21. Russell's Paradox - Stack Overflow Let X be the set of all sets that do not contain themselves. Is X a member of X? http://stackoverflow.com/questions/35339/russells-paradox

22. Russell's Paradox (Stanford Encyclopedia Of Philosophy/Spring 2004 Edition) This document uses XHTML1/Unicode to format the display. Older browsers and/or operating systems may not display the formatting correctly. http://stanford.library.usyd.edu.au/archives/spr2004/entries/russell-paradox/

This is a file in the archives of the Stanford Encyclopedia of Philosophy version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... Z This document uses XHTML-1/Unicode to format the display. Older browsers and/or operating systems may not display the formatting correctly. last substantive content change MAY Russell's Paradox 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 " R ." If 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.

23. Russell's Paradox - Definition Of Russell's Paradox By The Free Online Dictionar Disclaimer All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. http://www.thefreedictionary.com/Russell's paradox

24. Russell-Myhill Paradox [Internet Encyclopedia Of Philosophy] Although Frege was clearly devastated by the simpler “Russell’s paradox”, which Russell had related to Frege three months prior, Frege was not similarly impressed by the http://www.iep.utm.edu/par-rusm/

Internet Encyclopedia of Philosophy A B C ... Z Russell-Myhill Paradox The Russell-Myhill Antinomy, also known as the Principles of Mathematics about 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 w , consisting of all propositions that state the logical product of some class m in which they are not included. This w is itself a class of propositions, and so there is a proposition r , stating its logical product. The contradiction arises from asking the question of whether r is in the class w . It seems that r is in w just in case it is not. Principles of Mathematics Table of Contents History and Historical Importance Formulation and Derivation Possible Solutions References and Further Reading 1. History and Historical Importance Principles of Mathematics , Russell began searching for a solution. He soon came upon the Theory of Types, which he describes in Appendix B of the Principles Some authors have speculated that this antinomy was the first hint Russell found that what was needed to solve the paradoxes was something more than the simple theory of types. If so, then this antinomy is of considerable importance, as it might represent the first motivation for the ramified theory of types adopted by Russell and Whitehead in

25. Russell's Paradox: Definition From Answers.com The paradox concerning the concept of all sets which are not members of themselves which forces distinctions in set theory between sets and classes. http://www.answers.com/topic/russell-s-paradox

26. Russell's Paradox (logic) -- Britannica Online Encyclopedia Russell's paradox (logic), statement in set theory, devised by the English mathematicianphilosopher Bertrand Russell, that demonstrated a flaw in earlier efforts to axiomatize http://www.britannica.com/EBchecked/topic/513243/Russells-paradox

document.write(''); Search Site: Advanced Search With all of these words With the exact phrase With any of these words Without these words Home SHOP BROWSE BLOG ... HELP Username: Password: Remember me Forgot your password? Help Email " is the e-mail address you used when you registered. Password " is case sensitive. If you need additional assistance, please contact customer support Enter the e-mail address you used when enrolling for Britannica Premium Service and we will e-mail your password to you. Russells-paradox My Britannica CREATE MY Russell's pa... NEW ARTICLE ... SAVE Table of Contents: Article Article Related Articles Related Articles Citations Article Related Articles Citations Primary Contributor: Herbert Enderton ARTICLE from the statement in set theory , devised by the English mathematician-philosopher Bertrand Russell , that demonstrated a flaw in earlier efforts to axiomatize the subject. Russell found the paradox in 1901 and communicated it in a letter to the German mathematician-logician Gottlob Frege axiomatic system of set theory by deriving a paradox within it. (The German mathematician

27. Russell�s Paradox Russell’s Paradox (continued) Na vely, one might simply define a set as any collection of objects. This definition does not meet the rigors of formal mathematics (as we shall see http://student.ccbcmd.edu/~mdebonis/russell.htm

Russell�s Paradox (continued) For Russell�s paradox, we consider the following set: Now one might argue that this is not a reasonable set of objects to consider, for how can a set be an element of itself? Are not elements and sets two different types of objects? Not necessarily in mathematics. There are very reasonable examples in nature of sets being elements of themselves. Imagine sets as catalogues which have lists as their elements. One set could be the catalogue of all sporting goods (an element of which would be, for instance, a basketball); another could be the catalogue of all breeds of dogs, etc. Now consider the catalogue of all catalogues. This is a set which has as its elements other sets. In fact, this catalogue of catalogues is listed as one of its own elements. The set B defined above can be viewed as the catalogue of all catalogues which do not list themselves in their own catalogue. Now the paradox arises when one poses the following question: Is B, or is not B, an element of itself? In other words, does or does not B list itself in its own catalogue? The answer to this question turns out to be that neither is true (hence, the paradox). The proof of this fact is a mathematical tongue-twister. Lets first give the proof in the language of catalogues. Suppose B did list itself in its own catalogue. The fact that B is listed in B means, as an element, that B is a catalogue which does not list itself. But this contradicts what we originally supposed. Perhaps then, B does not list itself in its own catalogue. But since B is the catalogue of all catalogues which do not list themselves, it follows that B would be an element of the catalogue B, and hence B lists itself in its own catalogues, again contradicting our initial assumption.

28. Russells Paradox Article on Russells Paradox Russells Paradox. Suppose that for any coherent proposition, we can construct a set . http://myyn.org/m/article/russells-paradox/

29. [tw] : Russell's Paradox Russell's Paradox is a headache of logic and naive set theory. Russell's paradox is the most famous of the logical or settheoretical paradoxes. http://www.truerwords.net/2573

30. Russell's Paradox Undecidability of the Halting Up New proofs of old Previous Dynamic Programming. Russell's Paradox. In the middle of the night I got such a fright that woke me with a start, http://pages.cs.brandeis.edu/~mairson/poems/node4.html

Next: Undecidability of the Halting Up: New proofs of old Previous: Dynamic Programming Russell's Paradox In the middle of the night I got such a fright that woke me with a start, For I dreamed of a set that contained itself, in toto, not in part. If sets can thus contain themselves, then they might also fail To hold themselves as members, and this leads me to my tale. Now Frege thought he finally had the world inside a box, So he wrote a lengthy tome, but up popped paradox. Russell asked, ``You know that Epimenides said oft A Cretan who tells a lie does tell the truth, nicht war, dumkopf? And here's a poser you must face if continue thus you do, What make you of the following thought, tell me, do tell true. The set of all sets that contain themselves might cause a soul to frown, But the set of all sets that don't contain themselves will bring you down!'' Now Gottlob Frege was no fool, he knew his proof was fried. He published his tome, but in defeat, while in his beer he cried. And Bertrand Russell told about, in books upon our shelves

31. What Is Russell's Paradox? : Scientific American 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.scientificamerican.com/article.cfm?id=what-is-russells-paradox

32. Russells Paradox | Define Russells Paradox At Dictionary.com –noun Mathematics . a paradox of set theory in which an object is defined in terms of a class of objects that contains the object being defined, resulting in a logical http://dictionary.reference.com/browse/Russells paradox

33. Russell& Definition Of Russell& In The Free Online Encyclopedia. The paradox concerning the concept of all sets which are not members of themselves which forces distinctions in set theory between sets and classes. http://encyclopedia2.thefreedictionary.com/Russell's paradox

34. Russells Paradox Definition Of Russells Paradox In The Free Online Encyclopedia. Russell's paradox ′rəs əlz ′par ə‚d ks (mathematics) The paradox concerning the concept of all sets which are not members of themselves which forces distinctions http://encyclopedia2.thefreedictionary.com/Russells paradox

35. Russell& | Define Russell& At Dictionary.com a paradox of set theory in which an object is defined in terms of a class of objects that contains the object being defined, resulting in a logical http://dictionary.reference.com/browse/Russell's paradox

36. Russell's Paradox@Everything2.com The set of all and only those sets that are not members of themselves. If it's a member of itself, then by definition it is not a member of itself. http://www.everything2.com/title/Russell%27s paradox

Near Matches Ignore Exact Everything Russell's paradox cooled by bozon idea by brock Sat Nov 13 1999 at 8:51:29 The set of all and only those sets that are not members of themselves. If it's a member of itself, then by definition it is not a member of itself. But if it is not a member of itself, then by definition it is. Bertrand Russell 's formulation of this paradox pointed out the inconsistency in Gottlob Frege 's attempt to formalize set theory. I like it! idea by Gorgonzola Sun Jun 11 2000 at 2:37:09 Set Theorists get around this paradox by making a distinction between sets and classes The class of sets that do not contain themselves as elements is not represented by a set . (Neither is the class of sets that do contain themselves, for that matter). In certain axiom systems , a set cannot be a member of itself; in these systems the class of sets that do not contain themselves is equal to the class of all sets (which isn't represented by a set either). This may sound like circular logic or defining the problem out of existence with empty sophistry . It's not. If you don't understand why not

37. Russell - 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 http://www.newworldencyclopedia.org/entry/Russell's_paradox

38. Russell's Paradox@Everything2.com Bertrand Russell s formulation of this paradox pointed out the inconsistency in Gottlob Frege s attempt to formalize set theory. http://everything2.com/title/Russell%27s paradox

Near Matches Ignore Exact Everything Russell's paradox cooled by bozon idea by brock Sat Nov 13 1999 at 8:51:29 The set of all and only those sets that are not members of themselves. If it's a member of itself, then by definition it is not a member of itself. But if it is not a member of itself, then by definition it is. Bertrand Russell 's formulation of this paradox pointed out the inconsistency in Gottlob Frege 's attempt to formalize set theory. I like it! idea by Gorgonzola Sun Jun 11 2000 at 2:37:09 Set Theorists get around this paradox by making a distinction between sets and classes The class of sets that do not contain themselves as elements is not represented by a set . (Neither is the class of sets that do contain themselves, for that matter). In certain axiom systems , a set cannot be a member of itself; in these systems the class of sets that do not contain themselves is equal to the class of all sets (which isn't represented by a set either). This may sound like circular logic or defining the problem out of existence with empty sophistry . It's not. If you don't understand why not

39. Russells Paradox Russell's Paradox A logical contradiction in set theory discovered by Bertrand Russell. If R is the set of all sets which don't contain themselves, http://dictionary.die.net/russells paradox

Definition: russells paradox Search dictionary for Source: The Free On-line Dictionary of Computing (2003-OCT-10) Russell's Paradox set theory discovered by Bertrand Russell . 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 Frankel 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)

40. Some Paradoxes - An Anthology The barber paradox; Interesting and uninteresting numbers; Russell s paradox of classes; Grelling s paradox autological and heterological http://www.paradoxes.co.uk/

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