21. Dictionary.com/russell's Paradox Get the Top 10 Most Popular Sites for russell s paradox . Cannot connect 110 = Connection timed out. http://dictionary.reference.com/search?q=russell's paradox

22. Russell's Paradox Article on Russell s paradox from WorldHistory.com, licensed from Wikipedia, the free encyclopedia. Return Index Russell s paradox. http://www.worldhistory.com/wiki/R/Russell's-paradox.htm

World History (home) Encyclopedia Index Localities Companies Surnames ... This Week in History Russell's paradox Russell's paradox in the news 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

23. Notes To Russell's Paradox Stanford Encyclopedia of Philosophy Notes to Russell s paradox Citation Information. Notes. Notes to Russell s paradox Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/russell-paradox/notes.html

Stanford Encyclopedia of Philosophy Notes to Russell's Paradox Citation Information Notes Exactly when the discovery of the paradox took place is not completely clear. Russell initially states that he came across the paradox "in June 1901" (see Russell (1944), p. 13). Later he reports that the discovery took place "in the spring of 1901" (see Russell (1959), p. 58). Later still he reports that he came across the paradox, not in June, but in May of that year (see Russell (1967, 1968, 1969), vol. 3, p. 221). See Frege (1903), p. 127. It is worth noting that, even prior to Russell's discovery, this principle had not been universally accepted. Georg Cantor, for example, rejected it in favour of what was, in effect, a distinction between sets and classes, recognizing that some properties (such as the property of being an ordinal) produced collections that were too large to be sets, and that an assumption to the contrary would lead to inconsistency. For further details see Menzel (1984), Moore (1982), and Hallett (1984). One exception is paraconsistent set theory. Paraconsistent set theory retains an unrestricted comprehension axiom but abandons classical logic, substituting a paraconsistent logic in its place. For further information, see the entries on

24. Russell's Paradox [Internet Encyclopedia Of Philosophy] Russell s paradox. Russell s paradox represents either of two interrelated logical antinomies. See also the RussellMyhill paradox article in this encyclopedia. http://www.iep.utm.edu/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

25. Russell's Paradox - Wikipedia, The Free Encyclopedia Russell s paradox. From Wikipedia, the free encyclopedia. Russell s paradox does). Settheoretic responses to the Russell paradox. After http://en.wikipedia.org/wiki/Russell's_paradox

Russell's paradox From Wikipedia, the free encyclopedia. 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. Table of contents 1 History 2 Easy-to-understand version of the Paradox 3 Set-theoretic responses to the Russell Paradox 4 Easy-to-understand version of responses to the Paradox ... edit 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

26. Russell's Paradox - Wikipedia, The Free Encyclopedia Russell s paradox. (Redirected from Russell s paradox). Russell s paradox does). Settheoretic responses to the Russell paradox. After http://en.wikipedia.org/wiki/Russell's_Paradox

Russell's paradox From Wikipedia, the free encyclopedia. (Redirected from 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. Table of contents 1 History 2 Easy-to-understand version of the Paradox 3 Set-theoretic responses to the Russell Paradox 4 Easy-to-understand version of responses to the Paradox ... edit 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

27. Russell's Paradox Math puzzles. Interactive education. Logic and paradoxes. Selfreference. Russell s paradox. Russell s paradox. Poincaré disliked http://www.cut-the-knot.org/selfreference/russell.shtml

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.

28. Set Theory And Paradoxes Sets (Venn Diagrams) Set Theory Comparing Sets (Mappings and Cardinality, Power sets) Infinite sets (The Diagonal Argument) Russell s paradox Statement of the http://www.geocities.com/mathfair2002/school/logi/logi0.htm

home stands games about ... links Sets Picturing Sets (Venn Diagrams) Set Theory Comparing Sets (Mappings and Cardinality, Power sets) Infinite sets (The Diagonal Argument) Russell's Paradox Statement of the Paradox (The Barber of Seville, Set Theoretic Statement, Grelling's paradox, Resolving the Paradox) Gödel's Incompleteness Theorems Double Entendres and Gödelization Sets A set or class is a collection of distinct numbers or items. For example, we can define a set S of all stringed instruments. The item v, viola, is a member or element of S (v S), but t, trombone, is not (t S). Every member of S is also a member of M, the set of all musical instruments, so S is called a subset of M (S M). The set of all non-members of M is called the complement of M (M ). Given another set I, Indian instruments, the union of I and S, I S, is the set of those members that belong either to I, or to S, or to both. The intersection of I and S, I S, is the set of only those members that belong to both I and S. A set without any members is called the null or empty set (Ø). The set of musical instruments is a subset of the universal set, written e Picturing Sets Venn Diagrams A Venn diagram represents sets topologically, using intersecting circles. In the Venn diagram below, the outer rectangle represents the universal set

29. PlanetMath: Russell's Paradox Russell s paradox, (Definition). Suppose that for any coherent proposition , we can construct a set . Let . Russell s paradox is owned by Daume. http://planetmath.org/encyclopedia/RussellsParadox.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List Russell's paradox (Definition) Suppose that for any coherent proposition , we can construct a set Let . Suppose ; then, by definition, . Likewise, if , then by definition . Therefore, we have a contradiction. Bertrand Russell gave this paradox as an example of how a purely intuitive set theory can be inconsistent . The regularity axiom , one of the Zermelo-Fraenkel axioms , was devised to avoid this paradox by prohibiting self-swallowing sets. An interpretation of Russell paradox without any formal language "Russell's paradox" is owned by Daume full author list owner history view preamble View style: HTML with images page images TeX source See Also: Zermelo-Fraenkel axioms lambda calculus Keywords: set theory Cross-references: language interpretation Zermelo-Fraenkel axioms axiom ... proposition There are 8 references to this object.

30. MSN Encarta - Dictionary - Russells Paradox Russell 2000. Russells paradox. Russells viper. Russellville No thesaurus result for " Russells paradox" http://encarta.msn.com/encnet/features/dictionary/DictionaryResults.aspx?refid=5

31. Russell's Paradox Russell s paradox. mathematics A logical contradiction in set theory discovered by the British mathematician Bertrand Russell (18721970). http://burks.brighton.ac.uk/burks/foldoc/59/101.htm

The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: Russell's Attic Next: Russell, Bertrand Russell's Paradox mathematics set theory discovered by the British mathematician Bertrand Russell (1872-1970). 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

32. Russell's Attic Definition Of Russell's Attic In Computing. What Is Russell's At Russell s Attic. Word Word. (mathematics this. Some words with Russell s Attic in the definition http://computing-dictionary.thefreedictionary.com/Russell's Attic

Dictionaries: General Computing Medical Legal Encyclopedia Russell's Attic Word: Word Starts with Ends with Definition (mathematics) Russell's Attic - An imaginary room containing countably many pairs of shoes (i.e. a pair for each natural number ), and countably many pairs of socks. How many shoes are there? Answer: countably many (map the left shoes to even numbers and the right shoes to odd numbers, say). How many socks are there? Also countably many, we want to say, but we can't prove it without the Axiom of Choice , because in each pair, the socks are indistinguishable (there's no such thing as a left sock). Although for any single pair it is easy to select one, we cannot specify a general method for doing this. Some words with "Russell's Attic" in the definition: Axiom of Choice Axiom of Comprehension axiomatic set theory Bertrand ... Zermelo set theory Previous Computing Dictionary Browser Next run-time library run-time support run-time system ... S Full Dictionary Browser Russell L. Schweickart (enc.) Russell Leslie Brown (enc.)

33. Russell's Paradox Russell s paradox. The origins of Russell s paradox are even more controversial than the origins of BuraliForti and Cantor s paradoxes. http://www.u.arizona.edu/~miller/finalreport/node4.html

Next: Conclusion Up: An Historical Account of Previous: Cantor's Paradox Russell's Paradox Russell's paradox, also referred to as Russel's antinomy, Russell's problem, and Zermelo-Russellsches paradoxon ([ ], p. 21), is by far the most famous of the classical paradoxes of set theory, owing much of its fame to its simplicity and far-reaching implications. Russell is usually credited with its discovery in his The Principles of Mathematics ]), or during June of 1901 ([ ]); however, it is clear that Zermelo arrived at the paradox independently one or two years earlier ([ ]). The paradox is simpler than the paradoxes of Burali-Forti and Cantor because it relies only on the most elementary ideas of set theorythe notion of set, set membership, and the Axiom of Abstraction. The paradox can be formulated in the following way. Suppose that the the collection defined by the formula is a set. Then if , and if . Russell also used a statement about a barber to illustrate this principle: If a barber cuts the hair of exactly those who do not cut their own hair, does the barber cut his own hair? ([ ], p. 809). The first clear mention of this paradox occurred in Russell's letter to Frege, written on June 16, 1902. Russell realized that his paradox undermined Frege's theory set forth in

34. Cantor's Paradox The conclusion in the preceding proof that looks almost identical to the contradiction reached in Russell s paradox, and indeed, the most prominent theories on http://www.u.arizona.edu/~miller/finalreport/node3.html

Next: Russell's Paradox Up: An Historical Account of Previous: Burali-Forti's Paradox Cantor's Paradox Cantor's paradox, sometimes called the paradox of the greatest cardinal, expresses what its second name would implythat there is no cardinal larger than every other cardinal. There seems to be close consensus that Cantor discovered this paradox in 1899 or between 1895 and 1897 ([ ], p. 34), but there are some, including the authors who attribute the Burali-Forti paradox to Russell, who give credit to Russell in 1899 or 1901 ([ ], p. 343). The crux of Cantor's paradox is Cantor's Theorem, which states that for any set , where is the power set of and is the cardinality of . The typical, modern proof for this theorem is as follows, and can be found, among others, in [ ], and [ ]. Let be a fixed set. Then defined by , is an injection, so . It remains to show that , so by way of contradiction assume that is a surjection. Then , so there exists a with . Now implies and implies , so a contradiction has been reached. Thus, , so . The conclusion in the preceding proof that looks almost identical to the contradiction reached in Russell's paradox, and indeed, the most prominent theories on the origins of Russell's paradox suggest that his paradox was derived from Cantor's paradox alone or from a combination of Cantor's paradox and the proof of Cantor's Theorem. Given Cantor's Theorem, Cantor's paradox following almost immediately. Suppose that

35. Russell's Paradox - Reference Library Russell s paradox. Russell s paradox is closely related to the Liar paradox. Please Visit Our Sponsor. This article is from Wikipedia. http://www.campusprogram.com/reference/en/wikipedia/r/ru/russell_s_paradox.html

Reference Library: Encyclopedia 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

36. Russell's Paradox Russell s paradox. Russell s paradox trick. Russell s paradox is closely related to the Liar paradox. This article is from Wikipedia. All 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

37. One Hundred Years Of Russell's Paradox Translate this page Godehard Link. http://www.lrz-muenchen.de/~russell01/

Godehard Link Godehard Link

38. One Hundred Years Of Russell's Paradox - Menu Latest Info Schedule Invited Speakers Special Events Contributed Papers Abstracts Registration......Russell 2001. Photo Gallery http://www.lrz-muenchen.de/~russell01/menu.html

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39. Russell's Paradox Q A5. Q Russell s paradox is a model of the third valued proposition in four valued logic. So, what is a symbolic expression of the third valued proposition? http://www.elix.com/txQA5678.htm

text version Russell's paradox is a model of the third valued proposition in four valued logic. So, what is a symbolic expression of the third valued proposition? Which truth value have to be assigned to the assignment proposition of truth value to a certain proposition ? How to confirm the third valued proposition? Or, what is the identity of the third valued proposition very different from the identity of the true or false proposition? Is it necessary to position the third truth value between "the true" and "the false" like J.Lukasiewicz ? next Q: Russell's paradox is a model of the third valued proposition in four valued logic. So, what is a symbolic expression of the third valued proposition? ‚`F Russell's paradox "Set M which does not contain itself" is the third valued proposition which is neither true nor false. If set M which does not contain itself contain itself, set M is not set M. If set M which does not contain itself does not contain itself, then set M is set M. So If set M contains M, set M is not element of set M, and then if set M does not contain M, set M is element of set M. This means that Set M is composed of two dual implications between contradictory propositions. Let p= "M is not element of M".If not p, then p, and, if p, then not p. then p

40. Russell's Paradox Q A5. Q Russell s paradox is a model of the third valued proposition in four valued logic. So, what is a symbolic expression of the third valued proposition? http://www.elix.com/q&a5678.htm

Russell's paradox is a model of the third valued proposition in four valued logic. So, what is a symbolic expression of the third valued proposition? Which truth value have to be assigned to the assignment proposition of truth value to a certain proposition? How to confirm the third valued proposition? Or, what is the identity of the third valued proposition very different from the identity of the true or false proposition? Is it necessary to position the third truth value between "the true" and "the false" like J.Lukasiewicz ? Q F Russell's paradox is a model of the third valued proposition in four valued logic. So, what is a symbolic expression of the third valued proposition? ‚`F Russell's paradox "Set M which does not contain itself" is the third valued proposition which is neither true nor false. If set M which does not contain itself contain itself, set M is not set M. If set M which does not contain itself does not contain itself, then set M is set M. So If set M contains M, set M is not element of set M, and then if set M does not contain M, set M is element of set M. This means that Set M is composed of two dual implications between contradictory propositions. Let p= "M is not element of M".If not p, then p, and, if p, then not-p. Then p

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