41. Russell's Paradox Bertrand Russell (18721970) constructed a famous paradox (an antinomy ) topersuade the mathematical world that in developing consistent systems (systems 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

42. Russell's Paradox Russell s paradox is one of the classic math paradoxes, this time based on setsthat include themselves. The set of all sets, for example, includes itself. http://community.middlebury.edu/~dwalker/class/russell.html

Russell's Paradox Gödel Escher Bach Russell's paradox is one of the classic math paradoxes, this time based on sets that include themselves. The set of all sets, for example, includes itself. The set of all sets that do not include themselves, however, sparks the paradox. If the set does not include itself, then it is in the set, but since the set then includes itself, it is not in the set. This set exists IFF it does not exist, a contradiction in terms. The paradox was discovered in 1901 by Bertrand Russell. Cut The Knot includes an excerpt from Russell's autobiography about paradoxes. Erasing Russell's Paradox gives a group of axioms that allow the avoidance of Russell's Paradox. Read a poem about Russell's Paradox.

43. Russel's Paradox How does this tie in with Russell s paradox? Okay, so we ve established thatRussell s paradox exists because we are relying on incorrect premises. http://jhunix.hcf.jhu.edu/~blee27/essays/russels_paradox.htm

GEEK'S HAVEN GEEK'S HAVEN About News Free IQ Test Contact Info ... Links SOFTWARE Simulations Capture The Flag The Motion Applet QED Lens Applications AutoControl Rhyme Tagboy The GNU GPL ESSAYS Scientific Fun With Psi Russel's Paradox Ancient Mathematics Creative Satan Sophistry No Regrets Quiet Revolution POETRY About the Poems My Poems Things to do Wooded Forest My Song Rendezvous ... Kind of Slick Other Poems The Raven Song of Myself STORIES Blindman Kallista Story Ticket to Trouble Russel's Paradox This is a short little essay I wrote for a Discrete Mathematics class I took. Oh, and by "short little essay" I mean "hideously too long essay". Russell's Paradox Philosophical Ponderings of a Confused Discrete Student So, who is this Russell dude anyway, and what is his paradox? Russell was a mathematician that introduced an interesting concept sometime about the time the 19th century was rolling into the 20th. He asked us to consider a question. Something along the lines of, "Does the set of all sets that do not contain themselves contain itself?" A mouthful. How do we answer such a question? Let X be the set of all sets that do not contain themselves. The question now becomes "Is X a member of X?"

44. Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks What is Mathematics? Goedel s Theorem and Around. Textbook for students. Section 2.By K.Podnieks. http://www.ltn.lv/~podnieks/gt2.html

set theory, axioms, Zermelo, Fraenkel, Frankel, infinity, Cantor, Frege, Russell, paradox, formal, axiomatic, Russell paradox, axiom, axiomatic set theory, comprehension, axiom of infinity, ZF, ZFC Back to title page Left Adjust your browser window Right 2. Axiomatic Set Theory For a general overview and set theory links, see Set Theory by Thomas Jech in Stanford Encyclopedia of Philosophy 2.1. Origin of Cantor's Set Theory In the dates and facts of the real history I am following the excellent books by Fyodor Andreevich Medvedev F. A. Medvedev. Development of Set Theory in the XIX Century. Nauka Publishers, Moscow, 1965, 350 pp. (in Russian) F. A. Medvedev. The Early History of the Axiom of Choice. Nauka Publishers, Moscow, 1982, 304 pp. (in Russian) See also: Online paper "A history of set theory" in the MacTutor History of Mathematics archive A. Kanamori . Set Theory from Cantor to Cohen, Bulletin of Symbolic Logic, 1996, N2, pp.1-71 (online text at http://math.bu.edu/people/aki/cancoh.ps In XIX century, development process of the most basic mathematical notions led to the intuition of arbitrary infinite sets. Principles of the past mathematical thinking were developed up to their logical limits. Georg Cantor did the last step in this process, and this step was forced by a specific mathematical problem.

45. Russells Paradox Russell s paradox arises as a result of naive set theory s socalled Most attempts at resolving Russell s paradox have therefore concentrated on various http://www.literature-awards.com/nobelprize_winners/russells_paradox.htm

This site is for sale contact 1-904-260-7599 Russell's Paradox Bertrand Russell Nobel Prize 1950 Russell's Mathematical Contributions Prize Presentation Writings of Bertrand Russell Nobel Lecture Russell's Biography Over a long and varied career, Bertrand Russell made ground-breaking contributions to the foundations of mathematics and to the development of contemporary formal logic, as well as to analytic philosophy. His contributions relating to mathematics include his discovery of Russell's paradox, his defense of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his introduction of the theory of types, and his refining and popularizing of the first-order predicate calculus. Along with Kurt Gödel, he is usually credited with being one of the two most important logicians of the twentieth century The Autobiography of Bertrand Russell Hardcover Paperback Russell discovered the paradox which bears his name in May 1901, while working on his Principles of Mathematics (1903). The paradox arose in connection with the set of all sets which are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. The significance of the paradox follows since, in classical logic, all sentences are entailed by a contradiction. In the eyes of many mathematicians (including David Hilbert and Luitzen Brouwer) it therefore appeared that no proof could be trusted once it was discovered that the logic apparently underlying all of mathematics was contradictory. A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was thus prompted.

46. Russell's Paradox - Linix Encyclopedia Russell s paradox (also known as Russell s antinomy) is a paradox discovered Similarly, Russell s paradox proves that an encyclopedia entry titled List http://web.linix.ca/pedia/index.php/Russell's_paradox

Russell's paradox Russell's paradox (also known as Russell's antinomy ) is a paradox discovered by Bertrand Russell in which shows that the naïve 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. Table of contents showTocToggle("show","hide")

47. Read About Russell's Paradox At WorldVillage Encyclopedia. Research Russell's Pa Russell s paradox. Everything you wanted to know about Russell s paradox but hadno clue how to find it.. Learn about Russell s paradox here! http://encyclopedia.worldvillage.com/s/b/Russell's_Paradox

Culture Geography History Life ... WorldVillage Russell's paradox From Wikipedia, the free encyclopedia. (Redirected from Russell's Paradox Russell's paradox (also known as Russell's antinomy ) is a paradox discovered by Bertrand Russell in which shows that the naïve 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. Contents 1 History 2 Applied versions 3 Set-theoretic responses 3.1 Responses illustrated

48. Russell's Paradox The lesson that most mathematicians have drawn from Russell s paradox is that Thus, Russell s paradox can be thought of as a proof that there can be no http://www.cs.amherst.edu/~djv/pd/help/Russell.html

Russell's Paradox x x . This statement will be true for some values of x and false for others. It is tempting to think that we could form the set of all values of x for which the statement is true. In other words, it is tempting to think that the expression x x should be accepted as a definition of a set. However, the assumption that such expressions always name sets leads to a contradiction. This was first noticed by Bertrand Russell in 1901, and so it has come to be known as Russell's Paradox To see how the paradox is derived, suppose that all expressions of the type displayed above do name sets. Russell suggested that we consider the following definition of a set R R x x x According to this definition, an object x will be an element of R if and only if x x . But now suppose we ask whether or not R is an element of itself. Plugging in R for x in the definition of R , we come to the conclusion that R R if and only if R R . But this is impossible; whether R is an element of itself or not, this statement cannot be true. Thus we have reached a contradiction. The lesson that most mathematicians have drawn from Russell's Paradox is that definitions of the kind displayed above cannot always be trusted to define sets. To avoid the paradox, mathematicians use only a restricted form of this kind of definition. If

49. Russell’s Paradox One version of Russell s paradox, known as the barber paradox, Russell s paradox,in its original form considers the set of all sets that aren t members http://www.daviddarling.info/encyclopedia/R/Russells_paradox.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z entire Web this site Russell’s paradox A paradox uncovered by Bertrand Russell in 1901, which forced a reformulation of set theory S of all sets that aren't members of themselves? Somehow, S is neither a member of itself nor not a member of itself. Russell discovered this strange situation while studying a foundational work in symbolic logic by Gottlob Frege . After he described it, set theory had to be reformulated axiomatically in a way that avoided such problems. Russell himself, together with Alfred North Whitehead, developed a comprehensive system of types in Principia Mathematica . Although this system does avoid troublesome paradoxes and allows for the construction of all of mathematics, it never became widely accepted. Instead, the most common version of axiomatic set theory in use today is the so-called Zermelo-Fraenkel set theory , which avoids the notion of types and restricts the universe of sets to those that can be built up from given sets using certain axioms. Russell's paradox underlies the proof of

50. Russell's Paradox Russell s paradox. Russell s paradox. In the middle of the night I got sucha fright that woke me with a start, For I dreamed of a set that contained 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

51. Contexts Of Paradox In one popular version, Russell s paradox asks us to imagine a village where Russell s paradox signaled the end of the period of youthful innocence in http://www.maa.org/devlin/devlin_11_03.html

Search MAA Online MAA Home Devlin's Angle November 2003 Contexts of Paradox In one popular version, Russell's Paradox asks us to imagine a village where the barber shaves all men who do not shave themselves. The question then is, who shaves the barber? A straightforward attempt to answer this question leads you in a circle from which there seems to be no escape. If the barber shaves himself, then he does not shave himself. But then he does shave himself. But then he does not. And so on ad infinitum. (The possibility of a female barber did not arise at the start of the 20th Century, when Bertrand Russell formulated this puzzle.) Of considerably greater significance for the development of mathematics was the original set-theoretic version of the paradox: Let R be the set of all sets that are not members of themselves. Is R a member of itself or not? If it is, then it isn't, and if it isn't then it is. Russell's discovery of this paradox about set formation effectively destroyed the lifetime's work of Gottlob Frege in trying to establish a foundational framework for mathematics based on formal logic. The paradox was resolved or some would say sidestepped only by the formulation of axioms for set theory a few years later. Russell's Paradox signaled the end of the period of youthful innocence in Cantor's Set Theory, during which it was believed that to every property could be associated a well-defined set, namely the set of all objects having that property.

52. Russell's Paradox Re Russell s paradox by Henno Brandsma (May 3, 2005). From GMC; Date May 3, 2005;Subject Russell s paradox. Russell s paradox http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2005;task=show_msg;msg=2046

53. Re: Russell's Paradox In reply to Russell s paradox , posted by GMC on May 3, 2005 Russell s paradox Consider the set X of all sets that do not contain themselves. http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2005;task=show_msg;msg=2046.

54. Re: What Part Could Russell's Paradox Play In Quantam Mechanics? Russell s paradox is a mathematical aspect of set theory, seehttp//plato.stanford.edu/entries/russellparadox/ Russell s paradox was notdeveloped in a http://www.madsci.org/posts/archives/aug2000/966439824.Ph.r.html

MadSci Network : Physics Re: What part could Russell's Paradox play in quantam mechanics? Date: Tue Aug 15 11:44:53 2000 Posted By: Sidney Chivers, , Nuclear Engineering, retired Area of science: Physics ID: 964797763.Ph Message: Russell's Paradox is a mathematical aspect of set theory, see http://plato.stanford.edu/entries/russell-paradox/ Russell's Paradox was not developed in a search for alternatives to the Heisenberg Uncertainty Principle, and there is nothing inherent in the statement of Russell's Paradox that would suggest it holds promise as such an alternative. Though mathematics is very significant throughout physics, not every mathematical concept can be assumed to have significance in physics. That said, you may be interested is another concept, the measurement problem, http://plato.stanford.edu/entries/qt-measurement/ You can find at least one related response in the MadSci Archives using the search term measurement problem and many other by searching different combinations of the terms Heisenberg uncertainty quantum Thanks for your question. Current Queue Current Queue for Physics Physics archives Try the links in the MadSci Library for more information on Physics MadSci Home Information Search ... Join Us!

55. What Part Could Russell's Paradox Play In Quantam Mechanics? Subject What part could Russell s paradox play in quantam mechanics? Date FriJul 28 112243 2000 Posted by Anthony Mason http://www.madsci.org/posts/archives/aug2000/966439824.Ph.q.html

MadSci Network : Physics Subject: What part could Russell's Paradox play in quantam mechanics? Date: Fri Jul 28 11:22:43 2000 Posted by Anthony Mason Grade level: School: No school entered. City: No city entered. State/Province: No state entered. Country: No country entered. Area of science: Physics ID: 964797763.Ph Message: In theory, given that specific technology could 'negate' or 'bypass' the Heisenberg Uncertainty Principle, what part could Russell's Paradox have in the field of quantam observation and measurement? (I hesitate to use the word 'control') Re: What part could Russell's Paradox play in quantam mechanics? Current Queue Current Queue for Physics Physics archives Try the links in the MadSci Library for more information on Physics MadSci Home Information Search ... Join Us! MadSci Network, webadmin@www.madsci.org

56. Encyclopaedia Morningtonia: Russell's Paradox categories during the course of one game, their opponent will think that theyare trying to prove the Mornington Crescent version of Russell s paradox. http://kevan.org/morningtonia.pl?Russell's_Paradox

57. Russell's Paradox He converted Russell s paradox, the set version, into a statement in NumberTheory, and showed that Number Theory is inconsistent. 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.

58. E.W.Dijkstra Archive: Where Is Russell's Paradox? (EWD 923a) Why is Russell s paradox called a paradox? It is supposed to be rendered bythe example of the village with cleanshaven men in which the village barber is http://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD923a.html

EWD 923A Where is Russell's paradox? Why is "Russell's Paradox" called a paradox? It is supposed to be rendered by the example of the village with cleanshaven men in which the village barber is defined as the man such that the population he shaves is the population of villagers that don't shave themselves. And then comes the question "Who shaves the barber?". But where is the paradox? If we define in a certain context x to be equal to 3, we define x to satisfy x =3, i.e. to be a root of the equation y y y y y ≥0)? After the invention of the reals that equation has indeed one root. Did we only know the rationals, the equation would have no root and √2 "would not exist". For the barber of the village we have the equation y A i i is a villager : i shaver i y shaver i and that equation has no solution. Conclusion: the village has no barber. Where is the paradox? Probably I am very naive, but I also think I prefer to remain so, at least for the time being and perhaps for the rest of my life. Austin, 22 May, 1985

59. E.W.Dijkstra Archive: For Brevity's Sake (EWD 1070) Since the turn of the century, Russell s paradox is a standard ingredient of Today we are not interested in Russell s paradox, we are solely interested http://www.cs.utexas.edu/users/EWD/transcriptions/EWD10xx/EWD1070.html

EWD 1070 For brevity's sake Since the turn of the century, Russell's Paradox is a standard ingredient of all texts on the foundation of mathematics. Its purpose is to illustrate the trouble one can run into by admitting the seemingly innocent notion of "the set of all sets". We shall not pursue that trouble here, since this is not a book on the foundation of mathematics. Today we are not interested in Russell's Paradox, we are solely interested in its presentation. Being essentially a one-liner, it is a very simple example to make our point; it has the added advantage of being familiar to many. Its usual presentation is along the following lines. A set may be a member of itself or not. Consider now the set of all sets that are not a member of themselves. Calling this set R , its formal definition would be R x x x —to be read as "the set of all sets x such that x x —. The question we now try to answer is: "Is R a member of itself?". Suppose that R is not a member of itself"; since, according to (0), R contains all such sets

60. Russell's Paradox -- Facts, Info, And Encyclopedia Article Similarly, Russell s paradox proves that an encyclopedia entry titled List The whole point of Russell s paradox is that the answer such a set does not http://www.absoluteastronomy.com/encyclopedia/R/Ru/Russells_paradox.htm

var dc_UnitID = 10; var dc_PublisherID = 512; var dc_BackgroundColor1 = 'white'; var dc_BackgroundColor2 = 'white'; var dc_TitleColor = 'blue'; var dc_TextColor = 'black'; var dc_URLColor = 'blue'; var dc_URLVisitedColor = 'green'; var dc_sm_type = 'horizontal'; var dc_Width = 700; var dc_Height = 75; var dc_caption_font_bgColor = 'white'; var dc_caption_font_color = 'blue'; var dc_OutBorder = 'no'; var dc_adprod='TM'; Russell's paradox [Categories: Set theory, Paradoxes] Russell's paradox (also known as Russell's antinomy ) is a ((logic) a self-contradiction) paradox discovered by (English philosopher and mathematician who collaborated with Whitehead (1872-1970)) Bertrand Russell in 1901 which shows that the (Click link for more info and facts about naïve set theory) naïve set theory of (The official of a synagogue who conducts the liturgical part of the service and sings or chants the prayers intended to be performed as solos) Cantor and Frege is contradictory. Consider the set M to be "The set of all sets that do not contain themselves as members". Formally:

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