61. A Geometric Note On Russell's Paradox A geometric note on Russell s paradox. Welcome A lengthier discussion of Russell s paradox is found on my web page on vacuous truth. It http://www.angelfire.com/az3/nfold/russell.html

var cm_role = "live" var cm_host = "angelfire.lycos.com" var cm_taxid = "/memberembedded" 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.

62. Three Intrepretations Of Russell's Paradox - A Poem By T Newfields Three Intrepretations of Russell s paradox R is the set of all sets which are not members of themselves. I. We are members of each http://galileo.spaceports.com/~newfield/cyber/russell.htm

Three Intrepretations of Russell's Paradox "R is the set of all sets which are not members of themselves." I We are members of each other and part of an infinite plane set in motion by fierce algorithms and suppositions traversing the Suffolk Plains. II I have examined your member and decided it is not correct. I therefore took it to the Ministry of Education and exorcized it from the list of accepted absurdities. TOP SECRET: In Room 302B of the History Dept. among fossilized witch bones you are re- membered. III Let's set this straight - you never were p art of my groin, nor do your in finitely skewed contortions belong to my set. My parameters are parabolic yet y ours are ergonomic. Still, there is an admi ration for your logic which transc ends the totality of theorems w here suppositions sup posedly lie. http://galileo.spaceports.com/~newfield/cyber/russell.htm T Newfields

63. Russell's Paradox Russell s paradox. Background. Russell s paradox shows that these appearances are deceiving. 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?

64. Russell's Paradox Russell s paradox. Suppose This was first noticed by Bertrand Russell in 1901, and so it has come to be known as Russell s paradox. To 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

65. Russell S Paradox NebulaSearch Home NebulaSearch Encyclopedia Top Russell s paradox. Main _Illinois Russell s paradox, NebulaSearch article for Russell s paradox. http://www.nebulasearch.com/encyclopedia/article/Russell's_paradox.html

66. Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks Russell s paradox. A very simple way how to do this was invented by Bertrand Russell in 1901, and is now called Russell s paradox first published in. 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: The online paper "The 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.

67. Russell's Paradox - Wikipedia, The Free Encyclopedia Russell s paradox. Russell s paradox is a paradox found by Bertrand Russell in 1901 which shows that naive set theory in the sense of Cantor is contradictory. http://www.phatnav.com/wiki/wiki.phtml?title=Russells_paradox

68. Russell's Paradox Russell s paradox. Gödel Escher Bach. Russell s paradox is one of the classic math paradoxes, this time based on sets that include themselves. 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.

69. Science & Technology At Scientific American.com: Ask The Experts: Mathematics: W What is Russell s paradox? Russell s paradox is based on examples like this Consider a group of barbers who shave only those men who do not shave themselves. http://www.sciam.com/askexpert_question.cfm?articleID=0005DA51-B5F4-1C71-9EB7809

70. Russells Paradox This site is for sale contact 1904-260-7599. Russell s paradox. Bertrand Russell Nobel Prize 1950. Russell s Mathematical Contributions. Russell paradox. 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.

71. Russell's Paradox - Nathanael Thompson- The Examined Life On-Line Philosophy Jou Russell s paradox. by. Nathanael Thompson. The Examined Life OnLine Philosophy Journal, Vol. 01 Issue 01. http://examinedlifejournal.com/articles/template.php?shorttitle=russellparadox&a

72. Russell's Paradox - Nathanael Thompson - The Examined Life On-Line Philosophy Jo Russell s paradox. by. Nathanael Thompson. The Examined Life OnLine Philosophy Journal, Vol. 01 Issue 01. When Gottlieb Frege made http://examinedlifejournal.com/articles/printerfriendly.php?shorttitle=russellpa

73. The Paradigms And Paradoxes Of Intelligence, Part 1: Russell's Paradox Origin Kurzweil Archives The Paradigms and paradoxes of Intelligence, Part 1 Russell s paradox Permanent link to this article http//www.kurzweilai.net http://www.kurzweilai.net/articles/art0257.html?m=10

74. The Paradigms And Paradoxes Of Intelligence, Part 1: Russell's Paradox KurzweilAI.net, The Paradigms and paradoxes of Intelligence, Part 1 Russell s paradox The above is my version of what has become known as Russell s paradox. http://www.kurzweilai.net/articles/art0257.html?printable=1

75. Russell's Paradox: The Third Crisis In Mathematics Russell s paradox The Third Crisis in Mathematics. Michael Todd Huddleston 1991. Abstract. This 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

76. 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

77. Russell 2001 -- One Hundred Years Of Russell's Paradox Russell 2001 One Hundred Years of Russell s paradox. International Conference 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

78. Russells Paradox - Encyclopedia Article About Russells Paradox. Free Access, No Russells paradox. Word Word. Russell s paradox is a paradox. History. Exactly when Russell discovered the paradox is not clear. http://encyclopedia.thefreedictionary.com/Russells paradox

Dictionaries: General Computing Medical Legal Encyclopedia Russells paradox Word: Word Starts with Ends with Definition Russell's paradox is a paradox This article is about the logic concept. Borland Paradox is a database management tool, and Paradoxx is a hard rock band. A paradox is an apparently true statement that seems to lead to a logical self-contradiction, or to a situation that contradicts common intuition. The identification of a paradox based on seemingly simple and reasonable concepts has often led to significant advances in science, philosophy and mathematics. Click the link for more information. discovered by Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (May 18, 1872 - February 2, 1970) was one of the most influential mathematicians, philosophers and logicians working (mostly) in the 20th century, an important political liberal, activist and a populariser of philosophy. Millions looked up to Russell as a sort of prophet of the creative and rational life; at the same time, his stance Click the link for more information.

79. Erik Benson: Russell's Paradox Archives Yes. the yellow room (0). testing new nokia 6600 (1). Search. Previous Entry. « selfevident. Next Entry. Media Watch ». See More about Russell s paradox. http://erikbenson.com/entries/2002/07/30/russells_paradox.html

Erik Benson Self-acclaimed, self-published author of Man Versus Himself Russell's Paradox Jul , 10:14 PM ( Bertrand Russell constructed a contradiction within the framework of elementary logic itself, showing that the idea of sets or classes of numbers do not necessarily have to follow conventional rules of logic. Here's an excerpt from Godel's Proof Classes seem to be of two kinds: those which do not contain themselves and those which do. A class will be called "normal" if, and only if, it does not contain itself as a member; otherwise it will be called "non-normal." An example of a normal class is the class of mathematicians, for patently the class itself is not a mathematician and is therefore not a member of itself. An example of a non-normal class is the class of all thinkable things; for the class of all thinkable things is itself thinkable and is therefore a member of itself. Let "N" by definition stand for the class of all normal classes. We ask whether N itself is a normal class. If N is normal, it is a member of itself (for by definition N contains all normal classes); but, in that case, N is non-normal, because by definition a class that contains itself as a member is non-normal. On the other hand, if N is non-normal, it is a member of itself (by definition of "non-normal"); but, in that case, N is normal, because by definition the members of N are normal classes. In short, N is normal if, and only if, N is non-normal. It follows that the statement "N is normal" is both true and false. This fatal contradiction results from an uncritical use of the apparently pellucid notion of "class." Other paradoxes were found later, each of them constructed by means of familiar and semingly cogent modesof reasoning. Mathematicians came to realize that in developing consisten systems familiarity and intuitive clarity are weak reeds to lean on.

80. Erik Benson: Russell's Paradox Archives Yes. laundro (0). my drain (0). Search. Previous Entry. « selfevident. Next Entry. Media Watch ». See More about Russell s paradox. Related http://erikbenson.com/entries/2002/07/30/russells_paradox.html?highlight_keyword

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