101. Russell's Paradox | Jef's Web Files Russell s paradox. A logical contradiction in set theory discovered by BertrandRussell. If R is the set of all sets which don t contain themselves, http://www.jefallbright.net/node/1795

@import "misc/drupal.css"; @import "modules/sidecontent/sidecontent.css"; @import "themes/jwftheme/style.css"; home misc about empathy ... extropy Home Subcategories: Empathy Energy Efficiency Extropy ... Misc A 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. 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. Zermelo Fr¤nkel set theory is one "solution" to this paradox. Another, type theory, restricts sets to contain only elements of a single type, (e.g. integers or sets of integers) and no type is allowed to refer to itself so no set can contain itself. A message from Russell induced Frege to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway. Source: The Free On-line Dictionary of Computing Paradox login or register to post comments "...when you are a Bear of Very Little Brain, and you Think of Things, you find sometimes that a Thing which seemed very Thingish inside you is quite different when it gets out into the open and has other people looking at it." - A.A. Milne, "The House At Pooh Corner"

102. Lost And Found It is not, for example, Russell s paradox, nor is it Zeno s paradox, butthese names might give us a handle by which to come to know an aspect of this http://www3.telus.net/public/godog/lostfound.html

Lost and Found Back to Words

103. IIDB - Russell's Paradox Resolved This is an atheist / freethought discussion forum powered by vBulletin. To findout about vBulletin, go to http//www.vbulletin.com/ . http://www.iidb.org/vbb/archive/index.php/t-56645.html

IIDB II Philosophical Forums Philosophy PDA View Full Version : Russell's Paradox Resolved Witt June 26, 2003, 06:21 AM Stanford Encyclopedia of Philosophy Russell’s Paradox Russell’s paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox. 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. Proof: If we grant first order predicate logic, and add Russell's contextual definition of Classes, determined by some predicate, then the antinomy does not occur.

104. American Mathematical Monthly, The: Quine's NF--60 Years On This appears most simply in the form of Russell s paradox, This leads rapidlyto Russell s paradox, the paradox of the class of all sets that are not http://www.findarticles.com/p/articles/mi_qa3742/is_199711/ai_n8763902

@import url(/css/us/style1.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); @import url(/css/us/artHome1.css); Home Advanced Search IN free articles only all articles this publication Automotive Sports FindArticles American Mathematical Monthly, The Nov 1997 Content provided in partnership with 10,000,000 articles Not found on any other search engine. Featured Titles for ASA News ASEE Prism Academe African American Review ... View all titles in this topic Hot New Articles by Topic Automotive Sports Top Articles Ever by Topic Automotive Sports Quine's NF60 years on American Mathematical Monthly, The Nov 1997 by Forster, Thomas Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. Sixty years ago in this MONTHLY, the distinguished American philosopher W.V. Quine published a novel approach to set theory. The title was "New Foundations for Mathematical Logic" [6]. The diamond anniversary is being commemorated by a workshop in Cambridge (England) and comes at a time of rapid increase of interest in the alternatives to the hitherto customary Zemelo-Frankel set theory, which promises a new lease of life for the axiomatic system now known as 'NF'; its creator remains in good health too. Although he is best known to a wider public for his philosophical writings, his most enduring and most concrete legacy for the next fifty years may well turn out to be his most mathematical: he gave us NF. Since set theory first sprang on the scene about a hundred years ago there has been a tendency to attempt to use this simplicity to simplify and illuminate the rest of mathematics by translating (perhaps a better word is implementing) it into set theory. After all, if we can represent all of mathematics as facts about these delightfully simple things, some facts about mathematics might become clear that would otherwise remain obscure. This same simplicity means that set theory is always a good topic on which to try out any new mathematical idea.

105. Re: Russell's Paradox, Axioms Of Set Theory Russell s paradox was important for Russell, who wanted every wff with one freevariable (every propositional function ) to determine a set, http://bureau.philo.at/phlo/199604/msg00067.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Russell's Paradox, Axioms of Set Theory To phil-logic@bucknell.edu Subject : Re: Russell's Paradox, Axioms of Set Theory From rbj@campion.demon.co.uk (Roger Bishop Jones) Date : Fri, 12 Apr 96 19:24:28 GMT Organization : Myorganisation Reply-To rbj@campion.demon.co.uk http://www.rbjones.com/ at home rbj@campion.demon.co.uk Prev by Date: News from Sorites April 11 1996 Next by Date: No Subject Prev by thread: News from Sorites April 11 1996 Next by thread: forwarding Index(es): Date Thread

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