41. Barber Paradox - Wikipedia, The Free Encyclopedia The Barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell himself as an illustration of the paradox, though he attributes it to an unnamed http://en.wikipedia.org/wiki/Barber_paradox

Barber paradox From Wikipedia, the free encyclopedia Jump to: navigation search This article is about a paradox of self-reference. For an unrelated paradox in the theory of logical conditionals with a similar name, introduced by Lewis Carroll , see the Barbershop paradox The Barber paradox is a puzzle derived from Russell's paradox . It was used by Bertrand Russell himself as an illustration of the paradox , though he attributes it to an unnamed person who suggested it to him. It shows that an apparently plausible scenario is logically impossible. Contents The paradox History In Prolog In first-order logic ... edit The paradox Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves. Under this scenario, we can ask the following question: Does the barber shave himself? Asking this, however, we discover that the situation presented is in fact impossible:

42. (Blog&~Blog): Russell's Paradox, The Liar, The Barber And The Inclosure Schema ( Aug 2, 2010 In Parts I and II, I responded to the bulletbiting approach to Russell s Paradoxthat it shows that there really are true http://blogandnot-blog.blogspot.com/2010/08/russells-paradox-liar-barber-and.htm

Scattered Notes On Logic, Truth And Paradox ~ Updated Every Monday And Wednesday Blog Archive October Fun With Midterms A Quick Question For People Who Might Know More Th... Fun With Kant ... November About Me Ben Burgis I'm a PhD candidate in the Philosophy Department at the University of Miami (working dissertation title: "Truth Is A One-Player Game: A Defense of Monaletheism and Classical Logic"), a low-res MFA student in Creative Writing at the University of Southern Maine (Stonecoast), and the author of a a bunch of stuff . I also go out at night to fight crime, under the alias "the Caped Logician." Actually, that last sentence was a lie. (As is this one.) But, according to some people , I am an enemy of "Christendom." View my complete profile Monday, August 2, 2010 Russell's Paradox, the Liar, the Barber and the Inclosure Schema (The Russell's Paradox Series, Part III of IV) In Parts I and II All well and good, someone might argue, but these sorts of considerations clearly won't help us with the Liar Paradox. Any epistemic access we might have to sets is indirect and holistic, fair enough, but sentences are clearly a different matter. Thus, any plausible consistent solution to the Liar Paradox must surely be different from the solution I've offered here for Russell's Paradox. Given the deep structural similarity between the Liar and Russell's Paradox, though, aren't they both 'of a type'? And doesn't that mean that the right solution to the two paradoxes must be 'unified'?

43. What Is Bertrand Russell's Paradox?: The Theory Of Types And The Unrestricted Co Oct 2, 2009 Russell s paradox came about as a result of his concern with set theory s unrestricted comprehension axiom, and it s an important http://www.suite101.com/content/bertrand-russells-paradox-a153203

44. Russell's Paradox From FOLDOC Russell's Paradox mathematics A paradox (logical contradiction) in set theory discovered by Bertrand Russell. If R is the set of all sets which don't contain themselves http://foldoc.org/Russell's Paradox

Russell's Paradox mathematics paradox (logical contradiction) in set theory discovered by Bertrand Russell . If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa. The paradox stems from the acceptance of the following axiom : If P(x) is a property then is a set. This is the Axiom of Comprehension (actually an axiom schema ). By applying it in the case where P is the property "x is not an element of x", we generate the paradox, i.e. something clearly false. Thus any theory built on this axiom must be inconsistent. In lambda-calculus Russell's Paradox can be formulated by representing each set by its characteristic function - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself: If we now apply r to itself, An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of set theory suggest the existence of the paradoxical set R.

45. E.W.Dijkstra Archive: Where Is Russell's Paradox? (EWD 923a) Nov 11, 2004 Why is Russell s Paradox called a paradox? It is supposed to be rendered by the example of the village with cleanshaven men in which the http://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD923a.html

EWD 923A Where is Russell's paradox? Why is "Russell's Paradox" called a paradox? It is supposed to be rendered by the example of the village with cleanshaven men in which the village barber is defined as the man such that the population he shaves is the population of villagers that don't shave themselves. And then comes the question "Who shaves the barber?". But where is the paradox? If we define in a certain context x to be equal to 3, we define x to satisfy x =3, i.e. to be a root of the equation y y y y y ≥0)? After the invention of the reals that equation has indeed one root. Did we only know the rationals, the equation would have no root and √2 "would not exist". For the barber of the village we have the equation y A i i is a villager : i shaver i y shaver i and that equation has no solution. Conclusion: the village has no barber. Where is the paradox? Probably I am very naive, but I also think I prefer to remain so, at least for the time being and perhaps for the rest of my life. Austin, 22 May, 1985

46. Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks What is Mathematics? Goedel s Theorem and Around. Textbook for students. Section 2. By K.Podnieks. http://www.ltn.lv/~podnieks/gt2.html

set theory, axioms, Zermelo, Fraenkel, Frankel, infinity, Cantor, Frege, Russell, paradox, formal, axiomatic, Russell paradox, axiom, axiomatic set theory, comprehension, axiom of infinity, ZF, ZFC Back to title page Left Adjust your browser window Right 2. Axiomatic Set Theory The origin of Cantor's set theory Formalization of Cantor's inconsistent set theory Zermelo-Fraenkel axioms Around the continuum problem ... Ackermann's set theory (Church thesis for set theory?) For a general overview and set theory links, see Set Theory by Thomas Jech in Stanford Encyclopedia of Philosophy 2.1. The Origin of Cantor's Set Theory F. A. Medvedev. Development of Set Theory in the XIX Century. Nauka Publishers, Moscow, 1965, 350 pp. (in Russian) F. A. Medvedev. The Early History of the Axiom of Choice. Nauka Publishers, Moscow, 1982, 304 pp. (in Russian) See also: Online paper "A history of set theory" in the MacTutor History of Mathematics archive A. Kanamori . Set Theory from Cantor to Cohen, Bulletin of Symbolic Logic, 1996, N2, pp.1-71 (online text at

47. Russells Paradox - Discussion And Encyclopedia Article. Who Is Russells Paradox? Russells paradox. Discussion about Russells paradox. Ecyclopedia or dictionary article about Russells paradox. http://www.knowledgerush.com/kr/encyclopedia/Russells_paradox/

48. RUSSELL S PARADOX (PART II) Linton C. Freeman (University Of File Format PDF/Adobe Acrobat Quick View http://moreno.ss.uci.edu/34.pdf

49. The Paradox Of The Liar This paradox is a version of Russell s Paradox. It came about from Bertrand Russell thinking about the notion of a set, or class, or collection of things, http://www.philosophers.co.uk/cafe/paradox2.htm

Home Articles Games Portals ... Contact Us Paradoxes The second in Francis Moorcroft's series looking at some the classic philosophical paradoxes. No. 2 Russell's Paradox Francis Moorcroft The British Library sends out instructions that every library in the country has to make a catalogue of all its books. Each librarian makes their catalogue and are then faced with a choice: the catalogue is, after all, a book in their library; should the title of the catalogue be included in the catalogue itself or not? Some librarians decide to include it, others not to. don't include themselves the librarian is faced with a dilemma: should they include the title of the catalogue in the catalogue or not? if they do then it is not a catalogue that does not contain its own title and so it shouldn't be included; if they don't put it in then it is a catalogue that doesn't contains its own title and so should be included. Either way, it should contain itself if it doesn't and shouldn't contain itself if it does! This paradox is a version of Russell's Paradox not cats - dogs, chairs, books, violin sonatas, . . . and sets. This set is a member of itself. Now it is far more usual for a set

50. Russell's Paradox - Ask.com Top questions and answers about Russell'sParadox. Find 8 questions and answers about Russell's-Paradox at Ask.com Read more. http://www.ask.com/questions-about/Russell's-Paradox

51. No Match For Russell& In lambdacalculus Russell s Paradox can be formulated by representing each set by its characteristic function - the property which is true for members and http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Russell's Paradox

52. Russell’s Paradox (Stanford Encyclopedia Of Philosophy/Summer 2001 Edition) Russell’s Paradox Russell’s paradox is the most famous of the logical or settheoretical paradoxes. The paradox arises within naive set theory by considering the set of all http://seop.leeds.ac.uk/archives/sum2001/entries/russell-paradox/

This is a file in the archives of the Stanford Encyclopedia of Philosophy Stanford Encyclopedia of Philosophy A B C D ... Z Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox prompted much work in logic, set theory and the philosophy and foundations of mathematics during the early part of the twentieth century. History of the paradox Significance of the paradox Bibliography Other Internet Resources ... Related Entries History of the paradox Russell appears to have discovered his paradox in May of 1901 while working on his Principles of Mathematics (1903). Cesare Burali-Forti, an assistant to Giuseppe Peano, had discovered a similar antinomy in 1897 when he noticed that since the set of ordinals is well-ordered, it, too, must have an ordinal. However, this ordinal must be both an element of the set of all ordinals and yet greater than every such element. Russell wrote to Gottlob Frege f(x) may be considered to be both a function of the argument f and a function of the argument x.

53. Barber Paradox (Russell's Paradox) - Brain Teasers Forum Barber Paradox (Russell s Paradox) Back to the Paradoxes Analogue paradox to the paradox of liar formulated English logician, philosopher and math http://brainden.com/forum/index.php?/topic/205-barber-paradox-russells-paradox/

54. Russells Paradox A selection of articles related to russells paradox russells paradox Encyclopedia II Russell's paradox - Applications and related topics http://www.experiencefestival.com/russells_paradox

55. Russell Paradox Russell paradox. Paradoxes of set theory. Mathematical paradox. Mathematics, mathematical articles. We do mathematical research, math modeling and math http://www.suitcaseofdreams.net/Paradox_Russell.htm

Home Mathematics Database Programming ... Price List Possibly the greatest paradox is that mathematics has paradoxes... Bernoulli's sophism Paradox of Bernoulli and Leibniz Galileo's paradox Paradox of even (odd) and natural numbers ... Achilles and tortoise Russell paradox Formulation of the Russell's paradox Illustrations of the Russell's paradox Russell's paradox and naive set theory Avoiding Russell's paradox with type theory ... Avoiding Russell's paradox with axiomatic set theory Formulation of the Russell's paradox Russell's paradox: The set M is the set of all sets that do not contain themselves as members. Does M contain itself?

56. Russell's Paradox Russell s paradox is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. http://www.fact-index.com/r/ru/russell_s_paradox.html

Main Page See live article Alphabetical index Russell's paradox Russell's paradox is a paradox discovered by Bertrand Russell in which shows that the naive set theory of Cantor and Frege is contradictory. Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A In Cantor's system, M is a well-defined set. Does M contain itself? If it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, then it has to be a member of M , again according to the very definition of M . Therefore, the statements " M is a member of M " and " M is not a member of M " both lead to contradictions. In Frege's system, M corresponds to the concept does not fall under its defining concept . Frege's system also leads to a contradiction: that there is a class defined by this concept, which falls under its defining concept just in case it does not. History Exactly when Russell discovered the paradox is not clear. It seems to have been May or June 1901, probably as a result of his work on Cantor's theorem that the number of entities in a certain domain is smaller than the number of subclasses of those entities. In Russell's Principles of Mathematics (not to be confused with the later

57. Russell’s Paradox And Cloud Computing Jul 20, 2010 I am sure you ve heard of Bertrand Russell s paradox and one of its more widely known versions – Barber paradox. http://somic.org/2010/07/20/russell-paradox-and-cloud-computing/

58. Russell's Paradox | Facebook Welcome to the Facebook Community Page about Russell's paradox, a collection of shared knowledge concerning Russell's paradox. http://www.facebook.com/pages/Russells-paradox/112885282059692

23 people like this. to connect with Wall Info Fan Photos Just Others joined Facebook. March 26 at 10:48pm See More Posts English (US) Español More… Download a Facebook bookmark for your phone. Login Facebook ©2010

59. Resolving The Barber Paradox And The Russell's Paradox Resolving the Barber Paradox and the Russell s Paradox Philosophy discussion. The Russell paradox was based on the nowdiscredited axiom that every http://www.physicsforums.com/showthread.php?t=64685

60. Resolution Of Russell S Paradox (reflecting On Logicomix) File Format PDF/Adobe Acrobat Quick View http://www.dataweb.nl/~cool/Papers/ALOE/2010-02-14-Russell-Logicomix.pdf

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