41. Russell's Paradox Back. Russell s paradox. Russell s paradox can be put into everyday language in many ways. The most often repeated is the Barber Question. It goes like this http://fclass.vaniercollege.qc.ca/web/mathematics/real/russell.htm

Back Russell's Paradox Easy to state, yet difficult or impossible to resolve; self contradictory statements or paradoxes have presented a major challenge to Mathematics and Logic. Russell's Paradox can be put into everyday language in many ways. The most often repeated is the 'Barber Question.' It goes like this: In a small town there is only one barber. This man is defined to be the one who shaves all the men who do not shave themselves. The question is then asked, 'Who shaves the barber?' If the barber doesn't shave himself, then by definition he does. And, if the barber does shave himself, then by definition he does not. Another popular form of Russell's Paradox is the following: Consider the statement 'This statement is false.' If the statement is false, then it is true; and if the statement is true, then it is false. Let's look at this situation as mathematicians do. You may have noticed the remarkable similarity between logical symbols (like for ' and for ' or '; and ~ for ' not ') and the symbols used with sets. For example, compare Logic Set Theory p q P Q p q P Q p P' In logic a statement that has a single variable, like

42. Russell's Paradox Russell s paradox. In the middle of the night I got such a fright that woke me with a start, For I dreamed of a set that contained itself, in toto, not in part. http://www.cs.brandeis.edu/~mairson/poems/node4.html

Next: Undecidability of the Halting Up: New proofs of old Previous: Dynamic Programming Russell's Paradox In the middle of the night I got such a fright that woke me with a start, For I dreamed of a set that contained itself, in toto, not in part. If sets can thus contain themselves, then they might also fail To hold themselves as members, and this leads me to my tale. Now Frege thought he finally had the world inside a box, So he wrote a lengthy tome, but up popped paradox. Russell asked, ``You know that Epimenides said oft A Cretan who tells a lie does tell the truth, nicht war, dumkopf? And here's a poser you must face if continue thus you do, What make you of the following thought, tell me, do tell true. The set of all sets that contain themselves might cause a soul to frown, But the set of all sets that don't contain themselves will bring you down!'' Now Gottlob Frege was no fool, he knew his proof was fried. He published his tome, but in defeat, while in his beer he cried. And Bertrand Russell told about, in books upon our shelves

43. Russell's Paradox - Curiouser.co.uk Russell s paradox. All classes are either a member of themselves or not. The class of all ideas is an idea. The class of all classes is a class. http://www.curiouser.co.uk/paradoxes/russell.htm

Russell's Paradox All classes are either a member of themselves or not. The class of all ideas is an idea. The class of all classes is a class. Both these classes are members of themselves. The class of all men is not a man. The class of all illnesses is not an illness. Neither of these classes are members of themselves. Let S be the class of all S elf-membered classes. ie classes which are members of themselves. Let N be the class of all N on-self-membered classes. ie. classes which are not members of themselves. Consider N. CONTINUE N is itself a class and must therefore be either a member of N or S. Is N a member of itself? If it is not it must be a member of the class of non-self-members, which is N. But if N is a member of N, then it is a member of itself and therefore a member of S and not N. But if N is a member of S and not N, then it is not a member of its own class and must therefore be a member of N - which was where we began. CONTINUE The paradox may be easier to follow in the following form: If X is any class and N the class of all non-self-membered classes, then the following statement is true:

44. Russell's Paradox: The Achille's Heel Of Solipsism? - Physics Help And Math Help Posts 3,329. Russell s paradox The Achille s Heel of Solipsism? but i don t think solipsism leads to russell s paradox so that issue is moot. http://www.physicsforums.com/showthread.php?t=9546

45. Russell S Paradox The Achille S Heel Of Solipsism? - Technology View Thread Russell s paradox The Achille s Heel of Solipsism? Click Here Russell s paradox The Achille s Heel of Solipsism? http://www.physicsforums.com/archive/t-9546

Physics Help and Math Help - Physics Forums Philosophy Logic View Thread : Russell's Paradox: The Achille's Heel of Solipsism? Russell's Paradox: The Achille's Heel of Solipsism? Mentat I had always taken it for granted that nothing could disprove Solipsism, but now I think there may be an actual logical problem with it! I understand that I could easily be wrong, and that's why I'm posting it: for constructive criticism. Alright, now, the first think I might have gotten wrong is the name...Russell's paradox is the paradox that states that no set can contain itself, isn't it? If so, then isn't this a huge (possibly fatal) blow to Solipsism (which dictates that there is nothing that exists, except for what exists in my mind)? Any comments, constructive critiques, or corrections are appreciated. [:)] Register Now! Free! Talk Science! phoenixthoth first of all, that something contradicts an arbitrary contrivance such as logic means nothing. but i don't think solipsism leads to russell's paradox so that issue is moot. here's russell's paradox. one can view it as a theorem and not a paradox. it's based on the tautology

46. RussellsParadox Russell s paradox (English). Search for Russell s paradox in NRICH PLUS maths.org Google. Definition level 2. The paradox http://thesaurus.maths.org/mmkb/entry.html?action=entryById&id=1265

47. Russell's Paradox Definition Meaning Information Explanation Russell s paradox definition, meaning and explanation and more about Russell s paradox. FreeDefinition - Online Glossary and Encyclopedia, Russell s paradox. http://www.free-definition.com/Russells-paradox.html

A B C D ... Contact Beta 0.71 powered by: akademie.de PHP PostgreSQL Google News about your search term 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

48. Natural Religion > Glossary > Russell's Paradox Russell s paradox. The existence of russell s paradox points to a weakness in the assumptions made in the derivation of the paradox. http://www.naturaltheology.net/Glossary/russellParadox.html

Glossary Site Map Directory Search this site Home ... Synopsis Next: Previous: Glossary: Toc Development: toc Essays: toc Glossary: toc ... Supplementary: toc ... to restore theology to the mainstream of science Russell's paradox Hazewinkel , Russell's paradox] Borowski , Russell's paradox] An antinomy, contradiction or paradox is a situation in which two mutually contradictory statements (p and not-p) are demonstrated, each one having been deduced by means that are convincing from the point of view of the same theory. Russell worked on the paradox problem for years. He once wrote: 'Every morning', he later wrote, 'I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty. ... It seemed quite likely that the whole of the rest of my life might be consumed in looking at tha tblank sheet of paper.' Monk Cantor's Theorem Mendelson The existence of russell's paradox points to a weakness in the assumptions made in the derivation of the paradox. In particular, the assumption of an all inclusive (universal) set seems suspect. As Cantor himself proved, there appears to be no largest set. Click on an "Amazon" link in the booklist below to buy the book, see more details or search for similar items

49. The Paradox Of The Liar philosophical paradoxes. No. 2 Russell s paradox. Francis Moorcroft. The This paradox is a version of Russell s paradox. It came about 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 Russell s paradox. Bertrand Russell (18721970) constructed a famous paradox (an antinomy ) to persuade the mathematical world that http://users.forthnet.gr/ath/kimon/Russells_pdx.html

Russell's Paradox Bertrand Russell (1872-1970) constructed a famous paradox (an "antinomy") to persuade the mathematical world that in developing consistent systems (systems in which every statement is either true or false), familiarity and intuitive clarity are not solid bases. The argument goes on like this: There are sets than contain themselves (examples: "the set of all objects that can be described with exactly thirteen words", "the set of all thinkable things") Therefore, a set either contains itself or not. Let's call a set "non-normal" in the first case and "normal" in the second Let N be the collection of all normal sets, which of course, is itself a set Question: is N normal? If N is normal, then by definition of "normality" it does not contain itself. But N contains by construction all normal sets therefore itself too (contradiction) If N is not normal, then by definition of "non-normality" N is itself a member of N. But by construction, any member of N is a normal set (contradiction too) Conclusion: the statement "N is normal" is neither true nor false Famous Problems and Proofs Main Page

51. Arbitrarily Large Sets The Russell paradox on the Web. Considering Webmasters can appreciate the Russell paradox. As everyone knows, the web is about links. Any 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:

52. Online Encyclopedia - Russell's Paradox , Encyclopedia Entry for Russell s paradox. Russell s trick. Russell s paradox is closely related to the Liar paradox.......Encyclopedia http://www.yourencyclopedia.net/Russell's_Paradox.html

Encyclopedia Entry for Russell's Paradox Dictionary Definition of 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

53. Re: Russell's Paradox Re Russell s paradox. Posted by DickT on January 21, 2003 at 191947 In Reply to Russell s paradox posted by RocketMan on January 21, 2003 at 165805 http://superstringtheory.com/forum/qsboard/messages7/69.html

String Theory Discussion Forum String Theory Home Forum Index Re: Russell's paradox Follow Ups Post Followup Questions VII FAQ Posted by DickT on January 21, 2003 at 19:19:47: In Reply to: Russell's paradox posted by RocketMan on January 21, 2003 at 16:58:05: RocketMan, Have you come across the Spanish Barber? It was Russell's attempt to bring the paradox to the layman. In a certain Spanish town the barber (who is a man) shaves every man who does not shave himself. Who shaves the barber? So now. Some sets are members of themselves. For example the set of nonempty sets is a member of itself, because it is a nonempty set. Other sets are not members of themselves. For example the set of rusty anvils is not a member of itself since it is a set, not a rusty anvil. Consider then the set of all sets that are NOT members of themselves. Is it a member of itself? If it is, then it is like all the other members a set which is NOT a member of itself, so it cannot be a member of itself. Contradiction. Suppose it is NOT a member of itself, then by definition it IS a member of itself. Contradiction again. This sounds like a game to most of us, but it was deadly serious to Russell and the other early set theorists. They thought everything in math could be expressed through sets, and the fact that set theory could produce paradoxes was extremely shocking to them.

54. Re: Russell's Paradox Re Russell s paradox. Posted by sepiraph on January 25, 2003 at 223335 In Reply to Re Russell s paradox posted by DickT on January 21, 2003 at 191947 http://superstringtheory.com/forum/qsboard/messages7/78.html

String Theory Discussion Forum String Theory Home Forum Index Re: Russell's paradox Follow Ups Post Followup Questions VII FAQ Posted by sepiraph on January 25, 2003 at 22:33:35: In Reply to: Re: Russell's paradox posted by DickT on January 21, 2003 at 19:19:47: Just how was the paradox resolved? Did the set theorist resort to a different viewpoint in set theory? Does this have anything to do with axiom of choice that I keep hearing about? (Report this post to the moderator) Follow Ups: (Reload page to see most recent) Re: Russell's paradox DickT Post a Followup Follow Ups Post Followup Questions VII FAQ

55. Russell's Paradox :: Online Encyclopedia :: Information Genius Russell s paradox. Online Encyclopedia Russell s paradox is same trick. Russell s paradox is closely related to the Liar paradox. This http://www.informationgenius.com/encyclopedia/r/ru/russell_s_paradox.html

Quantum Physics Pampered Chef Paintball Guns Cell Phone Reviews ... Science Articles Russell's paradox Online 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. 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

56. Russell's Paradox Russell s paradox. The significance of Russell s paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. http://www.seop.leeds.ac.uk/archives/sum2000/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 Russell's Paradox The most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets which 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. Some sets, such as the set of 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 which 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 this century. History of the paradox Significance of the paradox Bibliography Other Internet Resources ... Related Entries History of the paradox Russell discovered his paradox in May 1901, while working on his

57. ABSTRACT For RUSSELL'S PARADOX Abstract. AD Irvine. Russell s paradox , The Stanford Encyclopedia of Philosophy (1995), http//plato.stanford.edu/entries/russellparadox/. Excerpt. http://www.philosophy.ubc.ca/faculty/irvine/wwwrp.htm

Abstract A.D. Irvine "Russell's Paradox" The Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/russell-paradox/ Excerpt 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 which 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. Some sets, such as the set of 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 which 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 this century. Summary This article consists of four short sections containing (1) a brief introduction to Russell's paradox, (2) the history of Russell's paradox, (3) the significance of Russell's paradox, and (4) a bibliography relating to the paradox. The article appears in The Stanford Encyclopedia of Philosophy . Unlike fixed-media reference works, this is an on-line reference work which is regularly revised in order that it not go out of date.

58. Russell's Paradox Russell s paradox. This is known as Russell s paradox, and was described in The Principles of Mathematics, chapter 10. To me it seems http://users.cgiforme.com/fbendz/messages/159.html

Russell's Paradox Post a new reply Back to the message board This message was posted by Fredrik Bendz , posted on September 24, 2000 at 17:33:07 coming from This message is a reply to Re: Bertrand Russel wrote something about this posted from Dan Proctor posted at September 15, 2000 at 05:17:55 The paradox of Russell is this: "I would never want to belong to any club that would have someone like me for a member." (Woodie Allen in Annie Hall Imagine a set M, which is the set of all sets which are not members (i.e. an element) of themselves. Now, ask yourself if M is such a set. By definition we have a paradox, which says that M is a set of itslef if and only if it is NOT a member of itself... :-) First imagine that M is a member of M. In that case M is a member of itself and should not be a member of M. Then imagine that M is not a member of M. In that case M is not a member of itself, and thus should be a member of M. This is known as Russell's Paradox, and was described in The Principles of Mathematics , chapter 10. To me it seems as an example of the liar paradox:

59. Re: Russell's Paradox Re Russell s paradox. This is known as Russell s paradox, and was described in The Principles of Mathematics, chapter 10. To me http://users.cgiforme.com/fbendz/messages/700.html

Re: Russell's Paradox Post a new reply Back to the message board This message was posted by Kartik , posted on August 01, 2002 at 03:02:57 coming from 203.200.224 This message is a reply to Russell's Paradox posted from Fredrik Bendz posted at September 24, 2000 at 17:33:07 > The paradox of Russell is this: > "I would never want to belong to any club that would have someone like me for a member." (Woodie Allen in Annie Hall > Imagine a set M, which is the set of all sets which are not members (i.e. an element) of themselves. Now, ask yourself if M is such a set. By definition we have a paradox, which says that M is a set of itslef if and only if it is NOT a member of itself... :-) > First imagine that M is a member of M. In that case M is a member of itself and should not be a member of M. > Then imagine that M is not a member of M. In that case M is not a member of itself, and thus should be a member of M. > This is known as Russell's Paradox, and was described in The Principles of Mathematics , chapter 10. To me it seems as an example of the liar paradox: > "This statement is a lie".

60. Russell's Paradox Russell s paradox. Russell s paradox. Russell proposed that it becomes paradoxical when the above mentioned gathering M is assumed to be a set. http://www.rinku.zaq.ne.jp/suda/incomplete/chap03_e.html

Russell's Paradox Previous Contents Japanese /English Definitions Definition 1. A set is the gathering of 'the one' which can be mutually identified to distinguish clearly and in case that the range to specify the whole of the gathering is clearly given. Definition 2. 'The one' which composes the gathering is called an element. Fundamental theory From definition 1, the set is the gathering of 'the one'. However, the mere gathering of 'the one' is not always a set. Even if it is a gathering of 'the one', if each of 'the one' cannot be mutually identified to distinguish clearly, it is not a set. Similarly, even if it is a gathering of 'the one', in case that the range to specify the whole of the gathering is not clearly given, it is not a set. Russell's Paradox Russell proposed that it becomes paradoxical when the above mentioned gathering M is assumed to be a set. The set which does not contain oneself as an element like the set of natural numbers, N exists. (A) Let the set M be the gathering of all sets which do not contain themselves as elements. (B) Either the set M contains oneself as an element or not.

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