1. Dedekind Cuts. dedekind cuts. If we choose a rational number q, we can use this to split the rationals in two sets, one larger than 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

2. 2.15.1 Dedekind Cuts 2.15.1 dedekind cuts. A real number is represented by a cut ,   http://www.dgp.toronto.edu/people/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

3. Dedekind Cuts Of Partial Orderings dedekind cuts of partial orderings dedekind cuts are a clever trick for defining the reals given the rationals. 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.

4. Dedekind Cuts dedekind cuts. The first construction of the Real numbers from the Rationalsis due to the German mathematician Richard Dedekind (1831 1916). http://www.gap-system.org/~john/analysis/Lectures/A3.html

5. 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

6. PlanetMath: Dedekind Cuts The purpose of dedekind cuts is to provide a sound logical foundation for the real number system. http://www.planetmath.org/encyclopedia/DedekindCuts.html

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports 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 property, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as

7. 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

8. Dedekind CutsThe First Construction Of The Real Numbers From The Rationals Is Du Biography of Richard Dedekind (18311916) was his redefinition of irrational numbers in terms of dedekind cuts which, as we mentioned above, first came to him as http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/A3.html

9. D D. De Morgan's laws; De Morgan, Augustus (18061871); decreasing function;dedekind cuts; Dedekind's Theorem; density principle; derivative http://www.shu.edu/projects/reals/gloss/index_d.html

D De Morgan's laws De Morgan, Augustus (1806-1871) decreasing function Dedekind Cuts ... domain Interactive Real Analysis , ver. 1.9.3 (c) 1994-2000, Bert G. Wachsmuth

10. Dedekind Cuts By Todd Trimble Subject dedekind cuts Author todd trimble trimble@math.luc.edu Date 3 Oct 98 185819 0400 (EDT) The idea is that http://forum.swarthmore.edu/epigone/alt.math.undergrad/dunyobe/x6e65pivbnzw@foru

dedekind cuts by todd trimble reply to this message post a message on a new topic Back to messages on this topic Back to alt.math.undergrad Subject: dedekind cuts Author: trimble@math.luc.edu Date: The Math Forum

11. 2.15 Real Representations next up previous notation contents Next 2.15.1 dedekind cuts Up2 Numbers Previous 2.14 Variants 2.15 Real Representations. To http://www.dgp.toronto.edu/people/mooncake/thesis/node60.html

Next: 2.15.1 Dedekind Cuts Up: 2 Numbers Previous: 2.14 Variants 2.15 Real Representations To perform computations with the aid of digital computers we must build the reals out of a discrete system. Mathematicians have historically built up the reals with different approaches; this section details some of these approaches. Some of these approaches lead to mechanical algorithms which may be contrasted with the interval approach. Readers interested solely in the interval approach should proceed to the next chapter. Next: 2.15.1 Dedekind Cuts Up: 2 Numbers Previous: 2.14 Variants Jeff Tupper March 1996

12. Dedekind Cut - Wikipedia The Dedekind cut is named after Richard Dedekind, who invented this constructionin order to represent the real numbers as dedekind cuts of the rational numbers http://www.wikipedia.org/wiki/Dedekind_cut

13. Real Number - Wikipedia This sense of completeness is most closely related to the construction of the realsfrom dedekind cuts, since that construction starts from an ordered field http://www.wikipedia.org/wiki/Real_number

14. Dedekind's Cuts 3 Oct 1998 dedekind cuts, by todd trimble http://mathforum.com/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

15. Math And Physics Transformations on vector spaces Implicit Methods Code for Fields in a computerHomotopy and Homotopy groups dedekind cuts of partial orderings Stowaways See http://www.cap-lore.com/MathPhys/

Math Imagining S(3) Turning in 3D Besikovich's problem Riemann programs Tiling Constant Curvature Spaces Elementary Geometry on a Complex ... Why Christoffel symbols are not tensors An ordinary introduction to abstract vector spaces Linear Transformations on vector spaces Implicit Methods Code for Fields in a computer Homotopy and Homotopy groups Dedekind cuts of partial orderings Stowaways See "Math Connection" in here Type equivalence , Infinite trees, Anti Foundation set theory Axioms for Sets Bug in Banach Crystals Idea The Platonic Polytope (applet) Area of Intersection of Polygons Conventional Sphere Packing A Functional Equation Factoring with the Quadratic Sieve Constructing a Brownian Function Some Baysian statistics and Likelihood note Counting constrained bit arrays Critical Density of Links Covariant and Contravariant ; Tensors and Types Correctness of Fast Fourier Transform Computing PI A glimpse at p-adic numbers Physics Outside Pages Relativity section Kinetic energy The Structure of Diamond and ice Vortices Notes on Tegmark's TOE 3D Elasto Dynamics Code Spinning Plastic Idea for ultra high resolution radio interferometry Six Axis Stress Gauge and Virtual touchy-feely The Lagrange Points Close Encounters Quantum Computers Fyzzy: Crazy Connections concerning Simultaneous Alternatives Edge of Chaos and Economics Understanding Things are as they are because they were as they were.

16. 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

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports 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 property, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as

17. PlanetMath: decomposition (complementary subspace) owned by rmilson. decomposition group ownedby djao. dedekind cuts owned by rmilson. Dedekind domain owned by saforres. http://planetmath.org/encyclopedia/D/

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports information Docs Classification News Legalese ... TODO List PlanetMath Encyclopedia (browse by subject) - 1 entry - 1 entry - 1 entry - 1 entry A - 215 entries B - 164 entries C - 534 entries D - 223 entries DAG (acyclic graph) owned by Logan Darboux's theorem (analysis) owned by mathwizard proof of Darboux's theorem owned by ariels Darboux's Theorem (symplectic geometry) owned by bwebste DCC (descending chain condition) owned by antizeus DCT (discrete cosine transform) owned by akrowne de Bruijn digraph owned by vampyr decide (decision problem) owned by Henry decision problem owned by Henry deck transformation owned by Dr_Absentius deck transformation (deck transformation) owned by Dr_Absentius decomposition (complementary subspace) owned by rmilson decomposition group owned by djao Dedekind cuts owned by rmilson Dedekind domain owned by saforres Dedekind-Hasse norm (Dedekind-Hasse valuation) owned by Henry Dedekind-Hasse valuation (Dedekind-Hasse valuation) owned by Henry Dedekind-Hasse valuation owned by Henry Dedekind infinite owned by Evandar Dedekind zeta function owned by bwebste deduction (deductions are ) owned by Henry deductions are delta 1 (deductions are ) owned by Henry deductions are owned by Henry definable owned by Timmy

18. 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.

19. Reals Via Dedekind Cuts Theorem Real Numbers as dedekind cuts. The proof isnot done, sorry. To Theory Glossary Map (bgw). http://pirate.shu.edu/projects/reals/infinity/proofs/r_dedek.html

Theorem: Real Numbers as Dedekind Cuts The proof is not done, sorry. Map (bgw)

20. Interactive Real Analysis Glossary D. De Morgan, Augustus (18061871); De Morgan's laws; decreasing function;dedekind cuts; Dedekind's Theorem; density principle; derivative http://pirate.shu.edu/projects/reals/gloss/index_d.html

D De Morgan, Augustus (1806-1871) De Morgan's laws decreasing function D De Morgan, Augustus (1806-1871) De Morgan's laws decreasing function ... domain

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