Real Paradoxes russell's paradox. russell's paradox is an example of a genuine selfreferringstatement, therefore it becomes particularly interesting in my investigation. http://www.algonet.se/~jen-tale/para_rea.html
Extractions: Date: Fri, Jan 18 2002 Next message: Jeff Heflin: "Original Slides from General Requirements Group" To: jjc@hplb.hpl.hp.com Cc: www-webont-wg@w3.org Message-Id: <20020118091722L.pfps@research.bell-labs.com> Date: Fri, 18 Jan 2002 09:17:22 -0500 From: "Peter F. Patel-Schneider" < pfps@research.bell-labs.com > Subject: Re: Patel-Schneider Paradox ... From: "Jeremy Carroll" < jjc@hplb.hpl.hp.com Next message: Jeff Heflin: "Original Slides from General Requirements Group" Previous message: Peter F. Patel-Schneider: "Re: antifoundation, flat, wellfounded"
Extractions: Date: Fri, Jan 18 2002 Next message: Frank van Harmelen: "Re: F2F: The requirements vote" From: "Jeremy Carroll" < jjc@hplb.hpl.hp.com www-webont-wg@w3.org Next message: Frank van Harmelen: "Re: F2F: The requirements vote" Previous message: Jeremy Carroll: "antifoundation, flat, wellfounded" Next in thread: Peter F. Patel-Schneider: "Re: Patel-Schneider Paradox ..." Reply: Peter F. Patel-Schneider: "Re: Patel-Schneider Paradox ..." Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Other mail archives: [this mailing list] [other W3C mailing lists] Mail actions: [ respond to this message ] [ mail a new topic ]
Department Of Mathematics And Statistics Bertrand russell's paradox. Dr Bertrand russell's paradox discoveredin 1901 surfaces at the very beginning of set theory. Several http://www.math.uregina.ca/seminars/j-20020212.html
Extractions: Professor Emeritus, A set or aggregate may be defined as a collection of some kind of other. In mathematics, the objects are usually mental objects or concepts; i.e. concepts and ideas. George Cantor (1845-1918) developed the sets in the latter part of the 19th century. Other mathematicians helped expand Cantor's work into a serious discipline, because it was seen that many branches of emerging mathematics depended to some extent on the notion and properties of sets. A critical scrutiny of the results obtained soon revealed contradictions. Antinomies appeared in the very advanced parts of set theory. Bertrand Russell's paradox discovered in 1901 surfaces at the very beginning of set theory. Several attempts have since been made to set up rules and axioms which would block Russell's paradox. No general method has yet been found that would guarantee absolute freedom from contradictions. It can be claimed, however, that contradictions are not encountered in the uses to which the theory of sets put in mathematics. In today's talk we will introduce the paradox put forward by Russell and analyse it with a view to suggesting means of avoiding it.
Kevin C. Klement: CV russell's paradox in Appendix B of the Principles of Mathematics Was Frege'sResponse Adequate? History and Philosophy of Logic 22 (2001) 1328. http://www-unix.oit.umass.edu/~klement/cv.html
Extractions: B.A. ( Philosophy / Peace Studies), University of Minnesota, Morris , June 1995 Languages: (in order of proficiency) English, German, French Scholarly Works: Ph. D. Dissertation "Redressing Frege's Failure to Develop a Logical Calculus for the Theory of Sinn and Bedeutung " (Advisor: Gregory Landini , University of Iowa, 2000.) Books
Owen Massey - Libraries - Maths For Librarians induction. 3. russell's paradox. The Library. See Francis Moorcroft,'russell's paradox' in The philosopher's magazine 3, 1998. AD http://owen.massey.net/libraries/maths.html
Extractions: Tracy Chapman Fast car Librarianship in the United Kingdom is now a graduate profession: it is necessary (though far from sufficient) to have a degree, whether in librarianship or a more conventional academic subject. Many people, when they've finished laughing, have expressed surprise that entry to the profession needs to be guarded so jealously. This page is a first attempt to indicate how my first degree in mathematics is highly relevant to my vocation as a librarian - and vice versa. 1. Information science and information theory
Web Links For Chapter 1 Page 45. The online Stanford Encyclopedia of Philosophy contains anexcellent discussion of russell's paradox. (russell's paradox). http://www.mhhe.com/math/advmath/rosen/student/webres/ch1links.mhtml
Extractions: Web Links for Chapter 1 Section 1.1 Logic Page 2 Information on logic and its applications can be found at http://www.rbjones.com/rbjpub/logic/index.htm Page 3 An extensive biography of George Boole, including a portrait, can be found at the Roger P@rsons_world of Lincolnshire site. http://homepages.enterprise.net/rogerp/george/boole.html (Roger P@rsons_world of George Boole) A biography and a portrait of George Boole can be found at the MacTutor History of Mathematics Archive at the University of St Andrews, Scotland. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Boole.html (Boole) Another source for biographical information and a portrait of Boole is http://www.digitalcentury.com/encyclo/update/boole.html#return (Jones Telecommunications and Multimedia Encyclopedia) A copy of George Boole's pioneering article "The Calculus of Logic," published in 1848, can be seen at the History of Mathematics Archive at the School of Mathematics, Trinity College, Dublin, Ireland. http://www.maths.tcd.ie/pub/HistMath/People/Boole/CalcLogic/ Page 9 Information on Boolean searching can found at the following sites: http://www.sps.edu/Academics/AIS/Library/Hypertext_tutorial/boolean.shtml
Extractions: An intuitivistic solution of the Continuum Hypothesis for definable sets and resolution of the set theoretic paradoxes numbers that can be described in a finite number of words of the English language! This apparent contradiction that on the one hand a set such as R cannot be said to exist but on the other hand R can be logically conceived as the set I, forces us to the startling conclusion that whereas a set such as R cannot be said to exist, it can be logically conceived! As mentioned earlier, our point of departure from the earlier held view is that WHEREAS THE STRONG ASSUMPTION 'R CAN BE DESCRIBED IN TERMS OF THE EXPRESSION BY WHICH IT IS LOGICALLY CONCEIVED' IS YET CONSIDERED EQUIVALENT TO THE ASSUMPTION 'R EXISTS', THIS IS NOT SO FOR THE WEAK ASSUMPTION 'R CAN BE LOGICALLY CONCEIVED', SO THAT IT IS NOT CONTRADICTORY TO MAINTAIN THAT ALTHOUGH 'R EXISTS' CANNOT BE SAID, 'R CAN BE LOGICALLY CONCEIVED' CAN BE SAID! Note that unlike the situation in Berry's paradox wherein the set F could actually be produced by listing its elements, the set I by comparison, can neither be produced nor defined: In fact, the assumption that I is definable/ describable, yields the contradiction that is Richard's paradox.
Russel's Paradox essay . russell's paradox Philosophical Ponderings of a Confused DiscreteStudent. How does this tie in with russell's paradox? Okay http://jhunix.hcf.jhu.edu/~blee27/essays/russels_paradox.htm
Extractions: So, who is this Russell dude anyway, and what is his paradox? Russell was a mathematician that introduced an interesting concept sometime about the time the 19th century was rolling into the 20th. He asked us to consider a question. Something along the lines of, "Does the set of all sets that do not contain themselves contain itself?" A mouthful. How do we answer such a question? Let X be the set of all sets that do not contain themselves. The question now becomes "Is X a member of X?"
Maths Thesaurus: Russell's Paradox Home russell's paradox The paradox which prompted Bertrand Russell andothers to rethink the theory of sets, early in the 20th century http://thesaurus.maths.org/dictionary/map/word/1371
Extractions: The paradox which prompted Bertrand Russell and others to re-think the theory of sets, early in the 20th century: A set can contain other sets as members. So it is possible for a set to have a member which is itself. A normal set is a set which does not contain itself. Now think of the set of all normal sets. Is it a normal set, or not?
Math Lair - Paradoxes Zeno's Paradox russell's paradox One can classify sets into one of two categories. Greeling'sParadox A version of russell's paradox using words. http://www.stormloader.com/ajy/paradoxes.html
Extractions: A paradox is a statement that goes against our intuition but may be true, or a statement that is self-contradictory. The paradoxes listed below and most other mathematical paradoxes fall into one of two categories: either they result from the counter-intuitive properties of infinity , or are a result of self-reference. Zeno's Paradox Russell's Paradox One can classify sets into one of two categories. The first category is sets that are not members of themselves. This contains most of the sets we run into in "real life". For example, the set of all penguins falls in the first category, because the set of all penguins is a set, not a penguin. On the other hand, some sets are members of themselves. The set of all non-penguins, for example, is a member of itself. So is the set of all sets. In which category would we find the set of all sets that are not members of themselves? If this set is not a member of itself, then it is a member of itself. If it is, then it isn't. So, this set is a member of itself if and only if it is not a member of itself, which is the paradox. This is similar in concept to the Cretan Liar paradox. An article about Russell's Paradox at the Stanford Encyclopedia of Philosophy Greeling's Paradox A version of Russell's Paradox using words. Some adjectives are self-descriptive, like "tiny", "unhyphenated", and "pentasyllabic". On the other hand, other adjectives are not self-descriptive, like "bisyllabic", "big", "tasty", and "incomplete". Call the self-descriptive adjectives
Russell's Antinomy -- From MathWorld russell's Antinomy, Bertrand Russell discovered this paradox and sent it in aletter to G. Frege just as Frege was completing Grundlagen der Arithmetik. http://mathworld.wolfram.com/RussellsAntinomy.html
Extractions: Let R be the set of all sets which are not members of themselves. Then R is neither a member of itself nor not a member of itself. Symbolically, let . Then iff Bertrand Russell discovered this paradox and sent it in a letter to G. Frege just as Frege was completing Grundlagen der Arithmetik. This invalidated much of the rigor of the work, and Frege was forced to add a note at the end stating, "A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press." Barber Paradox Catalogue Paradox Grelling's Paradox
Russell's Real Paradox: The Wise Man Is A Fool russell's Real paradox The Wise Man Is a Fool. Philip J. Davis. SIAMNews, Volume 26, Number 6, July 1994. Bertrand Russell A Life http://www.siam.org/siamnews/bookrevs/davis794.htm
Extractions: Reading this confession in Moorehead's excellent biography, I wondered just where Russell had picked up these factors. Was it as a young student, cramming for admission to Cambridge? I wondered whether he knew that this expression is the determinant of the $3 x 3$ circulant matrix whose first row is $[a, b, c]? And did he know that the factors, linear in $a, b, c,$ are the three eigenvalues of the matrix? Did he know that this factorization was historically the seed that, watered by Frobenius, grew into the great subject of group representation theory? I conjecture that he did not. To Russell, the algebraic expression was a mantra. He saw mathematics as the stabilizing force in the universe; it was the one place where absolute certainty reigned. In search of this certainty, groping for it, he said, as one might grope for religious faith, he devoted the first half of a very long life to an attempt to establish the identity of mathematics and logic. I first heard of Russell as an undergraduate. I did a chapter of Principia Mathematica, his masterwork, written (1910-1913) with Alfred North Whitehead, as a reading period assignment in a course in mathematical logic. At that time Russell was a celebrity, front-page news, having left the dots and epsilons and the "if-thens" of logic far behind. He had been appointed to a professorship of philosophy at CCNY in 1940, and almost immediately a charge of immorality was laid against him. It hit the papers. I, together with most undergraduates, sided with John Dewey, Albert Einstein, and Charlie Chaplin as they rushed in to defend Russell's right to teach epistemology.
