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
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
Extractions: 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'.
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
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
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
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Extractions: 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.
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
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
Extractions: 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!
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
Extractions: 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
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
Extractions: 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. 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.
A Contingent Russell's Paradox, Francesco Orilia A Contingent Russell s paradox. Source Notre Dame J. Formal Logic 37, no.1 (1996), 105111 Abstract. It is shown that two formally consistent typefree http://projecteuclid.org/Dienst/UI/1.0/Display/euclid.ndjfl/1040067319
Extractions: Current Issue Past Issues Search this Journal Editorial Board ... Note on Volumes 35-40 Francesco Orilia 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: Zentralblatt Math Identifier: To Table of Contents for this Issue [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
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
Extractions: 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.
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
Extractions: 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.
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
Extractions: 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
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
Extractions: 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.
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
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
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
Extractions: 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
