dedekind cuts for root2 and http://mathforum.org/epigone/math-history-list/foubuhal

Re: On Defining pi by John Conway reply to this message post a message on a new topic Back to math-history-list Subject: Re: On Defining pi Author: conway@math.Princeton.EDU Date: The Math Forum

19. [math/0305122] The Elementary Theory Of Dedekind Cuts In Polynomially Bounded St The elementary theory of dedekind cuts in polynomially bounded structures. AuthorsMarcus Tressl (University of Regensburg, Germany) Comments 16 pages. http://arxiv.org/abs/math.LO/0305122

Mathematics, abstract math.LO/0305122 From: Marcus Tressl [ view email ] Date ( ): Thu, 8 May 2003 13:15:56 GMT (40kb) Date (revised v2): Fri, 9 May 2003 10:23:08 GMT (40kb) The elementary theory of Dedekind cuts in polynomially bounded structures Authors: Marcus Tressl (University of Regensburg, Germany) Comments: 16 pages. The paper is a sequel to this http URL s/cutsa.ps Subj-class: Logic MSC-class: 03C (primary) Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M, which yields a description of all subsets of M^n, definable in the expanded structure. Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv math find abs

20. Dedekind Cut Dedekind Cut. Real Numbers can be defined using either dedekind cuts or CauchySequences. See also CantorDedekind Axiom, Cauchy Sequence. References. http://icl.pku.edu.cn/yujs/MathWorld/math/d/d068.htm

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 What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 71-72, 1996. Eric W. Weisstein

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