Dedekind, Richard 12, 1916, was a German mathematician known for his study of CONTINUITY and definitionof the real numbers in terms of dedekind cuts ; his analysis of the http://euler.ciens.ucv.ve/English/mathematics/dedekind.html
Dedekind's Real Numbers set of rational numbers ; Maddy1992, p. 81 ``by identifying real numbers withcertain sets (called `dedekindcuts ), dedekind misinterpretation. http://www.phil.cmu.edu/dschlimm/texts/reals.html
Extractions: email: dschlimm@andrew.cmu.edu February 22, 1999 Richard Dedekind's characterization of the real numbers as the system of cuts of rational numbers is by now the standard in almost every mathematical book on analysis or number theory. In the philosophy of mathematics Dedekind is given credit for this achievement, but his more general views are discussed very rarely and only superficially. For example, Leo Corry, who dedicates a whole chapter of his Modern Algebra and the Rise of Mathematical Structures (1996) writes: ``Dedekind defined the system of the real numbers as the collection of all cuts of rationals'' ([ ], p. 73). In this paper I will present Dedekind's own views of his ``definition'' and ``creation'' of the real numbers, and elucidate what he meant by saying that the real numbers ``correspond'' to the cuts. The upshot of this discussion will be that Corry's statement will be revealed as an obvious, but not uncommon (cf. [ ], p. 55: ``And every such cut, that corresponds to no rational number, defines an irrational number'' (my translation); [
Was Sind Und Was Sollen Die Zahlen?: Dedekind By November of 1858 dedekind had resolved the issue by showing how to obtain the andarithmetical operations) from the rational numbers by means of cuts in the http://www.thoralf.uwaterloo.ca/htdocs/scav/dedek/dedek.html
Extractions: Previous: Spectra Next: Cantor Up: Supplementary Text Topics Let us recall that by 1850 the subject of analysis had been given a solid footing in the real numbers infinitesimals had given way to small positive real numbers, the 's and In particular he was not satisfied with his geometrical explanation of why it was that a monotone increasing variable, which is bounded above, approaches a limit. By November of 1858 Dedekind had resolved the issue by showing how to obtain the real numbers (along with their ordering and arithmetical operations) from the rational numbers by means of cuts in the rationals for then he could prove the above mentioned least upper bound property from simple facts about the rational numbers. Furthermore, he proved that applying cuts to the reals gave no further extension. These results were first published in 1872, in Stetigkeit und irrationale Zahlen. In the introduction to this paper he points out that the real number system can be developed from the natural numbers: I see the whole of arithmetic as a necessary, or at least a natural, consequence of the simplest arithmetical act, of counting, and counting is nothing other that the successive creation of the infinite sequence of positive whole numbers in which each individual is defined in terms of the preceding one.
Yet Another Outline/kludgefest (fwd) system 11.1.1 real number as cut -1, 1 containing infinite cuts I thinkthat what you are designating as a cut is _not_ a dedekind cut, but I ll http://www.asifproductions.com/aleph/Jul93/msg00034.html
Extractions: Date Prev Date Next Thread Prev Thread Next ... Thread Index To Aleph@pyramid.com (Aleph) Subject : yet another outline/kludgefest (fwd) From Mitchell.Porter@lambada.oit.unc.edu Date : Wed, 30 Jun 1993 23:07:06 -0700 Cc Aleph@pyramid.com Prev by Date: [no subject] Next by Date: angels (fwd) Previous by thread: [no subject] Next by thread: angels (fwd) Index(es): Date Thread
Lista Keywords Boundary value problem. PA KRUTITSKII The skew derivative problemfor the Helmholtz equation outside cuts in a plane. dedekind domain. http://www.dmmm.uniroma1.it/~rendiconti/rol/keywords.htm
Intuition And Rigor rigorous introduction into the theory of real numbers. The claimed theoremhas been indeed proven on the foundation of dedekind s cuts. http://www.cut-the-knot.org/fta/bolzano.shtml
Extractions: Recommend this site One of the torch bearers of the formalization attempts in the 19th century was the Czech analyst Bernhard Bolzano (1781-1848.) In his critique of the attempts to prove the Fundamental Theorem of Algebra, he wrote The most common kind of proof depends on a truth borrowed from geometry, namely, that every continuous line of simple curvature of which the ordinates are first positive and then negative (or conversely) must necessarily intersect the x-axis somewhere at a point that lies in between those ordinates. There is certainly no question concerning the correctness, nor indeed the obviousness, of this geometrical proposition. But it is clear that it is an intolerable offense against correct method to derive truths of pure (or general) mathematics (i.e., arithmetic, algebra, analysis) from considerations which belong to a merely applied (or special) part, namely, geometry ... By which he of course meant that reliance on the geometrical intuition is an unacceptable tool in deriving analytic truths. He clearly accepts the statement as true but objects to the fact of its being used offhandedly, as a self-evident truth. In the article, Bolzano proceeds to justify the statement that is variably now known as Bolzano's or the Intermediate Value Theorem. His proof depends on the definition of continuity by Cauchy from which he derives the Sign Preserving Property of Continuous Functions . Assuming that at the left end of an interval the function is negative, he observes that it stays negative on a certain bounded set but not for the points near the second end of the interval. He continues
Dedekind remarkable piece of work was his redefinition of irrational numbers in terms of Dedekindcuts which, as we mentioned above, first came to him as early as 1858. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Dedekind.html
Extractions: Richard Dedekind 's father was a professor at the Collegium Carolinum in Brunswick. His mother was the daughter of a professor who also worked at the Collegium Carolinum. Richard was the youngest of four children and never married. He was to live with one of his sisters, who also remained unmarried, for most of his adult life. He attended school in Brunswick from the age of seven and at this stage mathematics was not his main interest. The school, Martino-Catharineum, was a good one and Dedekind studied science, in particular physics and chemistry. However, physics became less than satisfactory to Dedekind with what he considered an imprecise logical structure and his attention turned towards mathematics. Listing and Wilhelm Weber . The two departments combined to initiate a seminar which Dedekind joined from its beginning. There he learnt number theory which was the most advanced material he studied. His other courses covered material such as the differential and integral calculus, of which he already had a good understanding. The first course to really make Dedekind enthusiastic was, rather surprisingly, a course on experimental physics taught by
Maths@work - Famous Mathematicians Rigorously defined irrational numbers as classes of fractions using Dedekindcuts. ; Game a purely arithmetic definition of the essence of continuity. http://www.mathsatwork.com/famous_mathematicians/dedekind.html
Extractions: Julius Dedekind 1831-1916 . Born in Brunswick, Germany on October 6th, son of a jurist, profession and corporation lawyer. . Studied at Gymnasium Martino - Catharineum in Brunswick. Main interests were physics and chemistry. . Studied at Collegium Carolinum. Study included analytic geometry, algebraic analysis, calculus and higher mechanics. . Private tuition in mathematics by Hans Zincke. . Became close friends with George Riemann Gauss . Became friends with Peter Dirichlet. . Became Director of Polytechnic. . Elected correspondent member of Berlin Academy. . Promoted to Professor Emeritus. . Elected correspondent member of Paris Academy. . Received many scientific honours on 50th anniversary of doctorate. . Elected foreign member of Paris Academy.
Rhonda Roland Shearer / Marcel Duchamp's Impossible Bed... Also note that the English translation dedekind cut (Poincaré) is incorrect.Duchamp s original notes in French read Poincaré cut. . http://www.marcelduchamp.org/ImpossibleBed/PartI/note19.html
Documentclass{article} \usepackage{latexsym} \newtheorem {\it Well, almost. This is where we should start worrying about Dedekindcuts and Cauchy sequences. In fact we have disguised earlier http://www.bath.ac.uk/~masgcs/book1/amplifications/ch8q8_2.txt
Extractions: I highly recommend to anyone who has some spare time (but first finish the regular homework!) to construct the number system all the way from the beginning. In some places, this process takes the place of our class MAT 300 here we chose to have a broader focus, but I still invite you to trace out some of the major steps.
