1. Russell's Paradox - Wikipedia, The Free Encyclopedia In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard http://en.wikipedia.org/wiki/Russell's_paradox

Russell's paradox From Wikipedia, the free encyclopedia Jump to: navigation search Bertrand Russell series Russell in 1907 Bertrand Russell Views on philosophy Views on society Russell's paradox Russell's teapot Theory of Descriptions Logical atomism In the foundations of mathematics Russell's paradox (also known as Russell's antinomy ), discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard Dedekind and Frege leads to a contradiction. The very same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert Husserl and other members of the University of Göttingen. It might be assumed that, for any formal criterion, a set exists whose members are those objects (and only those objects) that satisfy the criterion; but this assumption is disproved by a set containing exactly the sets that are not members of themselves. If such a set qualifies as a member of itself, it would contradict its own definition as a set containing sets that are not members of themselves . On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox.

2. Russell's Paradox - ENotes.com Reference Get Expert Help. Do you have a question about the subject matter of this article? Hundreds of eNotes editors are standing by to help. http://www.enotes.com/topic/Russell's_paradox

3. Russell's Paradox - Iron Chariots Wiki Russell's paradox is a famous paradox arising in elementary set theory. In simplest terms, it points out the logical inconsistencies that arise from allowing a set to be considered a http://wiki.ironchariots.org/index.php?title=Russell's_paradox

4. Russell's Paradox - New World Encyclopedia Part of the foundation of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to http://www.newworldencyclopedia.org/entry/Russell's_paradox

Russell's paradox From New World Encyclopedia Jump to: navigation search Previous (Rus' Khaganate) Next (Russell Cave National Monument) Part of the foundation of mathematics, Russell's paradox (also known as Russell's antinomy ), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. Consider the set R of all sets that do not contain themselves as members. In set-theoretic notation: Assume, as in Frege's Grundgesetze der Arithmetik, that sets can be freely defined by any condition. Then R is a well-defined set. The problem arises when it is considered whether R is an element of itself. If R is an element of R, then according to the definition, R is not an element of R; if R is not an element of R, then R has to be an element of R, again by its very definition: Hence a contradiction. Russell's paradox was a primary motivation for the development of set theories with a more elaborate axiomatic basis than simple extensionality and unlimited set abstraction. The paradox drove Russell to develop type theory and Ernst Zermelo to develop an axiomatic set theory, which evolved into the now-canonical Zermelo–Fraenkel set theory.

5. Russell's Paradox - Academic Kids Russell's paradox (also known as Russell's antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. http://www.academickids.com/encyclopedia/index.php/Russell's_paradox

Russell's paradox From Academic Kids Russell's paradox (also known as Russell's antinomy ) 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 (but see Independence from Excluded Middle below). 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. Note: This article uses specialized mathematical symbols Contents showTocToggle("show","hide")

6. Russell's Paradox In English - Dictionary And Translation Russell's Paradox. Dictionary terms for Russell's Paradox in English, English definition for Russell's Paradox, Thesaurus and Translations of Russell's Paradox to English http://www.babylon.com/definition/Russell's_Paradox/English

7. Russell's Paradox - Conservapedia Bertrand Russell 's Paradox revealed a flaw in 1901 in early or naive set theory. (It is likely others saw this flaw before Bertrand Russell.) It is a form of the liar's http://www.conservapedia.com/Russell's_paradox

Russell's Paradox From Conservapedia (Redirected from Russell's paradox Jump to: navigation search Bertrand Russell 's Paradox revealed a flaw in 1901 in early or naive set theory . (It is likely others saw this flaw before Bertrand Russell .) It is a form of the liar's paradox expressed in the terms of set theory The defect is as follows. Let S be the set of all sets that do not contain themselves as members. In other words, set T is an element of S if, and only if, T is not an element of T: Here is the flaw. Is Set S a member of itself? If S is a member of itself, then it cannot be a member of itself by the very definition of S. But if S is not a member of itself, then it must be a member of itself, again by its very definition. Hence there is a fundamental logical contradiction in this type of set, and in any theory that allows it. In formal terms, Russel's paradox and others like it can be avoided only at the expense of giving up the idea that any criteria can be used to construct sets. For example in Zermelo-Fraenkel set theory the construction would not qualify as well-founded . Some formulations of set theory would allow such a construction but then S would be a proper class rather than a set.

8. Russell's Paradox - On Opentopia, Find Out More About Russell's Paradox Russell's paradox (also known as Russell's antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Frege is contradictory. http://encycl.opentopia.com/term/Russell's_paradox

About Opentopia Opentopia Directory Encyclopedia ... Tools Russell's paradox Encyclopedia R RU RUS : Russell's paradox Russell's paradox (also known as Russell's antinomy ) is a paradox discovered by Bertrand Russell in which shows that the naive set theory of 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 Frege's system, M would be 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 (but see Independence from Excluded Middle below). Contents 1 History 2 Applied versions 3 Set-theoretic responses 4 Applications and related topics ... 7 External links History Exactly when Russell discovered the paradox is not clear. It seems to have been May or June , probably as a result of his work on Cantor's theorem that the number of entities in a certain domain is smaller than the number of subclasses of those entities. In Russell's

9. Bambooweb: Russell´s Paradox Russell s paradox is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor a http://www.bambooweb.com/articles/r/u/Russell's_Paradox.html

Russell's Paradox Russell's paradox is a paradox discovered by Bertrand Russell in which shows that the naive set theory of Cantor and Cantor's theorem that the number of entities in a certain domain is smaller than the number of subclasses of those entities. In Russell's Principles of Mathematics (not to be confused with the later Principia Mathematica ) Chapter X, section 100, where he calls it "The Contradiction" he says that he was led to it by analyzing Cantor's proof that there can be no greatest cardinal. He also mentions it in a 1901 paper in the International Monthly, entitled "Recent work in the philosophy of mathematics" Russell mentioned Cantor's proof that there is no largest cardinal and stated that "the master" had been guilty of a subtle fallacy that he would discuss later. Famously, Russell wrote to Frege about the paradox in June , just as Frege was preparing the second volume of his Grundgesetze. Frege was forced to prepare an appendix in response to the paradox, but this later proved unsatisfactory. It is commonly supposed that this led Frege completely to abandon his work on the logic of classes. While Zermelo was working on his version of set theory, he also noticed the paradox, but thought it too obvious and never published anything about it! Zermelo's system avoids the difficulty through the famous

10. Russell’s Paradox Summary And Analysis Summary | BookRags.com Russell’s paradox summary with 12 pages of lesson plans, quotes, chapter summaries, analysis, encyclopedia entries, essays, research information, and more. http://www.bookrags.com/Russell's_paradox

11. Russell's Paradox (Stanford Encyclopedia Of Philosophy) by AD Irvine 2009 - Cited by 18 - Related articles http://plato.stanford.edu/entries/russell-paradox/

Cite this entry Search the SEP Advanced Search Tools ... Please Read How You Can Help Keep the Encyclopedia Free Russell's Paradox First published Fri Dec 8, 1995; substantive revision Wed May 27, 2009 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. R 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, while working on his Principles of Mathematics (1903). Cesare Burali-Forti, an assistant to Giuseppe Peano, had discovered a similar antinomy in 1897 when he noticed that since the set of ordinals is well-ordered, it too must have an ordinal. However, this ordinal must be both an element of the set of all ordinals and yet greater than every such element. Unlike Burali-Forti's paradox, Russell's paradox does not involve either ordinals or cardinals, relying instead only on the primitive notion of set.

12. Russell's Paradox The above story about the barber is the popular version of Russell s Paradox. The story was originally told by Bertrand Russell. And of course it has a http://www.jimloy.com/logic/russell.htm

Return to my Logic pages Go to my home page Russell's Paradox Let you tell me a famous story: There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved themselves or were shaved by the barber. And the barber only shaved the men who did not shave themselves. That is a nice story. But it raises the question: Did the barber shave himself? Let's say that he did shave himself. But we see from the story that he shaved only the men in town who did not shave themselves. Therefore, he did not shave himself. But we again see in the story that every man in town either shaved himself or was shaved by the barber. So he did shave himself. We have a contradiction. What does that mean? Maybe it means that the barber lived outside of town. That would be a loophole, except that the story says that he did live in the town, maybe in Seville. Maybe it means that the barber was a woman. Another loophole, except that the story calls the barber "he." So that doesn't work. Maybe there were men who neither shaved themselves nor were shaved by the barber. Nope, the story says, "All of the men in this town either shaved themselves or were shaved by the barber." Maybe there were men who shaved themselves AND were shaved by the barber. After all, "either ... or" is a little ambiguous. But the story goes on to say, "The barber only shaved the men who did not shave themselves." So that doesn't work either. Often, when the above story is told, one of these last two loopholes is left open. So I had to be careful, when I wrote down the story.

13. Russell's Paradox The set of all subsets of a given set. Selfreference. http://www.cut-the-knot.org/selfreference/russell.shtml

14. Russell’s Paradox [Internet Encyclopedia Of Philosophy] by KC Klement 2005 http://www.iep.utm.edu/par-russ/

Internet Encyclopedia of Philosophy A B C ... Z all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not. The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the property of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat ) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows. The paradox was named after Bertrand Russell (1872-1970), who discovered it in 1901. Table of Contents History Possible Solutions to the Paradox of Properties Possible Solutions to the Paradox of Classes or Sets References and Further Reading 1. History Principles of Mathematics Principles was dedicated to discussing the contradiction, and an appendix was dedicated to the theory of types that Russell suggested as a solution.

15. Russell's Paradox Russell s Paradox can be put into everyday language in many ways. The most often repeated is the Barber Question. It goes like this http://fclass.vaniercollege.qc.ca/web/mathematics/real/russell.htm

Back Russell's Paradox Easy to state, yet difficult or impossible to resolve; self contradictory statements or paradoxes have presented a major challenge to Mathematics and Logic. Russell's Paradox can be put into everyday language in many ways. The most often repeated is the 'Barber Question.' It goes like this: In a small town there is only one barber. This man is defined to be the one who shaves all the men who do not shave themselves. The question is then asked, 'Who shaves the barber?' If the barber doesn't shave himself, then by definition he does. And, if the barber does shave himself, then by definition he does not. Another popular form of Russell's Paradox is the following: Consider the statement 'This statement is false.' If the statement is false, then it is true; and if the statement is true, then it is false. Let's look at this situation as mathematicians do. You may have noticed the remarkable similarity between logical symbols (like for ' and for ' or '; and ~ for ' not ') and the symbols used with sets. For example, compare Logic Set Theory p q P Q p q P Q p P' In logic a statement that has a single variable, like

16. Russell's Paradox -- From Wolfram MathWorld Oct 11, 2010 Russell s Paradox. SEE Russell s Antinomy Send Contact the MathWorld Team © 19992010 Wolfram Research, Inc. Terms of Use http://mathworld.wolfram.com/RussellsParadox.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Last updated: Mon Oct 11 2010 Russell's Paradox SEE: Russell's Antinomy Contact the MathWorld Team Wolfram Research, Inc. Wolfram Research Mathematica Home Page ... Wolfram Blog

17. PlanetMath: Russell's Paradox An interpretation of Russell paradox without any formal language of set theory could be stated like ``If the barber shaves all those who do not themselves http://planetmath.org/encyclopedia/RussellsParadox.html

(more info) Math for the people, by the people. donor list find out how Encyclopedia Requests ... Advanced search Login create new user name: pass: forget your password? Main Menu sections Encyclopædia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Russell's paradox (Definition) Suppose that for any coherent proposition $P(x)$ , we can construct a set Let . Suppose ; then, by definition, . Likewise, if , then by definition . Therefore, we have a contradiction Bertrand Russell gave this paradox as an example of how a purely intuitive set theory can be inconsistent . The regularity axiom , one of the Zermelo-Fraenkel axioms , was devised to avoid this paradox by prohibiting self-swallowing sets. An interpretation of Russell paradox without any formal language of set theory could be stated like ``If the barber shaves all those who do not themselves shave, does he shave himself?''. If you answer himself that is false since he only shaves all those who do not themselves shave. If you answer someone else that is also false because he shaves all those who do not themselves shave and in this case he is part of that set since he does not shave himself. Therefore we have a contradiction. Remark . Russell's paradox is the result of an axiom (due to Frege) in set theory, now obsolete, known as the

18. Russell's Paradox 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 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

19. Russell's Paradox A paradox uncovered by Bertrand Russell in 1901, which forced a reformulation of set theory. One version of Russell s paradox, known as the barber paradox, http://www.daviddarling.info/encyclopedia/R/Russells_paradox.html

20. Russell S Paradox - Wikipedia, The Free Encyclopedia In the foundations of mathematics, Russell s paradox (also known as Russell s antinomy), discovered by Bertrand Russell in 1901, showed that the naive set http://en.wikipedia.org/wiki/Russell's_paradox

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