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Topic: Hausdorff paradox

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Vitali set

Superset

Lebesgue measure

Totally ordered set

Measurable function

Test function

Base (topology)

Axiom of choice

Open mapping theorem

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	PlanetMath: Hausdorff paradox
		This sounds paradoxical: wouldn't that mean that half of the sphere's area is equal to only a third?
		The theorem itself is a crucial ingredient to the proof of the so-called Banach-Tarski paradox.
		This is version 6 of Hausdorff paradox, born on 2005-05-15, modified 2006-12-24.
	planetmath.org /encyclopedia/HausdorffParadox.html (225 words)

	Hausdorff paradox - Wikipedia, the free encyclopedia
		In mathematics, the Hausdorff paradox, named after Felix Hausdorff, states that if you remove a certain countable subset of the sphere S², the remainder can be divided into three subsets A, B and C such that A, B, C and B ∪ C are all congruent.
		Sometimes the Hausdorff paradox refers to another theorem of Hausdorff which was proved in the same paper.
		Hausdorff described these constructions in order to show that there can be no non-trivial, translation-invariant measure on the real line which assigns a size to all bounded subsets of real numbers.
	en.wikipedia.org /wiki/Hausdorff_paradox (341 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.
		In 1924, Stefan Banach and Alfred Tarski described this paradox, building on earlier work by Felix Hausdorff who managed to "chop up" the unit interval into countably many pieces which (by translation only) can be reassembled into the interval of length 2.
		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 (943 words)

	Banach–Tarski paradox - Wikipedia, the free encyclopedia
		The paradox shows that it is impossible to define "volume" on all bounded subsets of Euclidean space such that equi-decomposable sets will have equal volume.
		The proof is based on the earlier work of Felix Hausdorff, who found a closely related paradox 10 years earlier; in fact, the Banach–Tarski paradox is a simple corollary of the technique developed by Hausdorff.
		Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
	en.wikipedia.org /wiki/Banach-Tarski_Paradox (1645 words)

	paradox information (Site not responding. Last check: 2007-10-29)
		A paradox is an apparently true statement or group of statements thatseems to lead to a contradiction or to a situation that defies intuition.The recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has often led to significant advances in science, philosophy and mathematics.
		Paradoxes which are not based on a hidden error generally happen atthe fringes of context or language, andrequire extending the context (or language) to lose their paradox quality.
		Hausdorff paradox : There exists a countable subset C of thesphere S such that S\C is equidecomposable with two copies of itself.
	www.vsearchmedia.com /paradox.html (1836 words)

	iqexpand.com (Site not responding. Last check: 2007-10-29)
		First stated by Stefan Banach and Alfred Tarski in 1924, the Banach-Tarski paradox or Hausdorff-Banach-Tarski paradox is the famous "doubling the ball" paradox, which states that by using the axiom of...
		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...
		Banach-Tarski paradox < mathematics > It is possible to cut a solid ball into finitely many pieces (actually about half a dozen), and then put the pieces together again to get two solid balls, each the...
	banach-tarski_paradox.iqexpand.com (380 words)

	Hausdorff paradox: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-29)
		The proof of the much more famous Banach-Tarski paradox First stated by stefan banach and alfred tarski in 1924, the banach-tarski paradox or hausdorff-banach-tarski paradox is the famous "doubling the ball" paradox,...
		In particular, it implies that if two open subsets of the plane (or the real line) are equi-decomposable First stated by stefan banach and alfred tarski in 1924, the banach-tarski paradox or hausdorff-banach-tarski paradox is the famous "doubling the ball" paradox,...
		Banach-Tarski paradox First stated by stefan banach and alfred tarski in 1924, the banach-tarski paradox or hausdorff-banach-tarski paradox is the famous "doubling the ball" paradox,...
	www.absoluteastronomy.com /h/hausdorff_paradox (1010 words)

	Paradoxes - curiouser.co.uk
		Below is a list of paradoxes which you might be interest to research, some of which may be featured at curiouser.co.uk in due course.
		Low birth weight paradox: low birth weight babies have a higher mortality rate, babies of smoking mothers have lower average birth weight, babies of smoking mothers have a higher mortality rate, but low birth weight babies of smoking mothers have a lower mortality rate than other low birth weight babies.
		Nihilist paradox: if truth does not exist, the statement "truth does not exist" is a truth, thereby proving itself incorrect.
	www.curiouser.co.uk /paradoxes (549 words)

	A Hyperbolic Interpretation of the Banach-Tarski Paradox -- from Mathematica Information Center
		The Banach-Tarski Paradox asserts that a solid ball in 3-space may be decomposed into five disjoint sets that can be rearranged to form two solid balls, each the same size as the original ball.
		However, the algebraic idea underlying the paradox can be given a constructive interpretation in the hyperbolic plane.
		We show how to combine the Hausdorff paradox in a certain free group with the Klein-Fricke tesselation of the hyperbolic plane.
	library.wolfram.com /infocenter/Articles/1996 (114 words)

	The world's top banach tarski paradox 1 websites
		However, such transformations in general are non-isometric or involve an uncountably infinite number of "pieces"—the surprising consequence of the Banach-Tarski paradox is that it can be done with only rotation and translation (isometric mapping) of a finite number of pieces (albeit infinitely convoluted/complicated pieces, which individually are not measurable).
		Its proof is based on the earlier work of Felix Hausdorff, who managed to "chop up" the unit interval into countably many pieces which (by translation only) can be reassembled into the interval of length 2.
		The Banach-Tarski paradox is also a corollary of the Hausdorff paradox.
	www.websbiggest.com /wiki-article-tab.cfm/banach_tarski_paradox_1 (1285 words)

	ddds.info Banach Tarski Paradoxical Decomposition (Site not responding. Last check: 2007-10-29)
		The paradox shows that it is impossible to define "volume" on all bounded subsets of Euclidean space such that eգui-decomposable sets will have eգual volume.
		The proof is based on the earlier work of Felix Hausdorff ; who found a Hausdorff paradox 10 years earlier; in fact; the Banach–Tarski paradox is a simple corollary of the techniգue developed by Hausdorff.
		Use the paradoxical decomposition of that ցroup and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
	ddds.info /3047 (1596 words)

	Non-measurable set: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-29)
		For a technical description see measure (mathematics)[For more info, click on this link] and the various constructions of non-measurable sets, Vitali set In mathematics, the vitali set is an elementary example of a set of real numbers that is not lebesgue measurable....
		[Follow this hyperlink for a summary of this subject], Hausdorff paradox[Click link for more facts about this topic], Banach-Tarski paradox First stated by stefan banach and alfred tarski in 1924, the banach-tarski paradox or hausdorff-banach-tarski paradox is the famous "doubling the ball" paradox,...
		Hausdorff paradox[Follow this hyperlink for a summary of this subject]
	www.absoluteastronomy.com /n/non-measurable_set (1364 words)

	``Linear'' chaos via paradoxical set decompositions
		The developement of paradoxical decompositions began with the formalization of measure theory at the beginning of the twentieth century.
		So, loosely speaking, a paradoxical set has two disjoint subsets, which can be taken apart and rearranged using G to cover all of the original set.
		And, of course, the sets to which the Banach Tarski paradox refers to must be nonmeasurable, otherwise there would be a flat contradiction to the additivity of the measure.
	tph.tuwien.ac.at /~svozil/publ/tarski.htm (4031 words)

	sci.math FAQ: Cutting a sphere
		This construction is known as the Banach-Tarski paradox or the Banach-Tarski-Hausdorff paradox (Hausdorff did an early version of it).
		The easiest decomposition ``paradox" was observed first by Hausdorff: * The unit interval can be cut up into countably many pieces which, by translation only, can be reassembled into the interval of length 2.
		This result is, nowadays, trivial, and is the standard example of a non-measurable set, taught in a beginning graduate class on measure theory.
	www.faqs.org /faqs/sci-math-faq/AC/cuttingSphere (694 words)

	Abebooks Search Results - Amenability
		This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, and logic.
		Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions: 9.
		The Banach Tarski paradox is a most striking mathematical construction: it asserts that a solid ball may be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large as the original.
	www.abebooks.co.uk /search/sortby/3/kn/Amenability (6752 words)

	Talk:Hausdorff paradox - TheBestLinks.com - Banach-Tarski paradox, ...
		Does anybody has any information on the history of the paradox and the reason for the name?
		The technical meaning of paradoxical intended is (presumably) that defined on this page: http://abel.math.umu.se/~frankw/articles/bt/node2.html.
		I moved a part from Banach-Tarski paradox, needs some work...
	www.thebestlinks.com /Talk__3A__Hausdorff_paradox.html (230 words)

	Earliest Known Uses of Some of the Words of Mathematics (P) (Site not responding. Last check: 2007-10-29)
		Paradox (from the Greek for contrary to received opinion) is an exasperatingly ambiguous word and it is not unusual to read statements like, "This is not a paradox at all, the only reason that it is given this name is that it is counter-intuitive." W.
		A falsidical paradox produces a result that not only appears false but actually is false; there is a fallacy in the argument.
		In the Middle Ages variants of the Liar paradox were studied under the heading insolubilia (W. and M. Kneale The Development of Logic (1962) pp.
	members.aol.com /jeff570/p.html (14013 words)

	Cutting a sphere into pieces of larger volume
		The full Banach-Tarski paradox is stronger than just doubling the ball.
		This is usually illustrated by observing that a pea can be cut up into finitely pieces and reassembled into the Earth.
		The unit interval can be cut up into countably many pieces which, by translation only, can be reassembled into the interval of length 2.
	www.cs.uwaterloo.ca /~alopez-o/math-faq/node70.html (753 words)

	ZhopkaRecords.com: Lyrics and MP3 Downloads for Paradox
		The Banach-Tarski paradox is made somewhat less bizarre by pointing out that there is always a function that can map one-to-one the points in one shape to another.
		Technically, they are not measurable, and so they do not have "reasonable" boundaries nor a "volume" in the ordinary sense.
		The proof is based on the earlier work of Felix Hausdorff, who found a closely related paradox 10 years earlier; in fact the Banach-Tarski paradox is a simple corollary of the technique developed by Hausdorff.
	www.zhopkarecords.com /Paradox/discography.html (694 words)

	The world's top hausdorff paradox websites
		In mathematics, the Hausdorff paradox in measure theory states that there is a countable subset C of the sphere S
		The more famous Banach-Tarski paradox is a corollary of this paradox.
		But if F is paradoxical and acts on X with no fixed points (except under action by the identity of G), then X is F-paradoxical, thus S
	www.websbiggest.com /dir-wiki.cfm?cat=hausdorff_paradox&tab=edit (225 words)

	Amazon.com: The Banach-Tarski Paradox (Encyclopedia of Mathematics and its Applications): Books: Stan Wagon,G.-C. ... (Site not responding. Last check: 2007-10-29)
		It has been known since antiquity that the notion of infinity leads very quickly to seemingly paradoxical constructions, many of which seem to change the size of objects by operations that appear to preserve size.
		If you're really interested in the Banach-Tarski Paradox this book is what you have to read.
		The straightness of thoughts of Stan Wagon and his beautiful ideas not only about the paradox but also set theorie in general were so impressing, that thi s book became one of my favourites.
	www.amazon.com /exec/obidos/tg/detail/-/0521457041?v=glance (810 words)

	Amazon.com: The Pea and the Sun: A Mathematical Paradox: Books: Leonard M. Wapner (Site not responding. Last check: 2007-10-29)
		The Pea And The Sun: A Mathematical Paradox is a fascinating introduction to the Banach-Tarski Paradox, a mathematical riddle that asserts it could be possible to create something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again.
		Written to be accessible to lay readers and non-mathematicians, The Pea And The Sun outlines the history of the paradox, introduces readers to the basics of such matters as set theory, isometrics, scissors congruence and equidecomposability, and walks the reader through the theorem and proof that object duplication is indeed mathematically possible.
		Written in a fresh, captivating, friendly style, The Pea And The Sun is remarkably engaging and will appeal to any reader with a discerning, inquisitive mind into the nature of the so-called impossible, regardless of their particular mathematical background.
	www.amazon.com /exec/obidos/tg/detail/-/1568812132?v=glance (1370 words)

	Earliest Known Uses of Some of the Words of Mathematics (B) (Site not responding. Last check: 2007-10-29)
		In this respect their attitude was like that of Hausdorff who had found a related result some years earlier.
		The tag paradox seems to have become attached to the result in the 1940s.
		had celebrated only 5 bithdays by the time he was 21: A most ingenious paradox.) For Blumenthal the theorem is a paradox, not because it embodies a contradiction, but because it goes against common sense notions about congruence.
	members.aol.com /jeff570/b.html (6229 words)

	Table of contents for Library of Congress control number 93246292 (Site not responding. Last check: 2007-10-29)
		Table of contents for Library of Congress control number 93246292
		Table of contents for The Banach-Tarski paradox / Stan Wagon.
		Bibliographic record and links to related information available from the Library of Congress catalog
	www.loc.gov /catdir/toc/cam028/93246292.html (107 words)

	paradoxo de Banach-Tarski (Site not responding. Last check: 2007-10-29)
		Asserting that a solid ball may be taken apart into many pieces that can be rearranged to form a ball twice as large as the original, the Banach-Tarski paradox is examined in relationship to measure and group theory, geometry and logic.
		The ULTIMATE place to find out about Banach-Tarski paradox!
		Everything you could possibly want is right here!
	www.mat.uc.pt /~jaimecs/btarski.html (172 words)

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