41. Russell's Paradox russell's paradox. Suppose contradiction. This was first noticed by BertrandRussell in 1901, and so it has come to be known as russell's paradox. 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

42. The New York Review Of Books: Russell's Paradox Gallery · NYR Books The New York Review of Books August 13, 1992.Review. russell's paradox. By Stuart Hampshire. The Selected Letters http://www.nybooks.com/articles/2834

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

43. Wo's Weblog: Idle Remarks On Russell's Paradox And Higher-order Entities Idle remarks on russell's paradox and higherorder entities. I will first describea general version of russell's paradox, of which Rieger's is a special case. http://www.umsu.de/wo/archive/1036163154

wo's weblog Idle remarks on Russell's paradox and higher-order entities philosophy Okay, as promised here comes the third and last part of my little series on Rieger's paradox. I will first describe a general version of Russell's paradox, of which Rieger's is a special case. Then I'll discuss whether Frege is already prey to the paradox by his admission of too many concepts. Whether he is will depend on whether it makes sense to say that there are entities which are not first-order entities. I'm sorry that there is probably nothing new in all this. First, the general version of Russell's paradox. Let R be any relation. Suppose there is some thing t such that all and only the (possibly zero) things which are not R -related to themselves are R -related to t . Then x( R(x,x) R(x,t)) . But then R(t,t) R(t,t) Contradiction. Hence there is no such thing. Examples. 1. Where R is the relation of class-membership, Russell's paradox proves that there is no class t of which all and only the classes that are not members of themselves are members. 2. Where R is the relation of satisfaction between things and predicates, Russell's paradox proves that there is no predicate

44. The Empty Set And Russell's Paradox The empty set and russell's paradox. Subject The empty set and russell's paradox;From Oz Oz@upthorpe.demon.co.uk ; Date Wed, 24 Oct 2001 210221 +0100; http://www.lns.cornell.edu/spr/2001-10/msg0036273.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index The empty set and Russell's paradox Subject : The empty set and Russell's paradox From Date : Wed, 24 Oct 2001 21:02:21 +0100 Approved : baez@math.ucr.edu Followups-To : sci.math Message-ID frZKzzCN5x17EwNI@upthorpe.demon.co.uk Newsgroups : sci.physics.research,sci.math Organization : State of Minnesota Intertech Prev by Date: Cech cohomology and bundles Next by Date: Re: Cech cohomology and bundles Prev by thread: Re: Cech cohomology and bundles Next by thread: pseudo gap in high Tc superconductivity Index(es): Date Thread

45. English Definition Of Russell's Paradox - WordReference.com For 'russell's paradox' we found Source The Collins English Dictionary© 1998 HarperCollins Publishers russell's paradox n. Logic. http://www.wordreference.com/English/definition.asp?en=russell's paradox

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

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

48. Russell's Paradox: The Third Crisis In Mathematics russell's paradox The Third Crisis in Mathematics. Michael Todd Huddleston 1991. AbstractThis thesis is concerning a mathematical problem, russell's paradox. http://www.nsula.edu/scholars_college/Thesis/Thesisabstracts/SItheses/Huddleston

Russell's Paradox: The Third Crisis in Mathematics Michael Todd Huddleston science theses author directory Abstract This thesis is concerning a mathematical problem, Russell's Paradox. What I have done with this problem is researched it throughly and attempted to find its solution. While during this, I also had to investigate Cantor's theory (the origin of Russell's Paradox) and three philosphies: Logicism, Intuitionism, and Formalism. Cantor's theory deals with sets in the infinite and the three philosphies are each a different mathematics that attempts to stay clear of contradictions such as Russell Paradox. After extensive research I chose Logicism to he the best solution. It is not the complete solution, yet it is the best one that I believe works. last update 1/11/03

49. Russell 2001 -- One Hundred Years Of Russell's Paradox Russell 2001 One Hundred Years of russell's paradox. InternationalConference in Logic and Philosophy. 0205 juin 2001 Munich, Germany http://conferences.atala.org/conferences/fiches/russell2001.html

Russell 2001 One Hundred Years of Russell's Paradox International Conference in Logic and Philosophy 02-05 juin 2001 Munich, Germany http://www.lrz-muenchen.de/~russell01/ source : Colibri

50. Russell's Paradox And The Law Of Excluded Middle Russell’s Paradox and the Law of Excluded Middle. On 31 Jan 1997, Chriswrote On 30 Jan 1997, Mark wrote . That is Russell’s Paradox. http://personal.bgsu.edu/~roberth/russell.html

Russell’s Paradox and the Law of Excluded Middle On 31 Jan 1997, Chris wrote: The Law of the Excluded Middle In addition to the two cases Mark mentioned – systems of logic with more than two truth-values and fuzzy concepts that aren’t sufficiently crisp to decide for all cases that something either is or is not “A,” there’s another interesting issue. Suppose that your term, A, is crisply defined. Still, to sensibly say that everything is either A or not-A, you need, at least implicitly, some kind of restriction to a domain or universe of discourse within which it applies. Whatever is not within that domain will be neither A nor not-A. Why can’t you just say, “I mean the domain to cover everything ”? Because you have to impose some restrictions on what gets included to avoid falling into various sorts of paradoxes. (Are impossibilities part of “everything”?) The most famous is Russell’s Paradox which deals with sets of sets that do or do not include themselves. Briefly, he proved that if you allow a set of all sets that do not include themselves, you can prove both that if it does include itself, then it doesn’t, and if it doesn’t, then it does. But to avoid Russell’s Paradox, you have to say that some things that are “sayable” don’t count as part of everything. So – back to the more restricted point – you always, whether explicitly or not, have to refer to a domain for “A or not-A” to have determinate sense. Then, anything outside that domain won’t count as either A or not-A.

51. November: RE: Russell's Paradox RE russell's paradox. Next message Ryan Jamieson russell's paradox Previous message Ryan Jamieson RE russell's paradox ; http://www.math.yorku.ca/Who/Faculty/Steprans/Courses/3500/m0211/0004.html

RE: Russell's paradox From: Ryan Jamieson ( yu273274@yorku.ca Date: Fri Nov 01 2002 - 01:40:11 EST Next message: Ryan Jamieson: "Russell's paradox" Previous message: Ryan Jamieson: "RE: Russell's paradox" Next in thread: Juris Steprans: "RE: Russell's paradox" Reply: Juris Steprans: "RE: Russell's paradox" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] > Penrose. It states, "R is the set of all sets that are not members of > themselves". This is a fascinating concept that, when one In "Truth, Proof, and Insight," there is a footnote indicating that a "set just means a collection of things physical objects or mathematical concepts that can be treated as a whole." We can define a set, S, to be some number of "pipes," say, 5 pipes. definition except that I have delimited the set to 5 pipes. I could change the definition of S to the set of "all students in Math 3500." Now that delimits the domain of discrete objects that I'm referring to in the universe to exactly the set of students in our class. In the

52. November: RE: Russell's Paradox RE russell's paradox. From EST. Next message Ryan Jamieson RE Russell'sparadox Previous message yu262646@yorku.ca (no subject) ; http://www.math.yorku.ca/Who/Faculty/Steprans/Courses/3500/m0211/0003.html

RE: Russell's paradox From: Ryan Jamieson ( yu273274@yorku.ca Date: Fri Nov 01 2002 - 01:39:38 EST Next message: Ryan Jamieson: "RE: Russell's paradox" Previous message: yu262646@yorku.ca: "(no subject)" Next in thread: Ryan Jamieson: "RE: Russell's paradox" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] > Penrose. It states, "R is the set of all sets that are not members of > themselves". This is a fascinating concept that, when one In "Truth, Proof, and Insight," there is a footnote indicating that a "set just means a collection of things physical objects or mathematical concepts that can be treated as a whole." We can define a set, S, to be some number of "pipes," say, 5 pipes. definition except that I have delimited the set to 5 pipes. I could change the definition of S to the set of "all students in Math 3500." Now that delimits the domain of discrete objects that I'm referring to in the universe to exactly the set of students in our class. In the article, I believe a similar example of all red things is used.

53. Mathenomicon.net : Reference : Russell's Paradox russell's paradox. noun. http://www.cenius.net/refer/display.php?ArticleID=russellsparadox

54. Untitled \documentstyle12pt{article} \begin{document} \title{russell's paradox} \author{JohnT. Baldwin and Olivier Lessmann \\ Department of Mathematics, Statistics http://www.math.uic.edu/~jbaldwin/pub/russ1.html

x=2 ') and mathematical properties (such as `even numbers'). In Frege's development, one could freely use any property to define further properties. Russell's paradox demonstrated a fundamental limitation of such a system. In modern terms, it is best described in terms of sets, using so-called 'set-builder' notation. For example, we can describe the collection of numbers 4, 5 and 6 by saying that x is the collection of integers n which are greater than 3 and less than 7; we write this formally as is an integer and < n . The objects don't have to be numbers. We might let is a male resident of the United States . Seemingly, any description of x could fill the space after the colon. But Russell (and independently, Ernst Zermelo) noticed that is not in leads to a contradiction in the same way as the description of the barber. Is x itself in the set x ? Either answer leads to a contradiction. When Russell discovered this paradox, Frege immediately saw that it had a devastating effect for his system. He was unable to resolve the paradox and there have been many further attempts in the last century to avoid it. Russell's own answer was to elaborate a `theory of types.' The problem in the paradox, he reasoned, is that we are confusing a description of sets of numbers with a description of sets of sets of numbers. So Russell introduced a hierarchy of objects: numbers, sets of numbers, sets of sets of numbers, etc. This system served as vehicle for the first formalizations of the foundations of mathematics and is still used in some philosophical investigations and in branches of computer science. Zermelo's solution to Russell's paradox is to replace the axiom: `for every formula

55. Russell's Paradox Q5. russell's paradox is a model of the third valued proposition in four valuedlogic. So, what is a symbolic expression of the third valued proposition? Q6. http://www.elix.com/ptxQA5678.htm

Plain text version 1998.4 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 ? 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? 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<=>not p.

56. Russell's Paradox 5. russell's paradox is a model of the third valued proposition in four valuedlogic. 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

57. Poincare's Concept Of Impredicativity, Russell's Paradox Poincare's concept of impredicativity, russell's paradox. To phillogic@bucknell.edu ;Subject Poincare's concept of impredicativity, russell's paradox; http://hhobel.phl.univie.ac.at/phlo/200009/msg00159.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Poincare's concept of impredicativity, Russell's paradox To phil-logic@bucknell.edu Subject : Poincare's concept of impredicativity, Russell's paradox From : "Drake O'Brien" < dobrien@lightspeed.bc.ca Date : Thu, 28 Sep 2000 12:36:42 -0700 References 200009270605.IAA12223@beta13.sm.luth.se Reply-To phil-logic@bucknell.edu Sender owner-phil-logic@bucknell.edu http://www.utm.edu/research/iep/p/poincare.htm References Feferman's completeness result From: Prev by Date: The polyvalued logic used by the 13,5 billion market? Next by Date: Polyvalued first order logic Prev by thread: Feferman's completeness result Next by thread: The polyvalued logic used by the 13,5 billion market? Index(es): Date Thread

58. Re: Kratoo CS1154 And Russell's Paradox (on Sets) CS1202 Re Kratoo CS1154 and russell's paradox (on sets). Iam not a mathemagician. Can this be compared in any way to russell's paradox ? http://www.vcharkarn.com/snippets/board/show_message.php?dtn=dtn20&ID=CS1202

59. Russell's Paradox russell's paradox. Background. russell's paradox shows that these appearances aredeceiving. The above ideas are not innocent, and in fact cannot all be correct. http://www.ilstu.edu/~kfmachin/phi281/russells_paradox.htm

Russell's Paradox Background A set (called a "class" by Russell) is a mathematical object. A set may be thought of informally as any collection of items. So, there would be a set of all the currently enrolled ISU students. Sets have members. The members of the set just mentioned are all the currently enrolled ISU students. Usually when people think of sets, they don't think of sets as having other sets as members. But there is no reason why a set can't have another set as a member. For example, we might think of the set that consists of all the sets of dishes currently for sale at Target. That set has other sets as its only members. In Frege's ground-breaking work on logic, there were five axioms. The fifth one basically said that any well-formed predicate expression with one gap defines a setnamely, the set whose members are all the items that the predicate expression is true of. So, '( ) is a set of dishes currently for sale at Target' would be an example of such a predicate expression, and its members would be all the sets of dishes currently for sale at Target. All this seems completely innocent, obvious, and perhaps even somewhat boring. Russell's paradox shows that these appearances are deceiving. The above ideas are not innocent, and in fact cannot all be correct. There is in fact a contradiction lurking in the very simple, innocent-looking ideas just described! How can this be?

60. 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

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