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Topic: Russells paradox

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	Russell's Paradox (Stanford Encyclopedia of Philosophy)
		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.
		Russell's type theory thus appears in two versions: the "simple theory" of 1903 and the "ramified theory" of 1908.
	plato.stanford.edu /entries/russell-paradox (1417 words)

	Bertrand Russell - Wikipedia, the free encyclopedia
		Russell and Moore strove to eliminate what they saw as meaningless and incoherent assertions in philosophy, and they sought clarity and precision in argument by the use of exact language and by breaking down philosophical propositions into their simplest components.
		Russell thought Wittgenstein's elevation of language as the only reality with which philosophy need be concerned was absurd, and he decried his influence and the influence of his followers, especially members of the so-called Oxford school, who he believed were promoting a kind of mysticism.
		Russell was an early critic of the official story in the John F. Kennedy assassination; his "16 Questions on the Assassination" from 1964 is still considered a good summary of the apparent inconsistencies in that case.
	en.wikipedia.org /wiki/Bertrand_Russell (8395 words)

	Russell's paradox: Definition and Links by Encyclopedian.com - All about Russell's paradox (Site not responding. Last check: 2007-10-10)
		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.
		After this paradox was described, set theory had to be reformulated axiomatically as axiomatic set theory in a way that avoided this and other related problems.
		The Barber paradox, in addition to leading to a cleaner set theory, has been used twice more with smashing success: Kurt Gödel proved his incompleteness theorem by formalizing the paradox, and Turing proved the undecidability of the Halting problem (and with that the Entscheidungsproblem) by using the same trick.
	www.encyclopedian.com /ru/Russells-paradox.html (1127 words)

	Encyclopedia: Russells paradox (Site not responding. Last check: 2007-10-10)
		Russell's paradox is a paradox discovered by Bertrand Russell in 1901 which shows that the naïve set theory of Cantor and Frege is contradictory.
		The whole point of Russell's paradox is that the answer "such a set does not exist" means that the definition of the notion of "set" within a given theory is unsatisfactory.
		The Barber paradox, in addition to leading to a tidier set theory, has been used twice more with great success: Kurt Gödel proved his incompleteness theorem by formalizing the paradox, and Turing proved the undecidability of the Halting problem (and with that the Entscheidungsproblem) by using the same trick.
	www.nationmaster.com /encyclopedia/Russells-paradox (1176 words)

	Russell's paradox - RecipeFacts (Site not responding. Last check: 2007-10-10)
		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.
		Russell, with Alfred North Whitehead, undertook to accomplish Frege's task, this time using a more restricted version of set theory that, they thought, would not admit Russell's Paradox, but would still produce arithmetic.
		There are some versions of this paradox which are closer to real-life situations and may be easier to understand for non-logicians: for example, the Barber paradox supposes a barber who shaves everyone who does not shave himself, and no one else.
	www.recipeland.com /encyclopaedia/index.php/Russells_paradox (1674 words)

	Russell's Paradox [Internet Encyclopedia of Philosophy]
		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.
		Russell discovered the contradiction from considering Cantor's power class theorem: the mathematical result that the number of entities in a certain domain is always smaller than the number of subclasses of those entities.
		Russell considered the simple mapping of classes onto themselves, and invoked the Cantorian approach of considering the class of all those entities that are not in the classes onto which they are mapped.
	www.utm.edu /research/iep/p/par-russ.htm (2709 words)

	Russells paradox - Wikipedia
		There are some versions of this paradox which are closer to real-life situations and may be easier to understand for non-logicians: For example, the story of the barber who shaves everyone who does not shave himself.
		Russell himself, together with Alfred North Whitehead, developed a system of types in his work Principia Mathematica.
		The Barber paradox, in addition to leading to a cleaner set theory, has been used twice with smashing success: Gödel proved his incompleteness theorem by formalizing the paradox, and Turing solved the Halting problem (and with that the Entscheidungsproblem) by using the same trick.
	nostalgia.wikipedia.org /wiki/Russells_paradox (377 words)

	Russells Paradox
		Grelling-Nelson paradox - The Grelling-Nelson paradox is a semantic paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to German philosopher and mathematician Hermann Weyl.
		Russell's philosophical and logical work Logic In mathematical logic, Russell established Russell's paradox, which exposed an inconsistency in naive set theory and led directly to the major texts with a fuller understanding of their language and an enhanced view of a play's theatrical potential.
		Paradox 2005 is a re-recorded version of the track Paradox which was a track originally on the album Hall Of The Mountain Grill.
	id20.homentertainsidesign.com /russellsparadox.html (917 words)

	Banach-Tarski Paradox - Wikipedia
		The Banach-Tarski Paradox is the famous "doubling the ball" paradox, which claims that by using the axiom of choice it is possible to take a solid ball in 3-dimensional space, cut it up into finitely many pieces and, using only rotation and translation, reassemble the pieces into two balls the same size as the original.
		Logicians most often use the term "paradox" for a statement in logic which creates problems because it causes contradictions, such as the Liar paradox or Russell's paradox.
		Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the unit sphere.
	nostalgia.wikipedia.org /wiki/Banach-Tarski_Paradox (951 words)

	Read about Russell's paradox at WorldVillage Encyclopedia. Research Russell's paradox and learn about Russell's paradox ... (Site not responding. Last check: 2007-10-10)
		Some of the various set-theoretic approaches to address and circumvent Russell's paradox can be illustrated in the context of Smartpedia, respecting the requirement that the content of each entry must be correct according to its entry name, and allowing the possibility of its entire contents to be correctly linked in turn:
		The Russell paradox arises from the supposition that one can meaningfully define a class in terms of any well-defined property P(x); that is, that we can form the set P = {x:P(x) is true }.
		The paradoxical argument like the one at the start of this article has the form of constructing a purported proposition P which would be true if and only if it were false, entailing that the construction is defective.
	encyclopedia.worldvillage.com /s/b/Russells_paradox (1714 words)

	Russells Paradox (Site not responding. Last check: 2007-10-10)
		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.
		The paradox arose in connection with the set of all sets which are not members of themselves.
		Russell's response to the second of these objections was to introduce, within the ramified theory, the axiom of reducibility.
	www.literature-awards.com /nobelprize_winners/russells_paradox.htm (1423 words)

	Russell's Paradox [Internet Encyclopedia of Philosophy]
		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.
		Russell discovered the contradiction from considering Cantor's power class theorem: the mathematical result that the number of entities in a certain domain is always smaller than the number of subclasses of those entities.
		Russell considered the simple mapping of classes onto themselves, and invoked the Cantorian approach of considering the class of all those entities that are not in the classes onto which they are mapped.
	www.iep.utm.edu /p/par-russ.htm (2714 words)

	Russell’s paradox
		A paradox uncovered by Bertrand Russell in 1901, which forced a reformulation of set theory.
		Russell's Paradox, in its original form considers the set of all sets that aren't members of themselves.
		Russell's paradox underlies the proof of Gödel's incompleteness theorem as well as Alan Turing's proof of the undecidability of the halting problem.
	www.daviddarling.info /encyclopedia/R/Russells_paradox.html (331 words)

	Russell's paradox : Russells paradox (Site not responding. Last check: 2007-10-10)
		There are some versions of this paradox which are closer to real-life situations and may be easier to understand for non-logicians: For example, the Barber paradox which considers a barber who shaves everyone who doesn't shave himself, and no one else.
		Russell made the following discovery: if every description determines a set, then so does the following description: "is a set which isn't a member of itself".
		So if X is a member of itself, it isn't; and if it isn't, it is. This is Russell's paradox, then: just the set of all sets that are not members of themselves.
	www.explainthis.info /ru/russells-paradox.html (1107 words)

	One Hundred Years of Russell's Paradox - Abstracts
		Russell made use of typical ambiguity in the theory of types in order to combine the assurance of its (apparent) consistency ("having the cake") with the freedom of the informal untyped theory of classes and relations (and "eating it too").
		Russell reified structures by adopting an ontology of propositions- mind and language independent 'states of affairs.' The thesis of Logicism advanced in the work held that the intuitions grounding all non-applied mathematics are logical intuitions of propositional structure.
		Russell's paradox arises within the system of Frege's Grundgesetze through the interplay between the second-order comprehension principle and the infamous basic law V. By weakening either of these principles, consistent subsystems of Frege's theory can be obtained.
	www.lrz-muenchen.de /~russell01/papers.html (8694 words)

	Bertrand Russell (Stanford Encyclopedia of Philosophy)
		Russell's contributions to logic and the foundations of 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 development of the theory of types, and his refining of the first-order predicate calculus.
		Russell's response was to introduce the axiom of reducibility, an axiom that lessened the vicious circle principle's scope of application, but which many people claimed was too ad hoc to be justified philosophically.
		Russell's social influence stems from three main sources: his long-standing social activism, his many writings on the social and political issues of his day, and his popularizations of technical writings in philosophy and the natural sciences.
	plato.stanford.edu /entries/russell (3965 words)

	The Limits of Logic - a Mathematical View
		Russell asked himself about the set consisting of all sets that are not members of themselves - in Cantor's system this is a well-defined set.
		Frege's stated his own version of this paradox in terms of concepts, in which case Russell's set is the concept that does not fall under its defining concept.
		What I understand from Russell's Paradox is the 'set of all sets', or the 'concept of all concepts' is in some sense too big or too powerful for everyday logic.
	mikefinch.com /md/fc/ll.htm (871 words)

	aufg-7
		Russell wurde von einer geistigen Lähmung ergriffen, die ihn an allem zweifeln ließ, und er versank schließlich gar in eine Depression.
		Russell war der Erste, der die Tragweite erkannte, obwohl er in Cambridge eine eher veraltete Mathematik studiert hatte, die weit hinter der damals führenden kontinentalen Lehre zurück war.
		Der Logiker Bertrand Russell dachte natürlich nicht über Friseure nach, sondern beschäftigte sich mit den abstrakten Gebilden der mathematischen Mengen.
	www.cl.uni-heidelberg.de /kurs/ss01/refer/aufg-7.mhtml (1122 words)

	Paradox Solutions (Site not responding. Last check: 2007-10-10)
		Russell introduced his own reply to the paradox the ‘theory of types’; in 1903 in his Principles of Mathematics.
		Russell’s ‘theory of types’; does resolve the paradox but it is seen in two totally different lights by others:
		Although it provides a uniform basis to the resolution of other paradox’s it did not resolve all of them.
	students.odl.qmul.ac.uk /~roger10/ODL122/page4.html (515 words)

	Russell's Paradox
		Russell's paradox shows that these appearances are deceiving.
		The result is said to be paradoxical simply because it refutes what seemed to be so obviously true, and calls for a radical revision in the ideas of what sorts of sets exist.
		Another version of basically the same paradox, again due to Russell: Let a "normal bibliography" be a bibliography that does not list itself.
	www.ilstu.edu /~kfmachin/phi281/russells_paradox.htm (926 words)

	► » invention of paradox (Site not responding. Last check: 2007-10-10)
		A paradox is an extreme case of a dichotomy.
		Paradoxes only occur relative to a logic for the model they are expressed
		In a boolean logic, this is certainly paradoxical.
	www.science-chat.org /detail-801542.html (2281 words)

	Text Götz Deutsch (Site not responding. Last check: 2007-10-10)
		Paradox droht es vielmehr und wie gesagt überall da zu werden, wo es aufs Prinzipielle geht und das Prinzipielle prinzipiell begründet werden soll.
		Ein Paradox hat nur zum Beispiel mindestens partiell auch ästhetische Qualitäten, insofern es gegebenenfalls eine Art,logischer Eleganz' besitzt, rhetorische Qualitäten, insofern es mindestens ambivalent und daher interessant ist, oder sogar praktische, insofern es von heuristischem Wert ist.
		Paradoxe haben ausser der logischen Bredouille, die sie darstellen, auch eine weitere Eigenschaft, die nicht zu verachten ist: Sie sind grundsätzlich interessant, interessanter und daher in gewissem Sinn auch gelegentlich wichtiger als Wahrheit, Echtheit, Wirklichkeit.
	home.snafu.de /jonasw/PARADOXGoetzD.html (5585 words)

	Russells paradox - rFind.net (Site not responding. Last check: 2007-10-10)
		Russells paradox (efter Bertrand Russell) visar att den till synes naturliga och självklara Abstraktionsprincipen ger upphov till motsägelser i mängdteorin.
		Russell upptäckte detta under läsning av första bandet av Gottlob Freges Grundgesetze.
		Russell meddelade Frege detta, varpå Frege gjorde ett tillägg i slutet på andra bandet av Grundgesetze där han skriver "En större olycka kan knappast drabba en vetenskaplig författare än att få en av grunderna för sitt verk raserad, när verket själv fullbordats".
	www.rfind.net /info/Russells_paradox (547 words)

	buffie the body (Site not responding. Last check: 2007-10-10)
		functions primarily as a few thousand years: As a young man, Russell had misrepresented what one means when one says The present Germanic king of France has a known specific cause or causes (called its etiology), from a syndrome, which is a wellknown behavior in many cultures.
		But, according to Russells theory seems to have been cases in which it is possible under certain conditions for remains to forestall position and restore a natural appearance; massage cream is also necessary.
		He died from his isomorphic requirement, but he was never realised, and Russell arranged a hasty divorce from Alys, boddy buffie model picture marrying Dora six days after the airstrike, body buffie dvd of injuries sustained in the status constructus or construct state is a man and obnoxious.
	buffie-the-body.writep.org (818 words)

	Psykosyntesakademin - Om Akademin
		På PsykosyntesAkademin i Stockholm kan du gå längre utbildningar och kortare kurser i psykosyntes - ett synsätt och en metod för mänsklig utveckling som formulerades av den italienske psykiatern Roberto Assagioli i början av 1900-talet.
		Margo Russell, 1939-2001, var rektor för PsykosyntesAkademin i Stockholm från starten 1989 till 2000.
		Våren 2001 avled Margo Russell hastigt efter en kortare tids sjukdom.
	www.psykosyntesakademin.se /om_akademin (2009 words)

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