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
Extractions: 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
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
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
Extractions: 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
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
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
Extractions: 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
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
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
Extractions: 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 edit 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 , +â); or a pair with A the interval a ], in which case
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
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
Extractions: 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
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
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
Extractions: 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
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
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
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
Extractions: (Farey sequences) 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. 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
Extractions: 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 2.5 Construction from surreal numbers 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
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
Extractions: 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.
