61. Russell's Paradox X Y Z russell's paradox. russell's paradox is the most famousof the logical or settheoretical paradoxes. The paradox arises http://www.seop.leeds.ac.uk/archives/fall2002/entries/russell-paradox/

Stanford Encyclopedia of Philosophy A B C D ... Z 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 has prompted much work in logic, set theory and the philosophy and foundations of mathematics. 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.

62. Russell’s Paradox s discovery. In this appendix Frege observes that the consequencesof russell's paradox are not immediately clear. For example, Is http://www.seop.leeds.ac.uk/archives/sum2001/entries/russell-paradox/

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.

63. FoRK Archive: Russell's Paradox On The Web russell's paradox on the Web. It occurred to me a couple of days ago that Russell'sParadox can be described quite nicely in terms of web pages and links. http://www.xent.com/FoRK-archive/may98/0192.html

Russell's Paradox on the Web James K. Tauber jtauber@jtauber.com Wed, 13 May 1998 01:40:47 +0800 Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Joe Barrera: "RE: Prisoner of cyberspace" Previous message: CobraBoy: "Re: Prisoner of cyberspace" Next in thread: Lloyd Wood: "Re: Russell's Paradox on the Web" It occurred to me a couple of days ago that Russell's Paradox can be described quite nicely in terms of web pages and links. It seems easier to conceptualize than sets containing themselves as members. It would go something like this: Imagine a web page paradox.html that has a link to every web page that doesn't link to itself. Does paradox.html link to itself? If not, then it should be included on the page in which case it does link to itself. If it does link to itself, then it shouldn't be on the page in which case it doesn't link to itself. James James Tauber / jtauber@jtauber.com Perth, Western Australia XML Pages: http://www.jtauber.com/xml/ Next message: Joe Barrera: "RE: Prisoner of cyberspace" Previous message: CobraBoy: "Re: Prisoner of cyberspace" Next in thread: Lloyd Wood: "Re: Russell's Paradox on the Web"

64. FoRK Archive: Re: Russell's Paradox On The Web Re russell's paradox on the Web. Previous message Lloyd Wood RE Prisoner ofcyberspace ; In reply to James K. Tauber russell's paradox on the Web . http://www.xent.com/FoRK-archive/may98/0208.html

Re: Russell's Paradox on the Web Lloyd Wood eep1lw@surrey.ac.uk Tue, 12 May 1998 21:26:51 +0100 (BST) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: CobraBoy: "Re: Prisoner of cyberspace" Previous message: Lloyd Wood: "RE: Prisoner of cyberspace" In reply to: James K. Tauber: "Russell's Paradox on the Web" On Wed, 13 May 1998, James K. Tauber wrote: by members of a mailing list who can't conceptualize sets? Surely you jest. L. L.Wood@surrey.ac.uk http://www.sat-net.com/L.Wood/ Next message: CobraBoy: "Re: Prisoner of cyberspace" Previous message: Lloyd Wood: "RE: Prisoner of cyberspace" In reply to: James K. Tauber: "Russell's Paradox on the Web"

65. Russell's Paradox russell's paradox. russell's paradox Let S be the set of all sets whichare not members of themselves. Is S a member of itself? Is S a set? http://web.math.fsu.edu/~pkirby/mad3107/SlideShow/Output/Section1_4/Slide11.html

change_script ("script.html#Link11") Russell's Paradox Russell's paradox: Let S be the set of all sets which are not members of themselves. Is S a member of itself? Is S a set? A similar paradox: Henry is a barber who shaves all people who do not shave themselves. Does Henry shave himself? Penelope Kirby Section 1.4: Sets - 11 of 13

66. Arbitrarily Large Sets The paradox exists in this finite universe. Modeling the Paradox. We have alreadydiscussed creating a web crawler that chokes on the russell's paradox. http://descmath.com/diag/russ.html

The Russell Paradox on the Web Considering the great amount of interest in the web, I think it is easier to introduce the reflexive paradox in the context of the internet than in the abstract realm of set theory. Webmasters can appreciate the Russell Paradox. As everyone knows, the web is about links. Any page worth its salt has links to other pages. Some pages (like my little Grand Junction Links Page ) have nothing but links. A web crawler is a program that crawls through pages on a web site. The typical web crawler reads a web page, then follows each of the links one that page. Web crawlers have to worry about infinite loops. The simplest infinite loop happens when a page contains a link back to itself. For example, this page has the name russ.html . I made the name hot. The page links back on itself. It is "self-referential." A web crawler needs to watch out for self-referential pages; Otherwise, it would fall into an infinite loop. If the bot was not programmed to handle recursive links, the bot would read a page, then follow the link back to the page, and read it again... To avoid infinite loops, the web crawler needs to maintain a database of all the places it has visited. Now, we get into the problem that caused Bertrand Russell such angst a century ago:

67. The Moon Rocket - We Are Russell's Paradox The Moon Rocket. Site navigation. http://www.aomr.co.uk/blog022002.html

The Moon Rocket Site navigation AOMR Home Page Whois AJL? Archives Current January 2002 December 2001 November 2001 ... October 2001 Other blogs inw David Chess Rebecca's Pocket Caterina ... Aron Hsiao Talk : Bigger Talk Box Stuff and Nonsense Personality INTP IQ Starwars Name : LeeAn' BraGr Films : Crouching Tiger, Hidden Dragon Dr Strangelove Dune eXistenZ Full Metal Jacket The Double Life Of Veronique The Empire Strikes Back The Wall -Pink Floyd Trois Couleurs - Bleu/Blanc/Rouge Directors: B. Bertolucci L. Besson K. Kieslowski D. Kronenberg S. Kubrick D. Lynch R. Scott Q. Tarantino Music : Bartok Bauhaus Mahler Marillion Opeth Pink Floyd Shostakovich The Sisters Of Mercy Tourniquet Authors : Douglas Adams Isacc Asimov Iain M. Banks Frank Herbert Robin Hobb James Joyce George R. R. Martin Sylvia Plath John Steinbeck Books: Dune - Herbert History Of Western Philosophy - Russel Lord Of the Rings - Tolkein Pendragon (Series) - Lawhead Robots and Empire - Asimov The Bible - God The Bell Jar - Plath The Last Days Of Socrates - Plato The Liveship Traders (Trilogy) - Hobb Instruments: Bass Drums Guitar Piano Fish: Guppy Tuna Thursday February 28, 2002

68. Russell's Paradox - Acapedia - Free Knowledge, For All russell's paradox. russell's paradox is a paradox found by Bertrand Russell in 1901which shows that naive set theory in the sense of Cantor is contradictory. http://acapedia.org/aca/Russell's_paradox

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69. RUSSELL'S PARADOX russell's paradox. D. Atkinson. Ulysses pontificated ‘The classicalstatement of the Russell paradox is in terms of the village http://www-th.phys.rug.nl/~atkinson/russell.html

RUSSELL'S PARADOX D. Atkinson Ulysses pontificated: ‘The classical statement of the Russell paradox is in terms of the village barber, who shaves all the men of the village who do not shave themselves. Does he then shave himself? Clearly so, for if he did not shave himself, he would be one of his own clients, since he shaves all the men who do not shave themselves. So he must shave himself. So far there is no paradox. Suppose though that we now add that he shaves only the men that do not shave themselves. Then it cannot be that he shaves himself, since he would be one of the self-shavers who, according to the terms of the conundrum, are not shaved by the barber. However, if he does not shave himself, he is, as mentioned above, one of his own clients.’ ‘But what about the men who don't shave at all?’ asked Helen coyly. ‘Men like my husband with a sexy red beard. Maybe the barber is one of them!’ ‘That is not the point at all!’ exclaimed Ulysses in exasperation. ‘The problem has to do with the definition or presumptive definition!

70. Metamath Proof Explorer - Ru russell's paradox. Proposition 4.14 of TakeutiZaring p. 14.......Browser slow? Try the Symbol font version. Theorem ru 1243 http://metamath.flatline.de/mpegif/ru.html

Home Metamath Proof Explorer Browser slow? Try the Symbol font version Theorem ru Description: Russell's Paradox. Proposition 4.14 of [ TakeutiZaring ] p. 14. Frege's Axiom of Comprehension, expressed in our notation as , asserted that any collection of sets is a set i.e. belongs to the universe of all sets. In particular, by substituting for , it asserted , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove . This contradiction was discovered by Russell in 1901 (published in 1903), invalidating Comprehension and leading to the collapse of Frege's system. In 1908 Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex asserting that is a set only when it is smaller than some other set . However, Zermelo was then faced with a "chicken and egg" problem of how to show is a set, forcing him to introduce the set-building axioms of Null Set zfnull , Pairing zfpaircl , Union , Power Set zfpowcl , and Infinity omex to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom

71. Hippias: Limited Area Search Of Philosophy On The Internet Paradox, Russell's (Internet Encyclopedia of Philosophy) russell's paradox Russell'sparadox represents either of two interrelated logical antinomies. http://hippias.evansville.edu/search.cgi?russell

72. [HM] Cantor's Theorem And Russell's Paradox By Richard E. Grandy HM Cantor's Theorem and russell's paradox by Richard E. Grandy. replyto this message post a message on a new topic Back to Historia http://mathforum.org/epigone/historia_matematica/cransezem

[HM] Cantor's Theorem and Russell's paradox by Richard E. Grandy reply to this message post a message on a new topic Back to Historia-Matematica Discussion Group Subject: [HM] Cantor's Theorem and Russell's paradox Author: rgrandy@rice.edu Date: The Math Forum

73. JavaScript Russell's Paradox NO russell's paradox in the computer world! This implements stop . Russell'sparadox is a clear problem for traditional logic. Why? Traditional http://clickenfind.com/pidmass/russell.html

NO Russell's paradox in the computer world! This implements the 'library catalogue' conundrum: does one put the name of the book of names of books that do not contain their name in itself? This is the finite form of "set of all sets that do not contain themselves". It is equivalent to Turing's and Penrose's "computer program that stops when it finds a program possibly itself which does not stop". Russell's paradox is a clear problem for traditional logic. Why? Traditional logic has an other worldly, timeless aspect. But the comfortable knish it has found in many human psyches has become a supernatural poison to others - this very timeless aspect requires the catalogues to be fixed, whereas their definition requires them to change. Book contents(letters after book name letter) are in 2nd column below. On clicking 'REcataloge' the program compares the first letter(book name) in each of 4 books below to the rest of book's contents and makes 2 catalogues which are THEN put in bottom 2 boxes AFTER the contents of ALL 4 boxes are analyzed: Note that repeated cataloging causes N(book of books that do Not self-cite) to simply cycle between catalogues: explicit, mechanical use of

74. Alternate Approaches And Axiomatizations Otherwise, it is a proper class. russell's paradox is avoided by showing that theclass Y is a proper class, not a set. V is also considered as a proper class. http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node4.html

Next: ZF Set Theory Up: CLASSICAL SET THEORY Previous: Earliest Developments Alternate Approaches and Axiomatizations The new axiomatizations took a common step for overcoming the deficiencies of the naive approach by introducing classes is a proper class, not a set. V is also considered as a proper class. The axioms of NBG are simply chosen with respect to the limitation of size constraint. Strengthening NBG by replacing the axioms of class existence with an axiom scheme, a new theory called Morse-Kelley (MK) is obtained (Morse 1965). MK is suitable if one is not interested in the subtleties of set theory. But its strength risks its consistency (Mendelson 1987). Ackermann (1956) also proposed an axiomatization again employing classes, but in which the central objects are sets. The main point of this axiomatization is that its axioms retain only the weakest consequences of the limitation of size constraint, i.e., a member of a set and a subclass of a set are sets. Other approaches against the deficiencies of the naive approach alternatively played with its language and are generally called type-theoretical approaches. Russell and Whitehead's

75. FOM: Simpson On Russell's Paradox For Category Theory FOM Simpson on russell's paradox for category theory. Robert M. Solovaysolovay@math.berkeley.edu Sat, 11 Mar 2000 190335 0800 (PST) http://www.cs.nyu.edu/pipermail/fom/2000-March/003913.html

FOM: Simpson on Russell's paradox for category theory Robert M. Solovay solovay@math.berkeley.edu Sat, 11 Mar 2000 19:03:35 -0800 (PST) Previous message: FOM: defining ``mathematics'' Next message: FOM: Simpson on Russell's paradox for category theory Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Previous message: FOM: defining ``mathematics'' Next message: FOM: Simpson on Russell's paradox for category theory Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]

76. FOM: Simpson On Russell's Paradox For Category Theory FOM Simpson on russell's paradox for category theory. Stephen G Simpsonsimpson@math.psu.edu Wed, 15 Mar 2000 110550 0500 (EST) http://www.cs.nyu.edu/pipermail/fom/2000-March/003918.html

FOM: Simpson on Russell's paradox for category theory Stephen G Simpson simpson@math.psu.edu Wed, 15 Mar 2000 11:05:50 -0500 (EST) Previous message: FOM: Simpson on Russell's paradox for category theory Next message: FOM: Survery Results Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Dear Bob, I just wanted to let you know that I am not ignoring your FOM posting of March 11, where you point out difficulties with my ``Russell paradox for naive category theory''. I think I will be able to answer your points, but I am busy with other things right now, and I will have to take some time off to get the details back into my brain. Steve Previous message: FOM: Simpson on Russell's paradox for category theory Next message: FOM: Survery Results Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]

77. ·¯¼¿ÀÇ ¿ª¼³ Russell's Paradox The summary for this Korean page contains characters that cannot be correctly displayed in this language/character set. http://www.seelotus.com/gojeon/cafe/phi/liberpro/phil/fjtpfdmldurtjf.html

·¯¼¿ÀÇ ¿ª¼³ Russell's paradox

78. Russel's Paradox russell's paradox. russell's paradox, named after its discoverer BertrandRussell, is a mathematical paradox based on set theory. http://www.freakytim.co.uk/stuff/paradox.asp

Hosted Sites Home College Stuff Random Stuff Digital Widgets ... Links Search WWW Search freakytim.co.uk Russell's Paradox A paradox is an apparently sound statement or proposition which leads to a logically unacceptable conclusion. One such statement, known as the barber's paradox, goes thus: In a village there is a barber. The barber cuts the hair of everybody in the village who doesn't cut their own hair. In other words: If someone cuts their own hair then the barber doesn't cut their hair. If the barber cuts their hair then they don't cut their own hair. The paradox arrises when you consider who cuts the barbers hair. If the barber cuts his own hair then the barber doesn't cut his hair. If the barber doesn't cut his own hair then then the barber cuts it. Russell's paradox, named after its discoverer Bertrand Russell, is a mathematical paradox based on set theory. Consider a set called S which is the set of sets that do not contain themselves. If S is not an element of S, then by definition it must be an element of S. If S is an element of S then it cannot be an element of S. n.b. the following does not display correctly in Microsoft Internet Explorer (the boxes should show the "not an element of" symbol):

79. Russell's Paradox First Previous Index Home Text. Slide 22 of 22. http://www.doc.ic.ac.uk/~twh1/mp4/notes2/sld022.htm

80. Re: Performative Inconsistency I did not think Boyle's explanation of why was particularly helpful, so I went intoan analysis of the Liar paradox, the Barber paradox, and russell's paradox. http://www.ccir.ed.ac.uk/~jad/vantil-list/archive-Apr-1999/msg00066.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Performative Inconsistency To Subject : Re: Performative Inconsistency From Date : Tue, 27 Apr 1999 18:04:38 -0700 (MST) Reply-To Prev by Date: Re: TA's and Necessary Truths Next by Date: Intensionality Prev by thread: Performative Inconsistency (Was TA's and Necessary Truths) Next by thread: Intensionality Index(es): Date Thread

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