21. One Hundred Years Of Russell's Paradox Translate this page Godehard Link. http://www.lrz-muenchen.de/~russell01/

Godehard Link Godehard Link

22. One Hundred Years Of Russell's Paradox - Speakers invited speakers John Bell (Western Ontario), Email, russell's paradoxand Diagonalization in a Constructive Context. Ulrich Blau http://www.lrz-muenchen.de/~russell01/speakers.html

Speakers The following persons have so far agreed to participate as invited speakers: John Bell (Western Ontario) Email Russell's Paradox and Diagonalization in a Constructive Context Ulrich Blau (Munich) c/o Russell01 The Significance of the Largest and Smallest Numbers for the Oldest Paradoxes Wilfried Buchholz (Munich) ... Andrea Cantini (Florence) Email Relating KF to NF Charles Chihara (Berkeley) Email Shapiro Objection to the Constructibility Theory in the Light of Russell's Theory of Types Solomon Feferman (Stanford) Email Typical Ambiguity: Having your cake and eating it too Harvey Friedman (Ohio) Email Research Program on Ways Out of Russell's Paradox Sy Friedman (Vienna) Email Completeness and Iteration in Modern Set Theory Nicholas Griffin (McMaster) Email The Prehistory of Russell's Paradox Allen Hazen (Melbourne) Email Interpreting the 1925 Logic Geoffrey Hellman (Minnesota) Email Russell's Absolutism vs(?) Structuralism Peter Hylton (Chicago) Email Presupposition and Type Theory Andrew Irvine (Vancouver) Email Russell on Method (Bern) Email On fixed point theories Hans Kamp (Stuttgart) Email Russell's Theory of Descriptions and Presuppositional Description Theories: How incompatible are they?

23. Re: Russell's Paradox Re russell's paradox. Posted by DickT on January 21, 2003 at 191947 In Replyto russell's paradox posted by RocketMan on January 21, 2003 at 165805 http://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.

24. Russell's Paradox russell's paradox. Follow Ups Post Followup Questions VII FAQ Posted by Anyone heard of russell's paradox. I recently http://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 Name : Save your login cookie Password : Delete your login cookie Subject : Comments: (The following are optional.) Link URL : Link Title : Image URL : Follow Ups Post Followup Questions VII FAQ

25. Russell's Paradox russell's paradox. russell's paradox. Russell proposed that it becomes paradoxicalwhen the above mentioned gathering M is assumed to be a set. http://www.rinku.zaq.ne.jp/suda/incomplete/chap03_e.html

Russell's Paradox Previous Contents Japanese /English Definitions Definition 1. A set is the gathering of 'the one' which can be mutually identified to distinguish clearly and in case that the range to specify the whole of the gathering is clearly given. Definition 2. 'The one' which composes the gathering is called an element. Fundamental theory From definition 1, the set is the gathering of 'the one'. However, the mere gathering of 'the one' is not always a set. Even if it is a gathering of 'the one', if each of 'the one' cannot be mutually identified to distinguish clearly, it is not a set. Similarly, even if it is a gathering of 'the one', in case that the range to specify the whole of the gathering is not clearly given, it is not a set. Russell's Paradox Russell proposed that it becomes paradoxical when the above mentioned gathering M is assumed to be a set. The set which does not contain oneself as an element like the set of natural numbers, N exists. (A) Let the set M be the gathering of all sets which do not contain themselves as elements. (B) Either the set M contains oneself as an element or not.

26. Russell's Paradox russell's paradox. In the middle of the night I got such a fright that woke me witha start, For I dreamed of a set that contained itself, in toto, not in part. http://www.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

27. NJPL Volume 5, Number 1 WORLD. NINO B. COCCHIARELLA russell's paradox OF THE TOTALITY OF PROPOSITIONS.JOHN CANTWELL TOWARDS AN ANALYSIS OF THE PROGRESSIVE. http://www.hf.uio.no/filosofi/njpl/vol5no1/

VOLUME 5 NUMBER 1 NOVEMBER 2000 CONTENTS F RANCIS Y. L IN EVENTS AND TIME IN A FINITE AND CLOSED WORLD N INO B. C OCCHIARELLA RUSSELL'S PARADOX OF THE TOTALITY OF PROPOSITIONS J OHN C ANTWELL TOWARDS AN ANALYSIS OF THE PROGRESSIVE NJPL Last modified: Mon Dec 18 14:24:38 CET 2000

28. Russell's Paradox russell's paradox Bertrand Russell (18721970) constructed a famousparadox (an antinomy ) to persuade the mathematical world that http://users.forthnet.gr/ath/kimon/Russells_pdx.html

Russell's Paradox Bertrand Russell (1872-1970) constructed a famous paradox (an "antinomy") to persuade the mathematical world that in developing consistent systems (systems in which every statement is either true or false), familiarity and intuitive clarity are not solid bases. The argument goes on like this: There are sets than contain themselves (examples: "the set of all objects that can be described with exactly thirteen words", "the set of all thinkable things") Therefore, a set either contains itself or not. Let's call a set "non-normal" in the first case and "normal" in the second Let N be the collection of all normal sets, which of course, is itself a set Question: is N normal? If N is normal, then by definition of "normality" it does not contain itself. But N contains by construction all normal sets therefore itself too (contradiction) If N is not normal, then by definition of "non-normality" N is itself a member of N. But by construction, any member of N is a normal set (contradiction too) Conclusion: the statement "N is normal" is neither true nor false Famous Problems and Proofs Main Page

29. ABSTRACT For RUSSELL'S PARADOX Abstract. AD Irvine. russell's paradox , The Stanford Encyclopedia of Philosophy(1995), http//plato.stanford.edu/entries/russellparadox/. Excerpt. http://www.arts.ubc.ca/philos/irvine/wwwrp.htm

Abstract A.D. Irvine "Russell's Paradox" The Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/russell-paradox/ Excerpt 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 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. Summary This article consists of four short sections containing (1) a brief introduction to Russell's paradox, (2) the history of Russell's paradox, (3) the significance of Russell's paradox, and (4) a bibliography relating to the paradox. The article appears in The Stanford Encyclopedia of Philosophy . Unlike fixed-media reference works, this is an on-line reference work which is regularly revised in order that it not go out of date.

30. Russells Paradox russell's paradox. russell's paradox arises as a result of naive set theory'ssocalled unrestricted comprehension (or abstraction) axiom. http://www.literature-awards.com/nobelprize_winners/russells_paradox.htm

Russell's Paradox Bertrand Russell Nobel Prize 1950 Russell's Mathematical Contributions Prize Presentation Writings of Bertrand Russell Nobel Lecture Russell's Biography Over a long and varied career, Bertrand Russell made ground-breaking contributions to the foundations of mathematics and to the development of contemporary formal logic, as well as to analytic philosophy. His contributions relating to mathematics include his discovery of Russell's paradox, his defense of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his introduction of the theory of types, and his refining and popularizing of the first-order predicate calculus. Along with Kurt Gödel, he is usually credited with being one of the two most important logicians of the twentieth century The Autobiography of Bertrand Russell Hardcover Paperback Russell discovered the paradox which bears his name in May 1901, while working on his Principles of Mathematics (1903). The paradox arose in connection with the set of all sets which are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. The significance of the paradox follows since, in classical logic, all sentences are entailed by a contradiction. In the eyes of many mathematicians (including David Hilbert and Luitzen Brouwer) it therefore appeared that no proof could be trusted once it was discovered that the logic apparently underlying all of mathematics was contradictory. A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was thus prompted.

31. A Geometric Note On Russell's Paradox A geometric note on russell's paradox. Welcome A lengthier discussionof russell's paradox is found on my web page on vacuous truth. It http://www.angelfire.com/az3/nfold/russell.html

A geometric note on Russell's paradox Welcome to N-fold Quirky notes to myself on math and science If you spot an error, please email me This page went online Dec. 28, 2001 When truth is vacuous (includes a section on Russell's paradox) Reach other N-fold pages A lengthier discussion of Russell's paradox is found on my web page on vacuous truth. It is there that I discuss banishing 'actual infinity' and replacing it with construction algorithms. However, such an approach, though countering other paradoxes, does not rid us of Russell's, which is summed up with the question: If sets are sorted into two types: those that are elements of themselves ('S is the name of a set that contains sets named for letters of the alphabet' is an example) and those that are not, then what type of set is the set that contains sets that are elements of themselves? Here we regard the null set as the initial set and build sets from there, as in: Using an axiom of infinity, we can continue this process indefinitely, leading directly to a procedure for building an abstraction of all countable sets; indirectly, noncountable sets can also be justified. In Russell's conundrum, some sets are elements of themselves.

32. Russell's Paradox russell's paradox Gödel Escher Bach russell's paradox is one of the classicmath paradoxes, this time based on sets that include themselves. http://community.middlebury.edu/~dwalker/class/russell.html

Russell's Paradox Gödel Escher Bach Russell's paradox is one of the classic math paradoxes, this time based on sets that include themselves. The set of all sets, for example, includes itself. The set of all sets that do not include themselves, however, sparks the paradox. If the set does not include itself, then it is in the set, but since the set then includes itself, it is not in the set. This set exists IFF it does not exist, a contradiction in terms. The paradox was discovered in 1901 by Bertrand Russell. Cut The Knot includes an excerpt from Russell's autobiography about paradoxes. Erasing Russell's Paradox gives a group of axioms that allow the avoidance of Russell's Paradox. Read a poem about Russell's Paradox.

33. Russell's Paradox russell's paradox. The origins of russell's paradox are even more controversialthan the origins of BuraliForti and Cantor's paradoxes. http://www.u.arizona.edu/~miller/finalreport/node4.html

Next: Conclusion Up: An Historical Account of Previous: Cantor's Paradox Russell's Paradox Russell's paradox, also referred to as Russel's antinomy, Russell's problem, and Zermelo-Russellsches paradoxon ([ ], p. 21), is by far the most famous of the classical paradoxes of set theory, owing much of its fame to its simplicity and far-reaching implications. Russell is usually credited with its discovery in his The Principles of Mathematics ]), or during June of 1901 ([ ]); however, it is clear that Zermelo arrived at the paradox independently one or two years earlier ([ ]). The paradox is simpler than the paradoxes of Burali-Forti and Cantor because it relies only on the most elementary ideas of set theorythe notion of set, set membership, and the Axiom of Abstraction. The paradox can be formulated in the following way. Suppose that the the collection defined by the formula is a set. Then if , and if . Russell also used a statement about a barber to illustrate this principle: If a barber cuts the hair of exactly those who do not cut their own hair, does the barber cut his own hair? ([ ], p. 809). The first clear mention of this paradox occurred in Russell's letter to Frege, written on June 16, 1902. Russell realized that his paradox undermined Frege's theory set forth in

34. Cantor's Paradox The conclusion in the preceding proof that looks almost identical to the contradictionreached in russell's paradox, and indeed, the most prominent theories on http://www.u.arizona.edu/~miller/finalreport/node3.html

Next: Russell's Paradox Up: An Historical Account of Previous: Burali-Forti's Paradox Cantor's Paradox Cantor's paradox, sometimes called the paradox of the greatest cardinal, expresses what its second name would implythat there is no cardinal larger than every other cardinal. There seems to be close consensus that Cantor discovered this paradox in 1899 or between 1895 and 1897 ([ ], p. 34), but there are some, including the authors who attribute the Burali-Forti paradox to Russell, who give credit to Russell in 1899 or 1901 ([ ], p. 343). The crux of Cantor's paradox is Cantor's Theorem, which states that for any set , where is the power set of and is the cardinality of . The typical, modern proof for this theorem is as follows, and can be found, among others, in [ ], and [ ]. Let be a fixed set. Then defined by , is an injection, so . It remains to show that , so by way of contradiction assume that is a surjection. Then , so there exists a with . Now implies and implies , so a contradiction has been reached. Thus, , so . The conclusion in the preceding proof that looks almost identical to the contradiction reached in Russell's paradox, and indeed, the most prominent theories on the origins of Russell's paradox suggest that his paradox was derived from Cantor's paradox alone or from a combination of Cantor's paradox and the proof of Cantor's Theorem. Given Cantor's Theorem, Cantor's paradox following almost immediately. Suppose that

35. Russell's Paradox - Nathanael Thompson- The Examined Life On-Line Philosophy Jou russell's paradox. by. Nathanael Thompson. http://examinedlifejournal.com/articles/template.php?shorttitle=russellparadox&a

36. Russell's Paradox - Nathanael Thompson - The Examined Life On-Line Philosophy Jo Back to Article. russell's paradox. by. Nathanael Thompson. When Gottlieb Fregemade predicate logic, He said for all p and A, p is in {x Ax} if and only if Ap. http://examinedlifejournal.com/articles/printerfriendly.php?shorttitle=russellpa

37. Erik Benson's Weblog: Russell's Paradox Related Nodes paradox. amazonid. number. definition. Russell. ether. will. AI.Notable. All Consuming. BlogNomic. Sister Cities. The Most Beautiful One. AmazonAPI. http://erikbenson.com/index.cgi?node=Russell's Paradox

38. Scientific American: Ask The Experts: Mathematics: What Is Russell's Paradox? What is russell's paradox? russell's paradox is based on examples like this Considera group of barbers who shave only those men who do not shave themselves. http://www.sciam.com/askexpert_question.cfm?articleID=0005DA51-B5F4-1C71-9EB7809

39. Russell's Paradox Back. russell's paradox. Easy to state, yet russell's paradox can beput into everyday language in many ways. The most often repeated 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

40. Russell's Paradox - Curiouser.co.uk russell's paradox. All classes are either a member of themselves or not.The class of all ideas is an idea. The class of all classes is a class. http://www.curiouser.co.uk/uk/russell.htm

Russell's Paradox All classes are either a member of themselves or not. The class of all ideas is an idea. The class of all classes is a class. Both these classes are members of themselves. The class of all men is not a man. The class of all illnesses is not an illness. Neither of these classes are members of themselves. Let S be the class of all S elf-membered classes. ie classes which are members of themselves. Let N be the class of all N on-self-membered classes. ie. classes which are not members of themselves. Consider N. CONTINUE N is itself a class and must therefore be either a member of N or S. Is N a member of itself? If it is not it must be a member of the class of non-self-members, which is N. But if N is a memember of N, then it is a member of itself and therefore a member of S and not N. But if N is a member of S and not N, then it is not a member of its own class and must therefore be a member of N - which was where we began. CONTINUE The paradox may be easier to follow in the following form: If X is any class and N the class of all non-self-membered classes, then the following statement is true:

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