61. One Hundred Years Of Russell's Paradox - Abstracts One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor s theorem that the cardinality of any http://www.lrz.de/~russell01/papers.html

Abstracts The following papers have so far been announced: We present an approximation space (U,R) which is an infinite (hypercontinuum) solution to the domain equation U isomorphic to C(R), (U, c: U-> U, i: U-> U), where c(u) = Union [u]_R and i(u)= Intersection [u]_R. John Bell Russell's Paradox and Diagonalization in a Constructive Context One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor's theorem that the cardinality of any set is strictly less than that of its power set. The other method of proving Cantor's theorem-employed by Cantor himself in showing that the set of real numbers is uncountable-is that of diagonalization. Typically, diagonalization arguments are used to show that function spaces are "large" in a suitable sense. Classically, these two methods are equivalent. But constructively they are not: while the argument for Russell's paradox is perfectly constructive, (i.e. employs on intuitionistically acceptable principles of logic) the method of diagonalization fails to be so. In my paper I shall describe the ways in which these two methods diverge in a constructive setting. Ulrich Blau The Significance of the Largest and Smallest Numbers for the Oldest Paradoxes Wilfried Buchholz On Gentzen's consistency proofs for arithmetic Gentzen has given three consistency proofs for arithmetic: "Der erste Widerspruchsfreiheitsbeweis fuer die klassische Zahlentheorie", Galley proof of sections IV and V of Gentzen 1936, Archiv Math.Logik 16(1974)

62. Russell S Paradox Russell s paradox, named after its discovery Bertrand Rusell, is a mathematical paradox based on set theory. Russell appears to have discovered his paradox http://www.mathresource.iitb.ac.in/project/russell.html

63. Free News, Magazines, Newspapers, Journals, Reference Articles And Classic Books Free Online Library Wittgenstein s Tractatus 3.333 and Russell s paradox.( Ludwig Wittgenstein and Bertrand Russell , Report) by Trames ; Social sciences, http://www.thefreelibrary.com/Wittgenstein's Tractatus 3.333 and Russell'

64. St. Anselm S Ontological Argument Succumbs To Russell S Paradox Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.springerlink.com/index/m43127273284521t.pdf

65. The Alligator Show: Episode 54: Russell's Paradox : The Alligator -- I Might Be This episode s description has nothing to do with its content; but maybe the title does. Subscribe to this podcast on iTunes, or at Blubrry. http://www.archive.org/details/TheAlligatorShowEpisode54RussellsParadox

66. Russell's Paradox: A Stepping Stone Towards Modern Set Theory Mathematical statements are either true or false. When they are neither, they remind mathematicians to formulate their theories more accurately. http://www.suite101.com/content/russells-paradox-a168549

67. ACA0, Russell's Paradox, ZF, Arithmetic Feb 14, 2006 That article appears to imply that ZF is open to Russell s paradox. That is not correct. However, my initial argument for it not being http://www.acooke.org/cute/ACA0Russel0.html

Andrew Cooke Contents Latest RSS ... Next ACA0, Russell's Paradox, ZF, Arithmetic Date: Tue, 14 Feb 2006 10:33:40 -0300 (CLST) I have joined the FOM mailing list - http://www.cs.nyu.edu/mailman/listinfo/fom/ Someone there mentioned ACA0, RCA0. ACA0 is, I think, the Axiom of Comprehension - http://planetmath.org/encyclopedia/ComprehensionAxiom.html - that every formula defines a set. That article appears to imply that ZF is open to Russell's paradox. That is not correct. However, my initial argument for it not being correct (from the Regularity Axiom) is wrong. There's a good comment here - http://en.wikipedia.org/wiki/Axiom_of_regularity http://en.wikipedia.org/wiki/Second-order_arithmetic Peano arithmetic - http://en.wikipedia.org/wiki/Peano_arithmetic An article on Frege that is beginning to make more sense, now that I understand more of the notation and history - http://plato.stanford.edu/entries/frege-logic/ Andrew Comment on this post I am always interested in offers/projects/new ideas. Eclectic experience in fields like: numerical computing; Java web/enterprise; functional languages; Python client GUI/web/database; etc. Based in Santiago, Chile; telecommute worldwide. CV email Last 1000 entries: Watching Wal-Mart at Midnight Also, From The Book

68. Russell's Paradox From FOLDOC Russell's Paradox mathematics A logical contradiction in set theory discovered by the British mathematician Bertrand Russell (18721970). If R is the set of all sets which don't http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Russell's paradox

69. A Geometric Note On Russell's Paradox A lengthier discussion of Russell s paradox is found on my web page on vacuous truth. It is there that I discuss banishing actual infinity and replacing http://www.angelfire.com/az3/nfold/russell.html

A geometric note on Russell's paradox Welcome to N-fold Quirky notes to myself on math and science If you spot an error, please email me This page went online Dec. 28, 2001 When truth is vacuous (includes a section on Russell's paradox) Reach other N-fold pages A lengthier discussion of Russell's paradox is found on my web page on vacuous truth. It is there that I discuss banishing 'actual infinity' and replacing it with construction algorithms. However, such an approach, though countering other paradoxes, does not rid us of Russell's, which is summed up with the question: If sets are sorted into two types: those that are elements of themselves ('S is the name of a set that contains sets named for letters of the alphabet' is an example) and those that are not, then what type of set is the set that contains sets that are elements of themselves? Here we regard the null set as the initial set and build sets from there, as in: Using an axiom of infinity, we can continue this process indefinitely, leading directly to a procedure for building an abstraction of all countable sets; indirectly, noncountable sets can also be justified. In Russell's conundrum, some sets are elements of themselves.

70. Legal Theory Blog: Carlson On Legal Positivism & Russell's Paradox Mar 11, 2009 David Gray Carlson (Yeshiva University Benjamin N. Cardozo School of Law) has posted Russell s Paradox and Legal Positivism on SSRN. http://lsolum.typepad.com/legaltheory/2009/03/carlson-on-legal-positivism-russel

71. Russell's Paradox - Definition Russell's Paradox definition from the mondofacto online medical dictionary http://www.mondofacto.com/facts/dictionary?Russell's Paradox

72. Russell's Paradox (Stanford Encyclopedia Of Philosophy/Winter 2008 Edition) Dec 8, 1995 Russell s paradox is the most famous of the logical or settheoretical paradoxes . The paradox arises within naive set theory by considering http://www.seop.leeds.ac.uk/archives/win2008/entries/russell-paradox/

Winter 2008 Edition Cite this entry Search this Archive Advanced Search Table of Contents ... Stanford University This is a file in the archives of the Stanford Encyclopedia of Philosophy Russell's Paradox First published Fri Dec 8, 1995; substantive revision Thu May 1, 2003 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. 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 the late spring of 1901

73. A Good Historical Example Is Russell S Paradox File Format PDF/Adobe Acrobat Quick View http://www2.sunysuffolk.edu/fultonj/MA21/Russells Paradox.pdf

74. Russells Paradox Explained - By Andrew Edge - Helium It would not be at all unusual for an individual to believe that there exist only two kinds of meaningful propositions. First, there are those pro , http://www.helium.com/items/1508763-russells-paradox-explained

Where Knowledge Rules Search Home Sciences Biology ... for this title @import url("/css/widget.css"); Russell's Paradox explained Top Article of 1 Write now Article Tools by Andrew Edge It would not be at all unusual for an individual to believe that there exist only two kinds of meaningful propositions. First, there are those propositions (or assertions) that are demonstrably true or false. An example of one of these is the proposition that it is currently raining in a particular locale. In order to determine whether this is or is not the case one must simply turn to various widely accepted methods of gathering contemporaneous weather data. Alternatively, there are those propositions that are not demonstrably true or false. These would be the propositions that indeed are either true or false but we, as mankind, do not yet have enough information to make the proper determination. A fine example of this, of course, is the proposition that God exists. Logically, whether or not God exists is demonstrable but the human race has not yet developed the means and/or had the opportunity to make a widely accepted determination as to which is the case. What follows from the above discussion is the idea that all meaningful propositions that assert a truth or falsehood are decidable if the totality of facts concerning the universe were readily available. It will be shown below, however, that strictly speaking this very idea is a false one. Enter: Bertrand Russell.

75. Arbitrarily Large Sets Webmasters can appreciate the Russell Paradox. As everyone knows, the web is about links. Any page worth its salt has links to other pages. http://descmath.com/diag/russ.html

The Russell Paradox on the Web Considering the great amount of interest in the web, I think it is easier to introduce the reflexive paradox in the context of the internet than in the abstract realm of set theory. Webmasters can appreciate the Russell Paradox. As everyone knows, the web is about links. Any page worth its salt has links to other pages. Some pages (like my little Grand Junction Links Page ) have nothing but links. A web crawler is a program that crawls through pages on a web site. The typical web crawler reads a web page, then follows each of the links one that page. Web crawlers have to worry about infinite loops. The simplest infinite loop happens when a page contains a link back to itself. For example, this page has the name russ.html . I made the name hot. The page links back on itself. It is "self-referential." A web crawler needs to watch out for self-referential pages; Otherwise, it would fall into an infinite loop. If the bot was not programmed to handle recursive links, the bot would read a page, then follow the link back to the page, and read it again... To avoid infinite loops, the web crawler needs to maintain a database of all the places it has visited. Now, we get into the problem that caused Bertrand Russell such angst a century ago:

76. Russell& - Definition Of Russell& By The Free Online Dictionary, Thesaurus And E (Philosophy / Logic) Logic the paradox discovered by British philosopher and mathematician Bertrand Russell (18721970) in the work of Gottlob Frege, http://www.thefreedictionary.com/Russell's paradox

77. Russell's Paradox - Paradox And Philosophy Discussion 5 posts 3 authors - Last post Jan 19Russell s Paradox is discussed at protheory.com - Paradox and Philosophy Discussion. http://www.fprotheory.com/showthread.php?t=45

78. Russell S Paradox From FOLDOC Nov 1, 2000 In lambdacalculus Russell s Paradox can be formulated by representing each set by its characteristic function - the property which is true http://foldoc.org/Russell's paradox

79. Russell's Paradox (Set Theory) Russell s paradox relates to a branch of philosophy called set theory. The sets were assumed to be hierarchical and this is where the paradox arises http://www.protheory.com/russell.html

Home Pro Theory Pro Answers Forum ... Discuss Russell's Paradox Russell's Paradox Introduction Is the ultimate set of all sets a member of itself or not? This paradox appears to be singularly unsolvable. Philosophy Russell's paradox relates to a branch of philosophy called set theory. Sets As its name suggests this theory uses the idea of categorising all parts of a problem into different sets or groups. Rules The rules for creating these sets are infinitely debatable, it depends what method you choose to divide your different objects, ideas, or answers. Relative All details are relative in other words. Top of Page Assumed The sets were assumed to be hierarchical and this is where the paradox arises from. Paradox The paradox within set theory and many similar theories is whether the set of all sets is a member of itself. Example Is the word "word" a word? Summary Russell noticed it was impossible to conclusively prove beyond all doubt that the highest possible set was or indeed was not a member of itself. The Problem Is the set of all sets a member of itself or not?

80. Mathematical Mysteries: The Barber's Paradox | Plus.maths.org May 1, 2002 So now we realise that Russell s Barber s Paradox means that there is a contradiction at the heart of naïve set theory. That is, there is a http://plus.maths.org/content/mathematical-mysteries-barbers-paradox

Skip to Navigation about Plus support Plus Plus sponsors ... subscribe to Plus Search this site: by Helen Joyce Issue 20 Submitted by plusadmin on April 30, 2002 in foundations of mathematics Mathematical mysteries set theory Theory of Types ... Zermelo-Fraenkel axiomatisation of set theory May 2002 A close shave for set theory Suppose you walk past a barber's shop one day, and see a sign that says "Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else." This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself? If he does, then he mustn't, because he doesn't shave men who shave themselves, but then he doesn't, so he must, because he shaves every man who doesn't shave himself... and so on. Both possibilities lead to a contradiction. This is the Barber's Paradox, discovered by mathematician, philosopher and conscientious objector Bertrand Russell, at the begining of the twentieth century. As stated, it seems simple, and you might think a little thought should show you the way around it. At worst, you can just say "Well, the barber's condition doesn't work! He's just going to have to decide who to shave in some different way." But in fact, restated in terms of so-called "naïve" set theory, the Barber's paradox exposed a huge problem, and changed the entire direction of twentieth century mathematics. In naïve set theory, a set is just a collection of objects that satisfy some condition. Any clearly phrased condition is thought to define a set - namely, those things that satisfy the condition. Here are some sets:

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