81. FOM: Re: Simpson On Russell's Paradox For Category Theory FOM Re Simpson on Russell s paradox for category theory. charles silver silver_1at mindspring.com Tue Apr 11 134657 EDT 2000 http://www.cs.nyu.edu/pipermail/fom/2000-April/003896.html

FOM: Re: Simpson on Russell's paradox for category theory charles silver silver_1 at mindspring.com Tue Apr 11 13:46:57 EDT 2000 Previous message: FOM: Orthogonal roles Next message: FOM: completeness theorem for stratification? Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] I would like an update on Solovay's post about Simpson's proof of Russell's Paradox for category theory. Charlie Silver Previous message: FOM: Orthogonal roles Next message: FOM: completeness theorem for stratification? Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... More information about the FOM mailing list

82. Information: Paradoxes And Antinomies What has been called Russell s paradox/antinomy was discovered by Russell in Russell s paradox/antinomy has come to be associated not with Russell s http://serendip.brynmawr.edu/local/scisoc/information/paradox.html

Paradoxes Center for Science in Society Working Group on Information (material stemming from and relevant to 1 July 2004 conversation) George Weaver There is a lot of confusion about what counts as a paradox. For some, a paradox is just some any conclusion that seems absurd and that has an argument to support it. Others talk of a person being in a paradox or facing a paradox when from assumptions that they believe, by methods of reasoning they find acceptable, they establish conclusions that they find unacceptable, perhaps because the conclusions are absurd. From this second view, if you don't accept the assumptions, or the methods of reasoning or the absurdity of the conclusion, then there is no paradox for you to face. Paradox, on this view, is in the eye of the beholder. There are those that distinguish between a paradox and an antinomy. For these folks an antinomy is a paradox that is absurd because the conclusion is a self-contradiction. Before I go further, a little history is in order. What has been called Russell's paradox/antinomy was discovered by Russell in 1901 and communicated by letter to Frege in 1902. There is a discussion of this discovery in Russell's Principles of Mathematics published in 1903. In 1908, Ernst Zermalo claimed to have discovered Russell's paradox/antinomy independently of Russell and to have written to Hillbert (among others) prior to 1903. Zermalo uses the term 'antinomy' rather than 'paradox'. Interestingly, no where in the 1902 letter nor in the 1903 book does Russell use the terms 'paradox' or 'antinomy' in discussing his discovery. In fact the heading in the section of Principles of Mathematics in which the discovery is discussed is 'The Contradiction'.

83. FoRK Archive: RE: Russell's Paradox On The Web It occurred to me a couple of days ago that Russell s paradox can be describedquite nicely in terms of web pages and links. It seems easier to http://www.xent.com/FoRK-archive/may98/0252.html

RE: Russell's Paradox on the Web Lisa Dusseault lisadu@exchange.microsoft.com Wed, 13 May 1998 10:18:42 -0700 Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Robert Harley: "Traitors, billionaires and sheep-shagging" Previous message: Joseph M. Reagle Jr.: "Re: FC: Responses to Why Microsoft hasn't violated Sherman Act" My fiance Eric: http://www.csclub.uwaterloo.ca/u/relipper/ used to maintain a page of links called "List of Pages with Links to This Page". A cool idea, and I immediately put up a page which linked to his page of links to that page, but he didn't bother to get many more links and dropped it eventually. I've always been fascinated by the idea of self-reference, so I maintain a page of self-referential (by self-modification) games: http://ofb.net/~lisa/games.html I've also started collecting meta-jokes, but haven't got enough to bother putting up a page of those yet. That was inspired by spending much time with my pure math friends. At Waterloo, mathie humour seemed to consist of: Meta-humour Null-set jokes Self-referential humour We were telling people this in a restaurant one night. Somebody gave Eric

84. FoRK Archive: Re: Russell's Paradox On The Web It occurred to me a couple of days ago that Russell s paradox can be describedquite nicely in terms of web pages and links. It seems easier to http://www.xent.com/FoRK-archive/may98/0208.html

Re: Russell's Paradox on the Web Lloyd Wood eep1lw@surrey.ac.uk Tue, 12 May 1998 21:26:51 +0100 (BST) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: CobraBoy: "Re: Prisoner of cyberspace" Previous message: Lloyd Wood: "RE: Prisoner of cyberspace" In reply to: James K. Tauber: "Russell's Paradox on the Web" On Wed, 13 May 1998, James K. Tauber wrote: by members of a mailing list who can't conceptualize sets? Surely you jest. L. L.Wood@surrey.ac.uk http://www.sat-net.com/L.Wood/ Next message: CobraBoy: "Re: Prisoner of cyberspace" Previous message: Lloyd Wood: "RE: Prisoner of cyberspace" In reply to: James K. Tauber: "Russell's Paradox on the Web"

85. Maverick Philosopher Russell's Paradox Explained The significance of Russell s paradox can be seen once it is realized that, Russell s paradox ultimately stems from the idea that any coherent condition http://maverickphilosopher.powerblogs.com/posts/1114014791.shtml

Maverick Philosopher Web Monkey's Tool Box Search This Site! HTML Color Codes Philosopher's Tool Box Athenaeum Library of Philosophy Butchvarov Papers Dennett's Dangerous Idea Internet Encyclopedia of Philosophy ... Stanford Encyclopedia of Philosophy Logician's and Linguist's Tool Box Online Etymology Dictionary Categories Main About Me, My Blog, and its Purposes Aboutness Academe ... You Call This Poetry? Contact William F. Vallicella Subscribe Titles RSS Full Content RSS Email Notification Get Posts by Email Recent Posts Consciousness Explained, or Explained Away? La Shawn Barber Man Invented Time; Seiko Perfected It Hocking on the Value of the Individual ... Does Dennett Deny Consciousness? Response to Pollack Archives Home Sep 2005 Aug 2005 Jul 2005 ... Mar 2005 E-Mail Policy Civil e-mail containing comments, constructive criticisms, and the like is gladly received, although I cannot promise to answer everything. I will, however, make an honest attempt. Offensive e-mail is deleted unread. Choose your subject headings carefully as I sometimes decide to delete from them. I do not open attached files from unknown parties. If you send a message not addressed to me in particular, I will be tempted to let someone in general answer it. Pith is king and neatness counts. E-mail is subject to posting in whole or in part at my discretion unless the sender requests otherwise. Comments Policy You must be pre-approved to leave comments on this site. If I know you and you have already established your bona fides, then you are in like Flynn, whether or not you hide behind a pseudonym. But if I don't know you, or you have proven yourself to be offensive in the past, then you are out like Stout — unless you reveal your real identity and provide me with some way of verifying it. In other words, no one I don't know will be approved who comments anonymously or pseudonymously. Full statement

86. One Hundred Years Of Russell's Paradox: Mathematics, Logic, Philosophy By Godehard Link (EDT), Walter De Gruyter Inc January 2004 ISBN 3110174383. http://www.thattechnicalbookstore.com/b3110174383.htm

Home Search Browse Shopping Cart ... Help QuickSearch (Words, Author, Subject, ISBN) Publishers Email a friend about this book One Hundred Years Of Russell's Paradox: Mathematics, Logic, Philosophy Format Hardcover Subject Mathematics / Research ISBN/SKU Author Godehard Link (EDT) Publisher Walter De Gruyter Inc Publish Date January 2004 Add to cart Price Qualified Frequent Buyer Price Ships from our store in 14 - 21 business days More delivery info here Review These 31 papers come from the June 2001 international conference held to commemorate the centenary of the discover of the famous "Russell's Paradox," and include contributions from Russell scholars, mathematical logicians, set theorists, and scholars in the philosophy of mathematics. Papers include an introduction by Godehard Link that credits Russell with the invention of the new mathematical philosophy, W. Hugh Woodin on set theory after Russell, Harvey Friedman on a way out of Russell's paradox, Sy Friedman on completeness and iteration in modern set theory, and John S. Bell on "Russell's Paradox and Diagonalization in a Constructive Context." Other papers include examinations of aspects of the Principia Mathematica, Russell on method, and critiques of works related to Russell's fields of study. Annotation ©2004 Book News, Inc., Portland, OR (booknews.com) QuickSearch (Words, Author, Subject, ISBN)

87. McMaster University: The Bertrand Russell Research Centre / Russell Journal ABSTRACT Russell discovered the classes version of Russell s paradox in spring1901, and the predicates version near the same time. There is a problem, http://www.humanities.mcmaster.ca/~russell/j2402.htm

Russell Russell: the Journal of Bertrand Russell Studies is published by The Bertrand Russell Research Centre , McMaster University. For ordering information, including prices, see the back issues table TABLE OF CONTENTS Editor's Notes Kevin C. Klement "The Origins of the Propositional Functions Version of Russell's Paradox" ABSTRACT: Russell discovered the classes version of Russell's Paradox in spring 1901, and the predicates version near the same time. There is a problem, however, in dating the discovery of the propositional functions version. In 1906, Russell claimed he discovered it after May 1903, but this conflicts with the widespread belief that the functions version appears in The Principles of Mathematics , finished in late 1902. I argue that Russell's dating was accurate, and that the functions version does not appear in the Principles . I distinguish the functions and predicates versions, give a novel reading of the Principles , section 85, as a paradox dealing with what Russell calls assertions , and show that Russell's logical notation in 1902 had no way of even formulating the functions version. The propositional functions version had its origins in the summer of 1903, soon after Russell's notation had changed in such a way as to make a formulation possible.

88. Resolving The Barber Paradox And The Russell S Paradox Resolving the Barber paradox and the Russell s paradox. Owen Holden Resolvingthe Barber paradox and the Russell s paradox http://www.physicsforums.com/archive/t-64685_Resolving_the_Barber_Paradox_and_th

Technology Services Philosophy Resolving the Barber Paradox and the Russell's Paradox Owen Holden - Resolving the Barber Paradox and the Russell's Paradox There is a man in a town that shaves all and only those that do not shave themselves. That is, this barber shaves x if and only if x does not shave x, for all x. Does the barber shave himself? No, he cannot shave himself! Where x and y are existent individuals and S is the relation 'shaves'... There is no existent individual that satisfies the description [an x such The 'barber' does not exist. He can't shave himself nor can he be shaven. All primary predications of 'the barber' are false. We cannot say what it is, but, we can say what it is not. The barber is not among the members of: those that do shave themselves, or, those that do not shave themselves. The barber cannot be: a man, a woman, a robot, etc. There is no entity that satisfies the description of 'the barber'. Also.. There is no existent individual class that satisfies the description [an x The 'Russell Class' does not exist.

89. ZFC And Russell S Paradox - Information Technology Services In particular, I m talking about Russell s paradox that shows {x x not in x} isnot a it s not clear how to prove Russell s paradox is impossible. http://www.physicsforums.com/archive/t-51980_ZFC_and_Russell's_Paradox.html

Technology Services Mathematics ZFC and Russell's Paradox Euclid - ZFC and Russell's Paradox How does ZFC manage to block Russell's paradox? I've read through the axioms extensively, and it's not clear how to prove Russell's paradox is impossible. Discuss ZFC and Russell's Paradox Here, Free! - ZFC and Russell's Paradox Basically, what ZFC does NOT have is an axiom that says every predicate can define a set. What it does have in its stead is an axiom that every predicate can define a subset of an existing set, plus a few other axioms to make up for the lost functionality. Hence there is no way to define Russell's set through the axioms of ZFC. Discuss ZFC and Russell's Paradox Here, Free! Hurkyl - ZFC and Russell's Paradox it's not clear how to prove Russell's paradox is impossible. I presume you see why Russel's construction isn't directly applicable: because the axiom of extensionality has been replaced with the axiom of subsets (and a few other axioms denoting how to "safely" construct sets). In particular, you are only allowed to say:

90. DIAMOND paradox Russell s paradox; Santa paradoxes; Game paradoxes; Diamond DiamondValues; Harmonic Functions; Diamond Circuits; Brownian Forms; Diamond Algebra http://www.worldscibooks.com/mathematics/3271.html

Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Bookshop New Titles Editor's Choice Bestsellers Book Series ... Series on Knots and Everything - Vol. 14 DIAMOND by N S Hellerstein (Lincoln University) This book is about "diamond", a logic of paradox. In diamond, a statement can be true yet false; an "imaginary" state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. Diamond is a new way to solve the dilemmas of higher mathematics. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book consists of two sections: Elementary; which covers the classic paradoxes of mathematical logic and shows how they can be resolved in this new system; and Advanced, which relates diamond to Boolean logic, three-valued logic, Gödelian meta-mathematics and dilemma games. Contents: Paradox: Russell's Paradox Santa Paradoxes Game Paradoxes Diamond: Diamond Values Harmonic Functions Diamond Circuits Brownian Forms Diamond Algebra: Laws Normal Forms Completeness and Categoricity Self-Reference: Re-entrance and Fixedpoints Phase Order The Outer Fixedpoints Fixedpoint Lattices: Relative Lattices Shared Fixedpoints Limit Logic: Limit Fixedpoints Diamond Computation Paradox Resolved: Russell's Paradox Santa Sentences Antistrephon Game Paradoxes The Continuum: Cantor's Paradox Dedekind Splices Zeno's Theorem Analytic Functions: Analytic Functions Dihedral Conjugation Harmonic Analysis: Harmonic Projection

91. Science - Conmathematical Resolution Of Russell's Paradox Russell s paradox A paradox uncovered by Bertrand Russell in 1901 that forced a One version of Russell s paradox, known as the barber paradox, http://www.articlehut.com/Goarticles/Science/conmathematical-resolution-of-russe

Free Articles..Health Articles..Computer Articles..ArticleHut.com Conmathematical Resolution of Russell's Paradox by Dr Kedar Joshi Russell's Paradox - ( See David Darling : The Universal Book of Mathematics, 2004 ) Conmathematical Resolution - The term 'Conmathematics' means conceptual mathematics ( invented by Dr. Kedar Joshi ( b. 1979 ), Cambridge, UK ). It is a meta - mathematical system that defines the structure of superultramodern mathematics. It essentially involves a heavy or profound conceptual approach which is in striking contrast with the traditional symbolic or set theoretic approach. Now conmathematically Russell's paradox is quite easy to resolve. The conmathematical resolution could be stated in just one sentence : As there is no barber who shaves every man who doesn't shave himself, and no one else, likewise there is no set of all sets that aren't members of themselves. This sentence is justified or explained below. Suppose there is a barber who shaves every man who doesn't shave himself, and no one else. Now the barber himself is a man and the supposition requires that the barber shave himself if and only if he does not ! This contradiction straightaway implies that the supposition is false. That is, there is no barber who shaves every man who doesn't shave himself, and no one else. The justification of the sentence 'there is no set of all sets that aren't members of themselves' goes on similar lines.

92. A Contingent Russell's Paradox, Francesco Orilia A Contingent Russell s paradox. Source Notre Dame J. Formal Logic 37, no.1 (1996), 105–111 Abstract. It is shown that two formally consistent typefree http://projecteuclid.org/Dienst/UI/1.0/Display/euclid.ndjfl/1040067319

Current Issue Past Issues Search this Journal Editorial Board ... Note on Volumes 35-40 Francesco Orilia A Contingent Russell's Paradox Source: Notre Dame J. Formal Logic Abstract: It is shown that two formally consistent type-free second-order systems, due to Cocchiarella, and based on the notion of homogeneous stratification, are subject to a contingent version of Russell's paradox. References Primary Subjects: Full-text: Open access Download the full-text in the following format: PDF (42 KB) Screen Optimized PDF (59 KB) Euclid Identifier: euclid.ndjfl/1040067319 Mathmatical Reviews number (MathSciNet): Digital Object Identifier (DOI): 10.1305/ndjfl/1040067319 Zentralblatt Math Identifier: To Table of Contents for this Issue References [1] Chierchia, G., Topics in the Syntax and Semantics of Infinitives and Gerunds , Ph.D. dissertation, University of Massachusetts, Amherst, 1984. Mathematical Reviews: [2] Cocchiarella, N., ``The theory of homogeneous simple types as a second order logic,'' Notre Dame Journal of Formal Logic Mathematical Reviews: Zentralblatt-MATH: Project Euclid: euclid.ndjfl/1093882656

93. Type-free Property Theory, Exemplification And Russell's Paradox., Francesco Ori Typefree property theory, exemplification and Russell s paradox. Source NotreDame J. Formal Logic 32, no. 3 (1991), 432–447 Primary Subjects 03A05 http://projecteuclid.org/Dienst/UI/1.0/Display/euclid.ndjfl/1093635839

Current Issue Past Issues Search this Journal Editorial Board ... Note on Volumes 35-40 Francesco Orilia Type-free property theory, exemplification and Russell's paradox. Source: Notre Dame J. Formal Logic Primary Subjects: Seconday Subjects: Full-text: Open access Download the full-text in the following format: PDF (1755 KB) Euclid Identifier: euclid.ndjfl/1093635839 Zentralblatt Math Identifier: Mathmatical Reviews number (MathSciNet): To Table of Contents for this Issue journals search login ... home

94. Set Theory:Naive - Wikibooks Russell s paradox. One of the most celebrated paradoxes is Russell s. BertrandRussell, the English philosopher and logician, discovered this paradox in http://en.wikibooks.org/wiki/Set_Theory:Naive

Set Theory:Naive From Wikibooks In the late 19th century, when Cantor proved his theorem and mathematicians' understanding of infinity developed, set theory was not the rigorously axiomatised subject it is today. It relied upon woolly intuitions about what sets were and their relationship with their members. This lack of rigour led to several paradoxes. In Naive Set Theory, something is a set if and only if it is a well-defined collection of objects. Sets count as objects. Members are anything contained by a set. Any two sets containing precisely the same members are the same set ( Principle of Extensionality Russell's Paradox One of the most celebrated paradoxes is Russell's. Bertrand Russell, the English philosopher and logician, discovered this paradox in 1901. It centres on the set containing all and only those sets that do not contain themselves. It is impossible to answer the question 'does this set contain itself?' without running into contradiction. If it contains itself, it is by definition a set that does not contain itself - contradiction. If it does not contain itself, it is a set that does not contain itself and so should contain itself - contradiction. A parallel semantic paradox is the Barber's paradox. The Barber shaves everyone in town who does not shave himself and only those people. Does the Barber shave himself? If so, he is not someone who does not shave himself. If he does not, he should. In this case, the Barber's contract is inconsistent and so, by analogy, is the Naive Set Theory upon which Russell's paradox is constructed.

95. Russell S Paradox. Russell s paradox. Take Frege s axiom (from his formalization of set theory) thatevery concept establishes the existence of the set of things that satisfy http://grimpeur.tamu.edu/~colin/Phil416/russpara.html

Russell's paradox. Take Frege's axiom (from his formalization of set theory) that every concept establishes the existence of the set of things that satisfy the concept. First consider the concept of sets that are members of themselves. For example, the set of abstract objects is itself an abstract object and hence a member of itself, but the set of cows is not a cow so it does not belong to itself. Now consider the concept of being a set that does not belong to itself, and consider the set of all sets that are not members of themselves. Ask "Does this set belong to itself?" If the answer is yes, then it shouldn't. If the answer is no, then it should. So there can be no such set. So not every concept guarantees the existence of a set.

96. Wo's Weblog: Idle Remarks On Russell's Paradox And Higher-order Entities I will first describe a general version of Russell s paradox, Where R is therelation of classmembership, Russell s paradox proves that there is no http://www.umsu.de/wo/archive/2002/11/01/Idle_remarks_on_Russell_s_paradox_and_h

wo's weblog Musings in analytical philosophy Friday, 01 November 2002 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

97. Wo's Weblog: What Does Russell's Paradox Teach In Semantics? (Russell s paradox is an independent argument for the same conclusion.) Now if, as I have claimed, Russell s paradox teaches a general lesson about http://www.umsu.de/wo/archive/2003/04/07/What_does_Russell_s_Paradox_Teach_in_Se

wo's weblog Musings in analytical philosophy Monday, 07 April 2003 What does Russell's Paradox Teach in Semantics? Philosophy On Friday, I wrote: Conclusion 2: If we want to avoid Bradley's regress, there is no reasonable way to defend the principle that every meaningful expression of our language has a semantic value. (Russell's paradox is an independent argument for the same conclusion.) Today, I was trying to prove the statement in brackets. This is more difficult than I had thought. Semantic paradoxes usually (always?) arise out of an unrestricted application of schemas like 'p is true' iff p; 'F' is true of x iff F(x); 't' denotes x iff t=x. The paradoxes prove that these schemas have false instances and therefore aren't generally correct. (Maybe they are correct only for a certain part of our language, the relevant 'object language'; or maybe they are correct only when "iff" is replaced with some non-standard operator; Anyway, the important thing is that, as they are, they are not generally correct.) So they can't be used to define the concepts "true", "true of", "denotes", etc.

98. Boards.ie/vbulletin - Russell's Paradox (aka Bash The Noob) View Full Version Russell s paradox (aka bash the noob) This is a paradoxbecause if the SOASWANMOT is a member of itself, it is disqualified, http://www.boards.ie/vbulletin/archive/index.php/t-246711.html

boards.ie/vbulletin Soc Philosophy PDA View Full Version : Russell's Paradox (aka bash the noob) ExOffender The set of all sets which are not members of themselves(SOASWANMOT): is it a member of itself? This is a paradox because if the SOASWANMOT is a member of itself, it is disqualified, and as soon as it is disqualified, it is eligible, and as soon as... you get the picture. It must be one or the other, it cannot be both and cannot be neither. So... What about the set of all sets which are members of themselves (SOASWAMOT)? Again, it seems to me, it must be one or the other, can't be both, can't be neither. And yet... there is no determining causal factor to say. Must be one or the other. No determining factor. Indeterminism? I have no direct philosophical background, as I'm sure is clear. The only possible solution I can come up with is that there's some default setting in cases like this - like the 'absolute value' law in maths that prevents mensurative calculations returnng negative amounts. Any thoughts? Be gentle...

99. Math Forum Discussions Topic Russell s paradox As Nuclear Logic Discovered. By AiyaOba Replies 0 Russell s paradox is a glimpse at the Absolute logic http://mathforum.org/kb/thread.jspa?threadID=1171028&tstart=0

100. Metamath Proof Explorer - Ru Russell s paradox. Proposition 4.14 of TakeutiZaring p. 14. Thus in a very real sense Russell s paradox spawned the invention of ZF set...... http://us.metamath.org/mpegif/ru.html

Home Metamath Proof Explorer Related theorems Unicode version Theorem ru Description: Russell's Paradox. Proposition 4.14 of [ TakeutiZaring ] p. 14. Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as , asserted that any collection of sets is a set i.e. belongs to the universe of all sets. In particular, by substituting for , it asserted , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove . This contradiction was discovered by Russell in 1901 (published in 1903), invalidating Comprehension and leading to the collapse of Frege's system. In 1908 Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex asserting that is a set only when it is smaller than some other set . However, Zermelo was then faced with a "chicken and egg" problem of how to show is a set, leading him to introduce the set-building axioms of Null Set , Pairing prex , Union uniex , Power Set pwex , and Infinity omex to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom

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