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A nuanced article on this paradox of naive set theory, happened upon by Russell in 1901. Includes Category Society Philosophy Philosophers Russell, Bertrandrussell's paradox. russell's paradox is the most famous of the logicalor settheoretical paradoxes. The paradox arises within naive

Extractions: JUN 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 S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S 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. Russell appears to have discovered his paradox in May 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.

2. Russell's Paradox [Internet Encyclopedia Of Philosophy]
russell's paradox. russell's paradox represents either of two interrelated logical antinomies.
http://www.utm.edu/research/iep/p/par-russ.htm

Extractions: Russell's Paradox Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of 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 propery 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, who discovered it in 1901. History Russell's discovery came while he was working on his Principles of Mathematics . Although Russell discovered the paradox independently, there is some evidence that other mathematicians and set-theorists, including Ernst Zermelo and David Hilbert, had already been aware of the first version of the contradiction prior to Russell's discovery. Russell, however, was the first to discuss the contradiction at length in his published works, the first to attempt to formulate solutions and the first to appreciate fully its importance. An entire chapter of the

http://www.jimloy.com/logic/russell.htm

Extractions: Go to my home page 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.

4. Notes To Russell's Paradox
Stanford Encyclopedia of Philosophy Notes to russell's paradox. Notes. Notesto russell's paradox Stanford Encyclopedia of Philosophy.

Extractions: Notes to Russell's Paradox Exactly when the discovery of the paradox took place is not completely clear. Russell initially states that he came across the paradox "in June 1901" (see Russell 1944, p. 13). Later he reports that the discovery took place "in the spring of 1901" (see Russell 1959, p. 58). Later still he reports that he came across the paradox, not in June, but in May of that year (see Russell 1967/1968/1969, volume 3 (1969), p. 221). See Frege 1903, p. 127. It is worth noting that even prior to Russell's discovery this principle had not been universally accepted. Georg Cantor, for example, rejected it in favour of what was, in effect, a distinction between sets and classes, recognizing that some properties (such as the property of being an ordinal) produced collections that were too big to be sets, and that an assumption to the contrary would result in an inconsistent theory. (For further details see Menzel 1984, Moore 1982, and Hallett 1984.) One exception is paraconsistent set theory. Paraconsistent set theory retains an unrestricted comprehension axiom but abandons classical logic, substituting a paraconsistent logic in its place. See the entries on

Latest Info Schedule Invited Speakers Special Events Contributed Papers Abstracts Registration/ Contact Travel/ Accommodation Media FundingRussell 2001 Photo Gallery
http://www.cut-the-knot.com/selfreference/russell.html

Extractions: Oxford University Press, 1997 Sets are defined by the unique properties of their elements. One may not mention sets and elements simultaneously, but one notion has no meaning without other. The widely used Peano's notation incorporates all the pertinent attributes: a set A, a property P, elements x. But, of course, one does not always use the formal notations. For example, it's quite acceptable to talk of the set of all students at the East Brunswick High or the set of fingers I use to type this sentence. The space being limited, some sets are described on this page and some are not. Let's call russell the set of all sets described on this page. Just driving the point in: russell's elements are sets described on this page. Note that this page is where you met russell. For it's where it was defined after all. So russell has an interesting property of being its own element: russell russell.

6. Russell-Myhill Paradox [Internet Encyclopedia Of Philosophy]
This antinomy was discovered by Bertrand Russell in 1902, a year afterdiscovering a simpler paradox usually called russell's paradox .
http://www.utm.edu/research/iep/p/par-rusm.htm

Extractions: Russell-Myhill Paradox The Russell-Myhill Antinomy, also known as the Principles of Mathematics Appendix B Paradox, is a contradiction that arises in the logical treatment of classes and "propositions", where "propositions" are understood as mind-independent and language-independent logical objects. If propositions are treated as objectively existing objects, then they can be members of classes. But propositions can also be about classes, including classes of propositions. Indeed, for each class of propositions, there is a proposition stating that all propositions in that class are true. Propositions of this form are said to "assert the logical product" of their associated classes. Some such propositions are themselves in the class whose logical product they assert. For example, the proposition asserting that all-propositions-in-the- class-of-all-propositions -are-true is itself a proposition, and therefore it itself is in the class whose logical product it asserts. However, the proposition stating that all-propositions-in-the- null-class -are-true is not itself in the null class. Now consider the class

In lambdacalculus russell's paradox can be formulated by representing each set by its characteristic function - the
http://burks.bton.ac.uk/burks/foldoc/59/101.htm

Extractions: The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: Russell's Attic Next: Russell, Bertrand mathematics 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 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 - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself: If we now apply r to itself, An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of

Math puzzles. Interactive education. Logic and Paradoxes. Selfreference.russell's paradox. russell's paradox. Poincaré disliked
http://www.cut-the-knot.com/selfreference/russell.shtml

Extractions: Oxford University Press, 1997 Sets are defined by the unique properties of their elements. One may not mention sets and elements simultaneously, but one notion has no meaning without other. The widely used Peano's notation incorporates all the pertinent attributes: a set A, a property P, elements x. But, of course, one does not always use the formal notations. For example, it's quite acceptable to talk of the set of all students at the East Brunswick High or the set of fingers I use to type this sentence. The space being limited, some sets are described on this page and some are not. Let's call russell the set of all sets described on this page. Just driving the point in: russell's elements are sets described on this page. Note that this page is where you met russell. For it's where it was defined after all. So russell has an interesting property of being its own element: russell russell.

9. One Hundred Years Of Russell's Paradox - Contributors
When and How Did Russell Become Vulnerable to russell's paradox?
http://www.lrz-muenchen.de/~russell01/contributors.html

Extractions: Contributed papers will be read by: Email Wolfgang Buschlinger (Braunschweig) Email The use of semantical paradox for graph theoretical problems: False-Systems Kai Hauser (Berlin) Email Two Versions of Realism Andreas Herberg-Rothe (Berlin) Email Tarski's "Magic Trick." The unrecognized antinomies of the meta-meta-language Email Inconsistency in the real world Phillip Keller (Geneva) Email What singletons could be James Levine (Dublin) Email When and How Did Russell Become Vulnerable to Russell's Paradox? Nikolay Milkov (Bielefeld) Email Russell's Turn of August-1900 and his Paradox Sebastiano Moruzzi (Bologna) Email Russell's Theory of Vagueness Karl-Georg Niebergall (Munich) Email Is ZF finitistically reducible? Francesco Orilia (Macerata) Email Logical Rules, Principles of Reasoning and Russell's Paradox Jacek Pasniczek (Lublin) Email Clark-Russell Paradox in Theories of Meinongian Objects Email Hilbert's Paradox Adolf Rami (Graz) Email Empty Names and Russell's Name Claim Hartley Slater (Western Australia) Email The Uniform Solution Holger Sturm (Konstanz) Email The relevance of Russells Paradox for a philosophical theory of properties Gabriel Uzquiano (Rochester) Email Categoricity Theorems and Conceptions of Set Russell Wahl (Iowa) Email Russell, Realism, Ramification and Reducibility

10. Russell's Paradox - Wikipedia
russell's paradox. russell's paradox is a paradox found by Bertrand Russell in 1901which shows that naive set theory in the sense of Cantor is contradictory.

Extractions: Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Other languages: Polski From Wikipedia, the free encyclopedia. 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. Initially, Russell discovered the paradox while studying a foundational work in symbolic logic by Gottlob Frege . While Zermelo was creating his version of set theory in which this paradox occured, he noticed it but thought it was too obvious and never published the inconsistency. 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 the sense of Cantor

11. Re: Russell's Paradox, Axioms Of Set Theory By John Conway
Subject Re russell's paradox, Axioms of Set Theory Author John Conway conway@math.Princeton.EDU Date Fri, 12 Apr
http://mathforum.com/epigone/math-history-list/clixprongli/Pine.3.89.9604120741.

12. Talk:Russell's Paradox - Wikipedia

Extractions: Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help From Wikipedia, the free encyclopedia. I have removed this: The Barber paradox doesn't point to any inconsistency or ambiguity: one simply has to conclude that there is no village where the barber shaves everyone who doesn't shave themselves. Such a village cannot exist, as the paradox shows. AxelBoldt 02:07 Nov 27, 2002 (UTC)

13. Re: Russell's Paradox, Axioms Of Set Theory By Samuel S. Kutler
Subject Re russell's paradox, Axioms of Set Theory Author Samuel S. Kutler skutler@sjca.edu Date Thu, 11 Apr 1996

14. Erasing Russell's Paradox
http://www.geocities.com/dblowe_47/sets.htm

Extractions: Cantor's Theorem The concept of set is so "naïve" and intuitive because it is directly related to the fundamental trait of human cognitive functioning to group and count in order to make sense of the constant bombardment of sensory stimuli. We group these stimuli based on common properties, and their distinctness allows us to count and relate how many items with common properties we encountered. What mathematicians have done is given this grouping concept "life" and the name of "set" and shifted focus away from the items with common properties that make up this mental grouping. Mathematicians have abstracted the concept and gone off to play with it without stopping to consider, at the most basic level, what it should and should not be. The first order of business is to establish what "exists" and what does not. Based on the general human cognitive ability to distinguish between and group "objects of thought", we can safely say that such objects of thought "exist". We generally agree upon the existence of objects since we can discuss the properties of objects among ourselves and conclude that we are discussing the same objects. And we can assume reasonably that there are many distinct objects. We can also reasonably assume that there are objects with common properties because the world would otherwise be in complete chaos. (Some might argue that the world is just that!) Philosophical questions aside, our first Axiom (1) is: There exist distinct objects that can be grouped according to a shared property (or properties).

russell's paradox. Let X and Y be sets. Define Y = {X X Ï X}, then Y Î Yif and only if Y Ï Y. We can find many variations of russell's paradox.
http://www.geocities.com/galois_e/page/russell.html

Extractions: Russell's Paradox Y if and only if Y Y. Let's illustrate the above statement with a discrete example. Let's say a doctor claims that he/she only treats all the people who do not treat themselves. A patient walks into this doctor's office seeking treatment does not contradict the doctor's claim. The patient does not treat himself and the doctor treats the patient, perfect. However, if the doctor is sick, who is going to treat him? If he treats himself, then he does not belong to the group "who do not treat themselves" and therefore he should not treat himself. However, if he goes to another doctor, then he belongs to the group "who do not treat themselves" and therefore he should treat himself. Whatever this doctor does, for sure he contradicts himself. This famous paradox was discovered by Bertrand Russell in 1902. This is probably one of the more dramatic story in the history of logic. As we know, our language is finite. The problem is that if we try to define everything, eventually we will run out of words and at the end we have a circular definition. For instance, the Random House Dictionary defines "left" as "...the side of a person...toward the west when the subject is facing north". If you look up "west", it is defined as "the direction to left of a person facing north". If someone does not know what "left" and "west" are at the beginning, after looking up the dictionary, he still does not know what they are. In mathematical logic, circular argument is not acceptable. This problem can be avoided if we use a axiomatic approach. That is to say, we accept certain undefined objects as our basis and then build a whole system based on these objects. Euclid had used this approach 2,500 years ago in his

russell's paradox. The significance of russell's paradox can be seen once it is realizedthat, using classical logic, all sentences follow from a contradiction.
http://setis.library.usyd.edu.au/stanford/archives/fall1997/entries/russell-para

Extractions: A B C D ... Z The most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets which 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. Some sets, such as the set of 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 which are not members of themselves S . If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox prompted much work in logic, set theory and the philosophy and foundations of mathematics during the early part of this century. Russell discovered his paradox in May 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 ordinals and yet greater than any such element.

17. Russell's Paradox From FOLDOC
russell's paradox. mathematics A logical contradiction in set theorydiscovered by Bertrand Russell. If R is the set of all sets

russell's paradox. mathematics A logical contradiction in set theorydiscovered by the British mathematician Bertrand Russell (18721970).
http://burks.brighton.ac.uk/burks/foldoc/59/101.htm

Extractions: The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: Russell's Attic Next: Russell, Bertrand mathematics 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 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 - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself: If we now apply r to itself, An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of

19. Russell's Paradox From FOLDOC
russell's paradox. mathematics A logical contradiction in set theorydiscovered by the British mathematician Bertrand Russell (18721970).