21. Russell S Paradox - BlueRider.com Other Services Word Index Contact Us About Links. russell s paradox listen domain availability russell s paradox. Your search results http://russell's paradox.bluerider.com/wordsearch/Russell's paradox

22. Russell's Paradox - Computing Reference - ELook.org In lambdacalculus Russell s paradox can be formulated by representing each setby its Previous Terms, Terms Containing Russell s paradox, Next Terms http://www.elook.org/computing/russell's-paradox.htm

eLook Home eLook Computing Reference Search Computing Reference More Resources Information US City Data Worldwide Weather Entertainment Video Game Cheats Food Nutrition Recipes Literature Literature Reference Computing Dictionary Internet Encyclopedia Programming Reference ... Thesaurus By Letter: Non-alphabet A B C ... Z Russell's Paradox 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

23. Archival Information For "Russell's Paradox" paradox Internet Encyclopedia of PhilosophyRussell s paradox represents either of two interrelated logical antinomies. This blocks Russell s paradox, because the formula used to define the http://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=russell-parado

24. Russell-Myhill Paradox [Internet Encyclopedia Of Philosophy] After discovering Russell s paradox in 1901 while working on his UnlikeRussell s paradox, this paradox cannot be blocked by the simple theory of http://www.iep.utm.edu/p/par-rusm.htm

Russell-Myhill Paradox The Russell-Myhill Antinomy, also known as the Principles of Mathematics Appendix B Paradox, is a contradiction that arises in the logical treatment of classes and "propositions", where "propositions" are understood as mind-independent and language-independent logical objects. If propositions are treated as objectively existing objects, then they can be members of classes. But propositions can also be about classes, including classes of propositions. Indeed, for each class of propositions, there is a proposition stating that all propositions in that class are true. Propositions of this form are said to "assert the logical product" of their associated classes. Some such propositions are themselves in the class whose logical product they assert. For example, the proposition asserting that all-propositions-in-the- class-of-all-propositions -are-true is itself a proposition, and therefore it itself is in the class whose logical product it asserts. However, the proposition stating that all-propositions-in-the- null-class -are-true is not itself in the null class. Now consider the class

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

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents 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.

26. Russell's Paradox - Wikipedia, The Free Encyclopedia Russell s paradox (also known as Russell s antinomy) is a paradox The wholepoint of Russell s paradox is that the answer such a set does not exist http://en.wikipedia.org/wiki/Russell's_paradox

Russell's paradox From Wikipedia, the free encyclopedia. Russell's paradox (also known as Russell's antinomy ) 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 disputed — see talk page . 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 (but see Independence from Excluded Middle below). 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. Contents History Applied versions Set-theoretic responses Responses illustrated ... edit History Exactly when Russell discovered the paradox is not clear. It seems to have been May or June

27. Barber Paradox - Wikipedia, The Free Encyclopedia The paradox considers a town with a male barber who shaves every man who doesnot shave a British logician who in 1901 constructed Russell s paradox to http://en.wikipedia.org/wiki/Barber_paradox

28. PlanetMath: Russell's Paradox This is version 5 of Russell s paradox, born on 200110-18, modified 2003-07-14.Object id is 316, canonical name is Russellsparadox. 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 ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About 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 of set theory could be stated like ``If the barber shaves all those who do not themselves shave, does he shave himself?''. If you answer himself that is false since he only shaves all those who do not themselves shave. If you answer someone else that is also false because he shaves all those who do not themselves shave and in this case he is part of that set since he does not shave himself. Therefore we have a contradiction. "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:

29. Erasing Russell's Paradox Axiomatically avoids the Russell paradox. Erasing Russell s paradox. Home Cantor s Theorem Numbers. The tacit and erroneous assumptions underlying 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).

30. Set Theory And Paradoxes Statement of the paradox. The Barber of Seville. Russell s paradox may be formulatedin several equivalent ways. Classically, it is known as the Barber 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

31. Russell's Paradox: Information From Answers.com Russell s paradox Russell s paradox (also known as Russell s antinomy ) is aparadox discovered by Bertrand Russell in 1901 which shows that the. http://www.answers.com/topic/russell-s-paradox

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Russell's paradox Wikipedia Russell's paradox Russell's paradox (also known as Russell's antinomy ) 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 (but see Independence from Excluded Middle below). 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.

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

Godehard Link Godehard Link

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

Russell 2001 Photo Gallery Description Latest Info Russell 2001 Photo Gallery Description Latest Info ... Funding

34. Russell's Paradox The significance of Russell s paradox can be seen once it is realized that, Zermelo s axioms were designed to resolve Russell s paradox by restricting http://setis.library.usyd.edu.au/stanford/archives/fall1997/entries/russell-para

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 Related Entries History of the paradox Russell discovered his paradox in May 1901, while working on his

35. Russell’s Paradox Russell’s paradox is the most famous of the logical or settheoretical paradoxes . Russell appears to have discovered his paradox in May of 19011 while http://setis.library.usyd.edu.au/stanford/archives/fall2001/entries/russell-para

This is a file in the archives of the Stanford Encyclopedia of Philosophy 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.

36. Russell's Paradox -- From MathWorld Russell s paradox. COMMENT On this Page. SEE Russell s Antinomy. Pages Linking Here http://mathworld.wolfram.com/RussellsParadox.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Russell's Paradox SEE: Russell's Antinomy [Pages Linking Here]

37. Russell's Paradox Russell s paradox can be put into everyday language in many ways. The most oftenrepeated is Another popular form of Russell s paradox is the following 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

38. Natural Religion > Glossary > Russell's Paradox Hazewinkel, Russell s paradox Borowski, Russell s paradox The existenceof russell s paradox points to a weakness in the assumptions made in the 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

39. Russell's Paradox Russell s paradoxRussell s paradox, also referred to as Russel s antinomy, Russell s problem, The origins of Russell s paradox are even more controversial than the http://www.seop.leeds.ac.uk/archives/sum2003/entries/russell-paradox/

This is a file in the archives of the Stanford Encyclopedia of Philosophy version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... Z This document uses HTML 4/Unicode to display special characters. Older browsers and/or operating systems may not display these characters 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.

40. Cantor's Paradox Next Russell s paradox Up An Historical Account of Previous origins ofRussell s paradox suggest that his paradox was derived from Cantor s paradox 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

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