1. Russell's Paradox By A. D. Irvine. http://plato.stanford.edu/entries/russell-paradox/

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... Z This document uses XHTML-1/Unicode to format the display. Older browsers and/or operating systems may not display the formatting correctly. last substantive content change MAY 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 " R ." If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R 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 the late spring of 1901

2. Russell's Paradox [Internet Encyclopedia Of Philosophy] Examines selfreferential linguistics used to describe properties and sets. http://www.utm.edu/research/iep/p/par-russ.htm

Russell's Paradox Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not. The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the propery of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat ) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows. The paradox was named after Bertrand Russell, who discovered it in 1901. Table of Contents (Clicking on the links below will take you to that part of this article) History Possible Solutions to the Paradox of Properties Possible Solutions to the Paradox of Classes or Sets Suggested Reading History Russell's discovery came while he was working on his Principles of Mathematics . Although Russell discovered the paradox independently, there is some evidence that other mathematicians and set-theorists, including Ernst Zermelo and David Hilbert, had already been aware of the first version of the contradiction prior to Russell's discovery. Russell, however, was the first to discuss the contradiction at length in his published works, the first to attempt to formulate solutions and the first to appreciate fully its importance. An entire chapter of the

3. Erasing Russell's Paradox Erasing russell's paradox. Home. Cantor's Theorem it is fraught with nagging inconsistencies, such as the Russell Paradox Let N denote the set of all sets which are not http://www.geocities.com/dblowe_47/sets.htm

Erasing Russell's Paradox Home Cantor's Theorem Numbers The concept of set is so "naïve" and intuitive because it is directly related to the fundamental trait of human cognitive functioning to group and count in order to make sense of the constant bombardment of sensory stimuli. We group these stimuli based on common properties, and their distinctness allows us to count and relate how many items with common properties we encountered. What mathematicians have done is given this grouping concept "life" and the name of "set" and shifted focus away from the items with common properties that make up this mental grouping. Mathematicians have abstracted the concept and gone off to play with it without stopping to consider, at the most basic level, what it should and should not be. The first order of business is to establish what "exists" and what does not. Based on the general human cognitive ability to distinguish between and group "objects of thought", we can safely say that such objects of thought "exist". We generally agree upon the existence of objects since we can discuss the properties of objects among ourselves and conclude that we are discussing the same objects. And we can assume reasonably that there are many distinct objects. We can also reasonably assume that there are objects with common properties because the world would otherwise be in complete chaos. (Some might argue that the world is just that!) Philosophical questions aside, our first Axiom (1) is: There exist distinct objects that can be grouped according to a shared property (or properties).

4. Russell's Paradox -- From MathWorld Order book from Amazon. R. russell's paradox. Russell's Antinomy Eric W. Weisstein. " russell's paradox." From MathWorldA Wolfram Web Resource http://mathworld.wolfram.com/RussellsParadox.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon R Russell's Paradox Russell's Antinomy search Eric W. Weisstein. "Russell's Paradox." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/RussellsParadox.html Wolfram Research, Inc.

5. Russell's Paradox russell's paradox. ©. Copyright 2000, Jim Loy. Let you tell me a famous story There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved http://www.jimloy.com/logic/russell.htm

Return to my Logic pages Go to my home page Russell's Paradox Let you tell me a famous story: There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved themselves or were shaved by the barber. And the barber only shaved the men who did not shave themselves. That is a nice story. But it raises the question: Did the barber shave himself? Let's say that he did shave himself. But we see from the story that he shaved only the men in town who did not shave themselves. Therefore, he did not shave himself. But we again see in the story that every man in town either shaved himself or was shaved by the barber. So he did shave himself. We have a contradiction. What does that mean? Maybe it means that the barber lived outside of town. That would be a loophole, except that the story says that he did live in the town, maybe in Seville. Maybe it means that the barber was a woman. Another loophole, except that the story calls the barber "he." So that doesn't work. Maybe there were men who neither shaved themselves nor were shaved by the barber. Nope, the story says, "All of the men in this town either shaved themselves or were shaved by the barber." Maybe there were men who shaved themselves AND were shaved by the barber. After all, "either ... or" is a little ambiguous. But the story goes on to say, "The barber only shaved the men who did not shave themselves." So that doesn't work either. Often, when the above story is told, one of these last two loopholes is left open. So I had to be careful, when I wrote down the story.

6. The New York Review Of Books: Russell's Paradox Volume 39, Number 14 · August 13, 1992. Review. russell's paradox. By Stuart Hampshire are indeed "the private years" of Bertrand Russell's long life, if they are compared with the http://www.nybooks.com/articles/2834

@import "/css/default.css"; Home Your account Current issue Archives ... August 13, 1992 Review Russell's Paradox By Stuart Hampshire The Selected Letters of Bertrand Russell Volume 1, The Private Years, 1884-1914 edited by Nicholas Griffin Houghton Mifflin, 553 pp., $35.00 The years between 1872 and 1914 are indeed "the private years" of Bertrand Russell's long life, if they are compared with the period following 1914, the years of his militant pacifism and imprisonment for opposing World War I. But even during his lonely childhood in the splendid late Victorian house, Pembroke Lodge, of his grandfather Lord John Russell and his grandmother the formidable Lady Stanley, he learned to take for granted the daily arguments about great affairs of state among those who were directly or indirectly involved in them as members of the aristocratic ruling class; and this included his own family and his numerous cousins. It was naturally assumed that he would in due time appear on the public stage as a leader in liberal politics, and perhaps also as publicly supporting the most advanced radical causes as his parents, Lord and Lady Amberley, did before they prematurely died, of diphtheria. 4066 words The full text of this piece is only available to subscribers of the Review 's electronic edition . To subscribe or learn more about the electronic edition, please

7. Russell's Paradox Math puzzles. Interactive education. Logic and Paradoxes. Selfreference. russell's paradox russell's paradox. Poincaré disliked Peano's work on a formal language for mathematics, then He wrote of russell's paradox, with evident satisfaction, "Logistic has finally proved http://www.cut-the-knot.com/selfreference/russell.html

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Russell's Paradox from R.Hersh, What is Mathematics, Really? Oxford University Press, 1997 Sets are defined by the unique properties of their elements. One may not mention sets and elements simultaneously, but one notion has no meaning without other. The widely used Peano's notation incorporates all the pertinent attributes: a set A, a property P, elements x. But, of course, one does not always use the formal notations. For example, it's quite acceptable to talk of the set of all students at the East Brunswick High or the set of fingers I use to type this sentence. The space being limited, some sets are described on this page and some are not. Let's call russell the set of all sets described on this page. Just driving the point in: russell's elements are sets described on this page. Note that this page is where you met russell. For it's where it was defined after all. So russell has an interesting property of being its own element: russell russell.

8. Wikipedia Russell's Paradox Wikipedia Free Encyclopedia's article on 'russell's paradox' russell's paradox is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor http://rdre1.inktomi.com/click?u=http://en.wikipedia.org/wiki/Russell's_paradox&

9. KurzweilAI.net Paradigms and Paradoxes of Intelligence, Part 1 russell's paradox. Permanent link to this article An exploration of russell's paradox, written for "The Futurecast " a monthly column http://www.kurzweilai.net/meme/frame.html?main=/articles/art0257.html?m=10

10. Russell's Paradox Definition Of Russell's Paradox In Computing. What Is Russell' Computer term of Russell s paradox in the Computing Dictionary and Thesaurus. Provides search by definition of Russell s Paradox. http://computing-dictionary.thefreedictionary.com/Russell's Paradox

Dictionaries: General Computing Medical Legal Encyclopedia Russell's paradox Word: Word Starts with Ends with Definition (mathematics) Russell's Paradox - 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. The paradox stems from the acceptance of the following axiom : If P(x) is a property then x : P 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

11. RUSSELL'S PARADOX - Meaning And Definition Of The Word RUSSELL S PARADOX Dictionary Entry and Meaning. Computing Dictionary. Definition A logical contradiction in set theory discovered by Bertrand Russell. http://www.hyperdictionary.com/computing/russell's paradox

English Dictionary Computer Dictionary Thesaurus Dream Dictionary ... Medical Dictionary Search Dictionary: RUSSELL'S PARADOX: Dictionary Entry and Meaning Computing Dictionary Definition: 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. The paradox stems from the acceptance of the following axiom : If P(x) is a property then x : P 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

12. Dictionary.com/russell's Paradox Get the Top 10 Most Popular Sites for russell s paradox . 1 entry found for russell s paradox. russell s paradox. mathematics A http://dictionary.reference.com/search?q=russell's paradox

13. Russell's Paradox From FOLDOC Russell s Paradox. mathematics A logical contradiction in set theory discovered by the British mathematician Bertrand Russell (18721970). http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Russell's Paradox

14. Russell's Paradox From FOLDOC Free Online Dictionary of Computing. Russell s Paradox. mathematics A logical contradiction in set theory discovered by Bertrand Russell. http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?Russell's paradox

15. The DICT Development Group: Online Dictionary Query- Russell's Paradox 1 definition found From The Free Online Dictionary of Computing (15Feb98) Russell s Paradox A logical contradiction in set theory discovered by the British http://mandrake.petra.ac.id/cgi-bin/Dict?Form=Dict2&Database=*&Query=Russell's p

16. No Match For Russell's Paradox No match for Russell s paradox. Sorry, the term Russell s paradox is not in the dictionary. Check the spelling and try removing suffixes like ing and -s . http://www.nue.org/foldoc/foldoc.cgi?Russell's paradox

17. No Match For Russell's Paradox No match for Russell s Paradox. Sorry, the term Russell s Paradox is not in the dictionary. Check the spelling and try removing suffixes like ing and -s . http://www.nue.org/foldoc/foldoc.cgi?Russell's Paradox

18. Russell's Paradox - Encyclopedia Article About Russell's Paradox. Free Access, N encyclopedia article about Russell s paradox. Russell s paradox in Free online English dictionary, thesaurus and encyclopedia. Russell s paradox. http://encyclopedia.thefreedictionary.com/Russell's paradox

Dictionaries: General Computing Medical Legal Encyclopedia Russell's paradox Word: Word Starts with Ends with Definition Russell's paradox is a paradox This article is about the logic concept. Borland Paradox is a database management tool, and Paradoxx is a hard rock band. A paradox is an apparently true statement that seems to lead to a logical self-contradiction, or to a situation that contradicts common intuition. The identification of a paradox based on seemingly simple and reasonable concepts has often led to significant advances in science, philosophy and mathematics. Click the link for more information. discovered by Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (May 18, 1872 - February 2, 1970) was one of the most influential mathematicians, philosophers and logicians working (mostly) in the 20th century, an important political liberal, activist and a populariser of philosophy. Millions looked up to Russell as a sort of prophet of the creative and rational life; at the same time, his stance Click the link for more information.

19. Index Of /~ml01078/fajlovi/Russell's Paradox [Internet Encyclopedia Of Philosoph Parent......Index of /~ml01078/fajlovi/Russell s Paradox Internet Encyclopedia of Philosophy_files. Name Last modified Size http://alas.matf.bg.ac.yu/~ml01078/fajlovi/Russell's Paradox [Internet Encyclope

Index of /~ml01078/fajlovi/Russell's Paradox [Internet Encyclopedia of Philosophy]_files Name Last modified Size Description ... Parent Directory 24-May-2004 17:56 - iepback.gif 28-Jan-2004 13:29 1k iepsmall.gif 28-Jan-2004 13:29 5k Apache/1.3.29 Server at alas.matf.bg.ac.yu Port 80

20. Gnome Türk Çeviri Terimleri Sorgu Sayfası- Russell's Paradox No definitions found for Russells paradox Sayfa hakkinda görüs ve önerileriniz için iletisim Onur Can ÇAKMAK ve Baris ÇIÇEK. http://gnome.uzem.itu.edu.tr/cgi-bin/Dict?Form=Dict2&Database=*&Query=Russell's

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