Re: Russell's Paradox, Axioms Of Set Theory By John Conway Re Russell s paradox, Axioms of Set Theory by John Conway. reply to this message post a message on a new topic Back to messages http://mathforum.org/epigone/math-history-list/clixprongli/Pine.3.89.9604120741.
Re: Russell's Paradox, Axioms Of Set Theory By Samuel S. Kutler Re Russell s paradox, Axioms of Set Theory by Samuel S. Kutler. reply to this message post a message on a new topic Back to messages http://mathforum.org/epigone/math-history-list/clixprongli/v01540b01ad9318b6ce1b
Puzzles & Riddles: Russell's Paradox Click Here. Solution Title Russell s paradox Author hongjun Points 500 Grade A Date 08/19/2003 1249AM PDT, Is this what you are looking for? http://www.experts-exchange.com/Miscellaneous/Puzzles_Riddles/Q_20713511.html
Extractions: (requires verification) Password Verify Password I agree to the Member Agreement (Step 1 of 2) Free School Matcher - Helping Tech Professionals Find the Right Schools! Online and Campus Schools are seeking candidates with Information Technology backgrounds. Companies are hiring well-rounded tech professionals with both technology and business degrees. See which of the top online and local schools can meet your needs.
FoRK Archive: Russell's Paradox On The Web Russell s paradox on the Web. It occurred to me a couple of days ago that Russell s paradox can be described quite nicely in terms of web pages and links. http://www.xent.com/FoRK-archive/may98/0192.html
FoRK Archive: Re: Russell's Paradox On The Web Re Russell s paradox on the Web. Previous message Lloyd Wood RE Prisoner of cyberspace ; In reply to James K. Tauber Russell s paradox on the Web . http://www.xent.com/FoRK-archive/may98/0208.html
Extractions: Russell's Paradox is a mathematical aspect of set theory, see http://plato.stanford.edu/entries/russell-paradox/ Russell's Paradox was not developed in a search for alternatives to the Heisenberg Uncertainty Principle, and there is nothing inherent in the statement of Russell's Paradox that would suggest it holds promise as such an alternative. Though mathematics is very significant throughout physics, not every mathematical concept can be assumed to have significance in physics. That said, you may be interested is another concept, the measurement problem, http://plato.stanford.edu/entries/qt-measurement/ You can find at least one related response in the MadSci Archives using the search term measurement problem and many other by searching different combinations of the terms Heisenberg uncertainty quantum Thanks for your question. Current Queue Current Queue for Physics Physics archives Try the links in the MadSci Library for more information on Physics MadSci Home Information Search ... Join Us!
Wo's Weblog: What Does Russell's Paradox Teach In Semantics? Monday, 07 April 2003. What does Russell s paradox Teach in Semantics? Philosophy. (Russell s paradox is an independent argument for the same conclusion.). 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.
Wo's Weblog: Idle Remarks On Russell's Paradox And Higher-order Entities Idle remarks on Russell s paradox and higherorder entities. I will first describe a general version of Russell s paradox, of which Rieger s is a special case. 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
Russell's Paradox And The Law Of Excluded Middle Russells paradox and the Law of Excluded Middle. On 31 Jan 1997, Chris wrote On 30 Jan 1997, Mark wrote . That is Russells paradox. http://personal.bgsu.edu/~roberth/russell.html
Extractions: Russells 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 arent sufficiently crisp to decide for all cases that something either is or is not A, theres 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 cant 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 Russells 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 doesnt, and if it doesnt, then it does. But to avoid Russells Paradox, you have to say that some things that are sayable dont 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 wont count as either A or not-A.
Encyclopedia4U - Russell's Paradox - Encyclopedia Article Russell s paradox. Russell s paradox is closely related to the Liar paradox. Content on this web site is provided for informational purposes only. http://www.encyclopedia4u.com/r/russell-s-paradox.html
Extractions: ENCYCLOPEDIA U com Lists of articles by category ... SEARCH : 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. Exactly when Russell discovered the paradox is not clear. It seems to have been May or June 1901, probably as a result of his work on
Poincare's Concept Of Impredicativity, Russell's Paradox Poincare s concept of impredicativity, Russell s paradox. To phillogic@bucknell.edu ; Subject Poincare s concept of impredicativity, Russell s paradox; http://hhobel.phl.univie.ac.at/phlo/200009/msg00159.html
Extractions: Date Prev Date Next Thread Prev Thread Next ... Thread Index http://www.utm.edu/research/iep/p/poincare.htm References Feferman's completeness result From: Prev by Date: The polyvalued logic used by the 13,5 billion market? Next by Date: Polyvalued first order logic Prev by thread: Feferman's completeness result Next by thread: The polyvalued logic used by the 13,5 billion market? Index(es): Date Thread
Russells Paradox - Definition By Dict.die.net Source The Free Online Dictionary of Computing (09 FEB 02) Russell s paradox mathematics A logical contradiction in set theory discovered by the British http://dict.die.net/russells paradox/
Extractions: Search dictionary for Source: The Free On-line Dictionary of Computing (09 FEB 02) Russell's Paradox set theory discovered by the British mathematician Bertrand Russell (1872-1970). If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa. The paradox stems from the acceptance of the following axiom : If P(x) is a property then x : P is a set. This is the Axiom of Comprehension (actually an axiom schema ). By applying it in the case where P is the property "x is not an element of x", we generate the paradox, i.e. something clearly false. Thus any theory built on this axiom must be inconsistent. In lambda-calculus Russell's Paradox can be formulated by representing each set by its characteristic function set theory suggest the existence of the paradoxical set R. Zermelo Fränkel set theory is one "solution" to this paradox. Another, type theory , restricts sets to contain only elements of a single type, (e.g. integers or sets of integers) and no type is allowed to refer to itself so no set can contain itself. A message from Russell induced Frege to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway. (2000-11-01)
Metamath Proof Explorer - Ru Russell s paradox. Proposition 4.14 of TakeutiZaring p. 14.Browser slow? Try the Unicode version. Theorem ru 1358. http://metamath.planetmirror.com/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
Rishabh Ðara - The World In My Brain Archives The paradox Archive Maths Russell s paradox archives - Russell s paradox is the most famous of the logical or set-theoretical paradoxes. http://www.rishabhdara.com/link.php?currentgrp=299
FOM: Simpson On Russell's Paradox For Category Theory FOM Simpson on Russell s paradox for category theory. Stephen G Simpson simpson@math.psu.edu Wed, 15 Mar 2000 110550 0500 (EST) http://www.cs.nyu.edu/pipermail/fom/2000-March/003918.html
Extractions: Wed, 15 Mar 2000 11:05:50 -0500 (EST) Dear Bob, I just wanted to let you know that I am not ignoring your FOM posting of March 11, where you point out difficulties with my ``Russell paradox for naive category theory''. I think I will be able to answer your points, but I am busy with other things right now, and I will have to take some time off to get the details back into my brain. Steve Previous message: FOM: Simpson on Russell's paradox for category theory Next message: FOM: Survery Results Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
FOM: Re: Simpson On Russell's Paradox For Category Theory FOM Re Simpson on Russell s paradox for category theory. charles silver silver_1@mindspring.com Tue, 11 Apr 2000 104657 0700 http://www.cs.nyu.edu/pipermail/fom/2000-April/003945.html
Extractions: Tue, 11 Apr 2000 10:46:57 -0700 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 ]
The Challenge Of Self-reference. puzzles. The prototype of these is Russell s paradox. One version of this goes as follows Sal is a barber in the town of Seville. http://www.atarimagazines.com/creative/v11n6/22_The_challenge_of_selfref.php
Extractions: CREATIVE COMPUTING VOL. 11, NO. 6 / JUNE 1985 / PAGE 22 The challenge of self-reference. (Recreational computing) Michael W. Ecker. The Challenge of Self-Reference Greetings! Welcome to "Recreational Computing,' anew column which will appear regularly in Creative Computing. We will play, explore, challenge, as well as form conjectures, test them, and just plain have good programming fun. I would especially like to invite all readers to try the challenges, come up with your own solutions, and your own problems as well. They should be recreational in nature. Digit delving and other forms of "microcomputer mathemagic' in Basic are what I have foremost in mind, but I wish to hear from you. Your ideas, if used, will be acknowledged in this column. I solicit your new problems, programs, improved solutions, etc. I will also answer readers who have pertinent questions and who supply a SASE. You may write to me directly: Michael W. Ecker, Ph.D., Contributing Editor, Creative Computing, 129 Carol Dr., Clarks Summit, PA 18411. Please try to keep your solutions in generic Basic (Microsoft), and as machine-independent as possible. For those who wish to send longer programs, please note that I do not have time to type in long listings, so if you have one of the following machines, magnetic media submissions are welcome: I have a TRS-80 Model 3 with 48K and two disk drives and tape, a TRS-80 Model 4P with 128K and two disk drives, and a Sanyo MBC-555 with 256K and two single-sided drives under MS-DOS 1.25 only. The Sanyo can read IBM PC Basic programs if you save them in ASCII format on one side only. I do have access to other machines, including IBM PCs, but these are less convenient.
November: RE: Russell's Paradox RE Russell s paradox. From EST. Next message Ryan Jamieson RE Russell s paradox Previous message yu262646@yorku.ca (no subject) ; http://www.math.yorku.ca/Who/Faculty/Steprans/Courses/3500/m0211/0003.html
RUSSELL S PARADOX RUSSELL S paradox. D. Atkinson. Ulysses pontificated The classical statement of the Russell paradox is in terms of the village http://www-th.phys.rug.nl/~atkinson/russell.html
Extractions: 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!
Russells Paradox Russell s paradox is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. http://www.knowledgerush.com/kr/jsp/db/facts.jsp?title=Russells paradox
