61. Epimenides Paradox -- Facts, Info, And Encyclopedia Article link for more info and facts about Russell s paradox) Russell s paradox, the BuraliForti paradox and the paradox now called Russell s paradox. http://www.absoluteastronomy.com/encyclopedia/E/Ep/Epimenides_paradox.htm

Epimenides paradox [Categories: Paradoxes] The Epimenides paradox is a problem in (The branch of philosophy that analyzes inference) logic . This problem is named after the (A native or inhabitant of Crete) Cretan philosopher (Click link for more info and facts about Epimenides) Epimenides of (An ancient town on Crete where Bronze Age culture flourished from about 2000 BC to 1400 BC) Knossos (flourished circa 600 BC), who stated , "Cretans, always liars". There is no single statement of the problem; a typical variation is given in the book (Click link for more info and facts about Gödel, Escher, Bach) Gödel, Escher, Bach (page 17), by (Click link for more info and facts about Douglas R. Hofstadter) Douglas R. Hofstadter Epimenides was a Cretan who made one immortal statement: "All Cretans are liars." Did Epimenides speak the truth? We will first consider the logical status of his statement and then review the history of this famous quote. Logical analysis If we define "liar" to mean that every statement made by a liar is false (so that Epimenides' statement amounts to "Anything said by a Cretan is false"), then the statement "All Cretans are liars," if uttered by the Cretan Epimenides, cannot be consistently true. (And, as will be noted below, according to one interpretation it also cannot be consistently false, either.) The conjunction of "Epimenides said all Cretans are liars" and "Epimenides is a Cretan" would, if true, imply that a Cretan has truthfully asserted that no Cretan has ever spoken the truth; the truth of Epimenides' statement would be a counterexample (some Cretan has told the truth at least once) and would mean that not all Cretans are necessarily always liars, which would contradict Epimenides' statement and thus would render it false.

62. Russell's Paradox Anyone heard of Russell s paradox. I recently encountered this set theory contradiction and am having trouble processing it. I ve studied Venn diagrams in http://www.superstringtheory.com/forum/qsboard/messages7/68.html

String Theory Discussion Forum String Theory Home Forum Index Russell's paradox Follow Ups Post Followup Questions VII FAQ Posted by RocketMan on January 21, 2003 at 16:58:05: Anyone heard of Russell's paradox. I recently encountered this "set theory" contradiction and am having trouble processing it. I've studied Venn diagrams in my high school Finite class and some algebra/geometry in first year but that's about it. I don't understand how some sets cannot be members of themselves, or for that matter why some sets are members of themselves. What exactly does that mean anyway? Can anyone explain the paradox in lay? Thanks. Chris (Report this post to the moderator) Follow Ups: (Reload page to see most recent) Re: Russell's paradox DickT Re: Russell's paradox sepiraph ... RocketMan Post a Followup Follow Ups Post Followup Questions VII FAQ

63. Re: Russell's Paradox It was Russell s attempt to bring the paradox to the layman. In a certain Spanishtown the barber (who is a man) shaves every man who does not shave himself http://www.superstringtheory.com/forum/qsboard/messages7/69.html

String Theory Discussion Forum String Theory Home Forum Index Re: Russell's paradox Follow Ups Post Followup Questions VII FAQ Posted by DickT on January 21, 2003 at 19:19:47: In Reply to: Russell's paradox posted by RocketMan on January 21, 2003 at 16:58:05: RocketMan, Have you come across the Spanish Barber? It was Russell's attempt to bring the paradox to the layman. In a certain Spanish town the barber (who is a man) shaves every man who does not shave himself. Who shaves the barber? So now. Some sets are members of themselves. For example the set of nonempty sets is a member of itself, because it is a nonempty set. Other sets are not members of themselves. For example the set of rusty anvils is not a member of itself since it is a set, not a rusty anvil. Consider then the set of all sets that are NOT members of themselves. Is it a member of itself? If it is, then it is like all the other members a set which is NOT a member of itself, so it cannot be a member of itself. Contradiction. Suppose it is NOT a member of itself, then by definition it IS a member of itself. Contradiction again. This sounds like a game to most of us, but it was deadly serious to Russell and the other early set theorists. They thought everything in math could be expressed through sets, and the fact that set theory could produce paradoxes was extremely shocking to them.

64. Math Lair - Paradoxes An article about Russell s paradox at the Stanford Encyclopedia of Philosophy.Greeling s paradox A version of Russell s paradox using words. http://www.stormloader.com/ajy/paradoxes.html

Paradoxes A paradox is a statement that goes against our intuition but may be true, or a statement that is self-contradictory. The paradoxes listed below and most other mathematical paradoxes fall into one of two categories: either they result from the counter-intuitive properties of infinity , or are a result of self-reference. Some Famous Paradoxes: Zeno's Paradox Russell's Paradox One can classify sets into one of two categories. The first category is sets that are not members of themselves. This contains most of the sets we run into in "real life". For example, the set of all penguins falls in the first category, because the set of all penguins is a set, not a penguin. On the other hand, some sets are members of themselves. The set of all non-penguins, for example, is a member of itself. So is the set of all sets. In which category would we find the set of all sets that are not members of themselves? If this set is not a member of itself, then it is a member of itself. If it is, then it isn't. So, this set is a member of itself if and only if it is not a member of itself, which is the paradox. This is similar in concept to the Cretan Liar paradox. An article about Russell's Paradox at the Stanford Encyclopedia of Philosophy Greeling's Paradox A version of Russell's Paradox using words. Some adjectives are self-descriptive, like "tiny", "unhyphenated", and "pentasyllabic". On the other hand, other adjectives are not self-descriptive, like "bisyllabic", "big", "tasty", and "incomplete". Call the self-descriptive adjectives

65. The Paradox Of The Liar This paradox is a version of Russell s paradox. Readers may wonder whetherRussell s paradox is a problem only in the foundations of mathematics or if 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

66. RUSSELL'S PARADOX RUSSELL S paradox. D. Atkinson. Ulysses pontificated ‘The classical statementof the Russell paradox is in terms of the village barber, who shaves all the http://atkinson.fmns.rug.nl/public_html/russell.html

RUSSELL'S PARADOX D. Atkinson Ulysses pontificated: ‘The classical statement of the Russell paradox is in terms of the village barber, who shaves all the men of the village who do not shave themselves. Does he then shave himself? Clearly so, for if he did not shave himself, he would be one of his own clients, since he shaves all the men who do not shave themselves. So he must shave himself. So far there is no paradox. Suppose though that we now add that he shaves only the men that do not shave themselves. Then it cannot be that he shaves himself, since he would be one of the self-shavers who, according to the terms of the conundrum, are not shaved by the barber. However, if he does not shave himself, he is, as mentioned above, one of his own clients.’ ‘But what about the men who don't shave at all?’ asked Helen coyly. ‘Men like my husband with a sexy red beard. Maybe the barber is one of them!’ ‘That is not the point at all!’ exclaimed Ulysses in exasperation. ‘The problem has to do with the definition or presumptive definition!

67. The Paradigms And Paradoxes Of Intelligence, Part 1: Russell's Paradox An exploration of Russell s paradox, written for The Futurecast, a monthly The above is my version of what has become known as Russell s paradox. http://www.kurzweilai.net/articles/art0257.html?m=10

68. 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. Russell s paradox concerned set A, defined as containing all sets that are not http://www.kurzweilai.net/articles/art0257.html?printable=1

69. Russell's Paradox And The Law Of Excluded Middle The most famous is Russell’s paradox which deals with sets of sets that do or But to avoid Russell’s paradox, you have to say that some things that are 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.

70. The New York Review Of Books: Russell's Paradox Preview of an article by Stuart Hampshire from The New York Review of Books,August 13, 1992. http://www.nybooks.com/articles/2834

@import "/css/default.css"; Home Your account Current issue Archives ... 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. 4066 words The full text of this piece is available only to subscribers of the Review 's electronic edition . To subscribe or learn more about the electronic edition, please

71. Department Of Mathematics And Statistics Bertrand Russell s paradox. Dr. HN Gupta, Professor Emeritus, Mathematics Statistics, University of Regina. Abstract. A set or aggregate may be defined http://www.math.uregina.ca/sem2002/j-20020212.html

Junior Seminar - Winter 2002 Tuesday, February 12, 3:30p.m., CW307.18 Contact People Undergraduate Graduate ... Local Access only Bertrand Russell's Paradox Dr. H. N. Gupta, Professor Emeritus, Abstract: A set or aggregate may be defined as a collection of some kind of other. In mathematics, the objects are usually mental objects or concepts; i.e. concepts and ideas. George Cantor (1845-1918) developed the sets in the latter part of the 19th century. Other mathematicians helped expand Cantor's work into a serious discipline, because it was seen that many branches of emerging mathematics depended to some extent on the notion and properties of sets. A critical scrutiny of the results obtained soon revealed contradictions. Antinomies appeared in the very advanced parts of set theory. Bertrand Russell's paradox discovered in 1901 surfaces at the very beginning of set theory. Several attempts have since been made to set up rules and axioms which would block Russell's paradox. No general method has yet been found that would guarantee absolute freedom from contradictions. It can be claimed, however, that contradictions are not encountered in the uses to which the theory of sets put in mathematics. In today's talk we will introduce the paradox put forward by Russell and analyse it with a view to suggesting means of avoiding it.

72. Russell's Paradox: The Third Crisis In Mathematics Louisiana Scholars College Thesis by Michael Todd HuddlestonAbstract. http://www.nsula.edu/scholars/Thesis/Thesisabstracts/SItheses/Huddleston.html

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

73. Science & Technology At Scientific American.com: Ask The Experts: Mathematics: W Russell s paradox is based on examples like this Consider a group of Russell s paradox, which he published in Principles of Mathematics in 1903, http://www.sciam.com/askexpert_question.cfm?articleID=0005DA51-B5F4-1C71-9EB7809

74. \documentstyle[12pt]{article} \begin{document} \title{Russell S Russell s paradox demonstrated a fundamental limitation of such a system. Zermelo s solution to Russell s paradox is to replace the axiom `for every http://www2.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

75. Re: Russell's Paradox This message is a reply to Russell s paradox posted from Fredrik Bendz posted This is known as Russell s paradox, and was described in The Principles of http://users.cgiforme.com/fbendz/messages/700.html

Help keep this site alive. Please visit our sponsors. Sex. Relationships. Friendships. Things that matter to us. Get A Word of Advice at INPursuit. Re: Russell's Paradox Post a new reply Back to the message board This message was posted by Kartik , posted on August 01, 2002 at 03:02:57 coming from 203.200.224 This message is a reply to Russell's Paradox posted from Fredrik Bendz posted at September 24, 2000 at 17:33:07 > The paradox of Russell is this: > "I would never want to belong to any club that would have someone like me for a member." (Woodie Allen in Annie Hall > Imagine a set M, which is the set of all sets which are not members (i.e. an element) of themselves. Now, ask yourself if M is such a set. By definition we have a paradox, which says that M is a set of itslef if and only if it is NOT a member of itself... :-) > First imagine that M is a member of M. In that case M is a member of itself and should not be a member of M. > Then imagine that M is not a member of M. In that case M is not a member of itself, and thus should be a member of M. > This is known as Russell's Paradox, and was described in

76. Russell's Paradox This is known as Russell s paradox, and was described in The Principles of In Russell s paradox, then, M would be a class if it was not a member of M, http://users.cgiforme.com/fbendz/messages/159.html

Help keep this site alive. Please visit our sponsors. Sex. Relationships. Friendships. Things that matter to us. Get A Word of Advice at INPursuit. Russell's Paradox Post a new reply Back to the message board This message was posted by Fredrik Bendz , posted on September 24, 2000 at 17:33:07 coming from This message is a reply to Re: Bertrand Russel wrote something about this posted from Dan Proctor posted at September 15, 2000 at 05:17:55 The paradox of Russell is this: "I would never want to belong to any club that would have someone like me for a member." (Woodie Allen in Annie Hall Imagine a set M, which is the set of all sets which are not members (i.e. an element) of themselves. Now, ask yourself if M is such a set. By definition we have a paradox, which says that M is a set of itslef if and only if it is NOT a member of itself... :-) First imagine that M is a member of M. In that case M is a member of itself and should not be a member of M. Then imagine that M is not a member of M. In that case M is not a member of itself, and thus should be a member of M. This is known as Russell's Paradox, and was described in

77. Russell's Paradox - Nathanael Thompson- The Examined Life On-Line Philosophy Jou Nathanael Thompson uncovers a possible solution to Russell s paradox. http://examinedlifejournal.com/articles/template.php?shorttitle=russellparadox&a

78. Russell's Paradox Russell s paradox. by Nathanael Thompson. When Gottlieb Frege made predicatelogic, He said for all p and A, p is in {x Ax} if and only if Ap. Well, http://examinedlifejournal.com/archives/vol1ed1/nat.html

Russell's Paradox by Nathanael Thompson The resolution of this problem that I can see, is to do away with the above notation. we can still use set but we'll have to define what's in that set by formulas. So, in Bertrands case, he can say well for all x, x is in the set P iff x isn't in x. then later on he'd say, oops that's a contradiction, I guess I'll have to what is in that set a different way. No biographical information is available for Nathanael at this time. To comment on this article, click here

79. Article About "Russell's Paradox" In The English Wikipedia On 24-Jul-2004 The Russell s paradox reference article from the English Wikipedia on 24Jul-2004 Similarly, Russell s paradox proves that, on Wikipedia, http://july.fixedreference.org/en/20040724/wikipedia/Russell's_paradox

The Russell's paradox reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference : snapshots of Wikipedia from wikipedia.org) Russell's paradox Russell's paradox is a paradox discovered by Bertrand Russell in which shows that the naive set theory of Cantor and Frege is contradictory. Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A In Cantor's system, M is a well-defined set. Does M contain itself? If it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, then it has to be a member of M , again according to the very definition of M . Therefore, the statements " M is a member of M " and " M is not a member of M " both lead to contradictions. In Frege's system, M corresponds to the concept does not fall under its defining concept . Frege's system also leads to a contradiction: that there is a class defined by this concept, which falls under its defining concept just in case it does not. Table of contents showTocToggle("show","hide")

80. FOM: Simpson On Russell's Paradox For Category Theory FOM Simpson on Russell s paradox for category theory. Robert M. Solovay solovayat math.berkeley.edu Sat Mar 11 220335 EST 2000 http://www.cs.nyu.edu/pipermail/fom/2000-March/003864.html

FOM: Simpson on Russell's paradox for category theory Robert M. Solovay solovay at math.berkeley.edu Sat Mar 11 22:03:35 EST 2000 Previous message: FOM: defining ``mathematics'' Next message: FOM: Simpson on Russell's paradox for category theory Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Previous message: FOM: defining ``mathematics'' Next message: FOM: Simpson on Russell's paradox for category theory Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... More information about the FOM mailing list

Page 4     61-80 of 105    Back | 1  | 2  | 3  | 4  | 5  | 6  | Next 20