81. Science Forum - Dedekind Cuts Make Dedekind cuts of the ordered set of rationals, and note that, as one of the resulting subsets is open at the cut, then that set will contain the http://thescienceforum.com/Dedekind-cuts-684t.php

The Science Forum - Scientific Discussion and Debate Live Chat FAQ Search Usergroups ... Dedekind cuts Goto page Next  Dedekind cuts View previous topic View next topic Author Message Guitarist Posted: Wed Jun 29, 2005 12:40 pm    Post subject: Dedekind cuts Moderator Joined: 08 Jun 2005 Posts: 1125 Normally, I avoid number theory, but here's a nifty one I discovered recently. Take the real number line, R, and cut it any any point. Let's say at 10. Then we have two subsets of R, let's call them L and U (for upper and lower cuts). Then, obviously, 10 must either be the upper bound of L or the lower bound of U. Let's say the latter. Thus L is open on the right, and U is closed on the left. This means that, having made the cut, we can never glue it back together again, for the upper bound of L can only ever approach abitrarily close to 10. I think that's pretty cool. Apparently it says something profound about the rational and irrationals. If I can bear more number theory, I'll dig a bit more. Back to top Posted: Wed Jun 29, 2005 1:22 pm    Post subject: Forum Ph.D.

82. Dedekind Cut - Need Help With Proof 2 posts 1 author(p is a positive rational number x is a Dedekind cut) - Eq(qex ~ p +q e x) First note that a Dedekind cut is not all of the rationals -this http://www.mathkb.com/Uwe/Forum.aspx/math-logic/3955/Dedekind-cut-need-help-with

83. Dedekind Cut - Definition Of Dedekind Cut By The Free Online Dictionary, Thesaur (Mathematics) a method of according the same status to irrational and rational numbers, devised by Julius Wilhelm Dedekind (18311916) http://www.thefreedictionary.com/Dedekind cut

84. Information 2 posts 1 author - Last post Oct 10show that {r IN Q r^3 2} is a dedekind cut. {r in Q r^3 2} = {r in Q r ( 2^(1/3))} i) http://www.mymathforum.com/viewtopic.php?f=22&t=16303

85. JSTOR: An Error Occurred Setting Your User Cookie Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.jstor.org/stable/1995137

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86. The Riemann Conjecture And The Advanced Calculus Methods For File Format PDF/Adobe Acrobat Quick View http://www.worldscibooks.com/etextbook/6856/6856_riemann.pdf

87. Dedekind, Richard Apr 2, 2008 While teaching calculus for the first time at the ETH Zürich Polytechnic, Dedekind came up with the notion now called a Dedekind cut (in http://www.newworldencyclopedia.org/entry/Richard_Dedekind

Dedekind, Richard From New World Encyclopedia Jump to: navigation search Previous (Richard Daley) Next (Richard Felton Outcault) Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was one of the major German mathematicians in the late nineteenth century who did important work in abstract algebra, algebraic number theory, and laid the foundations for the concept of the real numbers. He was one of the few mathematicians who understood the importance of set theory developed by Cantor Dedekind argued that the numbers system can be independently developed from geometrical notations and that they are grounded in and derived from a certain inherent creative capacity of the mind, which were some of those issues debated by Bolzano, Cantor Frege , and Hilbert. Contents Life Work Quotation Notes ... Credits Life Dedekind was the youngest of four children of Julius Levin Ulrich Dedekind. He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English). In 1848, he entered the Collegium Carolinum in Braunschweig, where his father was an administrator, obtaining a solid grounding in mathematics. In 1850, he entered the University of Göttingen. Dedekind studied number theory under Moritz Stern. Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled

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