1. Dedekind Cut -- From MathWorld Real numbers can be defined using either dedekind cuts or Cauchy sequences.SEE ALSO CantorDedekind Axiom, Cauchy Sequence. Pages Linking Here http://mathworld.wolfram.com/DedekindCut.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team 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 SEE ALSO: Cantor-Dedekind Axiom Cauchy Sequence [Pages Linking Here] 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. CITE THIS AS: Eric W. Weisstein. "Dedekind Cut." From

2. PlanetMath Dedekind Cuts dedekind cuts (Definition) The purpose of dedekind cuts is to provide a sound logical foundation for the real number system. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

3. PlanetMath: Dedekind Cuts The purpose of dedekind cuts is to provide a sound logical foundation for thereal number In the construction of the real numbers from dedekind cuts, 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 ... Classification 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

4. Dedekind One remarkable piece of work was his redefinition of irrational numbers in terms of dedekind cuts which, as we mentioned above, first came to http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. PlanetMath: Dedekind Cuts The purpose of dedekind cuts is to provide a sound logical foundation for thereal number dedekind cuts are particularly appealing for two reasons. http://planetmath.org/encyclopedia/Schnitt.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 ... Classification 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

6. Math Forum - Ask Dr. Math dedekind cuts. 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 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Dedekind Cut - Wikipedia, The Free Encyclopedia It is more symmetrical to use the (A,B) notation for dedekind cuts, For exampleit is shown that the typical Dedekind cut in the real numbers is either http://en.wikipedia.org/wiki/Dedekind_cut

Dedekind cut From Wikipedia, the free encyclopedia. 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 Ordering Dedekind cuts The cut construction of the real numbers Additional structure on the cuts ... edit 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 , +∞); or a pair with A the interval a ], in which case

8. Dedekind, Richard study of CONTINUITY and definition of the real numbers in terms of Dedekind _QUOTATION_cuts_QUOTATION_, the nature of number and mathematical http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. Dedekind Cuts. dedekind cuts. If we choose a rational number q, we can use this to split therationals in We call 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

10. Dedekind Cuts. Previous Irrational Numbers, Algebraic Numbers. Up Contents. Next Real numbers, other definitions. dedekind cuts. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. The Reals. The set {x Q x 2}={x x 2}c will thus be a Dedekind s cut. We do now definethe 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.

12. Dedekind Cut Analysis In particular, John said in a discussion of covering dedekind cuts It (dedekind cuts) was really cool, awesome, and nobody challenged him. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. The Elementary Theory Of Dedekind Cuts (ResearchIndex) @misc{ tresslelementary, author = "Marcus Tressl", title = "The Elementary Theory of dedekind cuts", url = "citeseer.ist.psu.edu/423172.html" } http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

14. Dedekind Cuts dedekind cuts. The first construction of the Real numbers from the Rationals isdue to the Such a pair is called a Dedekind cut (Schnitt in German). http://www-history.mcs.st-and.ac.uk/~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

15. 2.15.1 Dedekind Cuts 2.15.1 dedekind cuts http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

16. Farey Sequences Previous page (dedekind cuts), Contents Next page (Cardinal numbers) Previous page (dedekind cuts), Contents Next page (Cardinal numbers) http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/A4.html

MT2002 Analysis Previous page (Dedekind cuts) Contents Next page (Cardinal numbers) Farey sequences The set Q of rational numbers was proved countable in an earlier section . Here is a different way of seeing the same thing. Define the Farey sequence F n to be the ascending sequence of fractions in the interval [0, 1] whose denominators are n F F F F F One nice thing about these sequences is the way they can be made. To get the sequence F n from the sequence F n , take an adjacent pair of fractions h k and h k (say) and provided k k n , insert the mediant fraction h h k k ) between them. So, for example, to get F from F , insert between and and insert between and Some results about Farey sequences You will be able to verify these for the examples above, but will probably not be able to prove them. If h k and h k are successive fractions in a Farey sequence, then k h h k Given any three successive terms in a Farey sequence, the middle one is the mediant (see above) of the outer two. ( If h k and h k are successive fractions in the Farey sequence F n , then

17. Dedekind Cuts Math reference, dedekind cuts. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

18. \documentclass[a4paper]{article} \usepackage{amssymb} \pagestyle{ textwidth 15.3cm \textheight 22cm % to fit our printers \begin{document} \begin{center}{\huge On dedekind cuts in Polynomially Bounded }\end http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

19. Construction Of Real Numbers - Definition Of Construction Of Real Numbers In Enc The quotient of two dedekind cuts, in case {\mathbf \forall b_x \in B_x 0 \leqb_x \and. {\mathbf (A_x, B_x) / (A_y, B_y) = ( { (\{ a_\mathrm{quot} \in http://encyclopedia.laborlawtalk.com/Construction_of_real_numbers

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law In mathematics , there are a number of 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 all of these models are built using the basic properties of the rational number system as an ordered field. Contents showTocToggle("show","hide") 1 Synthetic approach 2 Explicit constructions of models 2.1 Construction from Cauchy sequences 2.2 Construction by Dedekind cuts ... 2.5 Construction from surreal numbers Synthetic approach The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R , two distinct elements and 1 of R , two binary operations + and * on R (called addition and multiplication , resp.), a

20. Dedekind Cuts Math reference, dedekind cuts. Also, most people find cauchy sequences moreintuitive, so dedekind cuts are primarily of historical interest. http://www.mathreference.com/top-ms,dcuts.html

Metric Spaces, Dedekind Cuts Search Site map Contact us Download this Site Main Page Topology Metric Spaces Use the arrows at the bottom to step through Metric Spaces. 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.

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