Where results make sense

About us   |   Why use us?   |   Reviews   |   PR   |   Contact us

Topic: Banach Tarski paradox

    Note: these results are not from the primary (high quality) database.

Banach-Tarski paradox - Wikipedia, the free encyclopedia 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. Banach and Tarski intended for this proof to demonstrate that the axiom of choice was incorrect, but the nature of the proof is such that most mathematicians take it to mean that the axiom of choice merely results in bizarre and unintuitive consequences. 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   (1277 words)

Talk:Banach-Tarski paradox - Wikipedia, the free encyclopedia The paradox in Banach-Tarski paradox is that if you take a (solid) sphere, cut it in 5 pieces (although non-measurable, weird beast pieces), then move and rotate these to their places, you get two exact copies of the original sphere. The paradoxical decomposition of the free group in two generators, which underlies the proof, could maybe be visualized by depicting its (infinite) Cayley graph and showing how it consists of four pieces that look just like the whole graph. This is a popular paradox, and I believe that there are (relatively) many laypersons who may have a passing interest in what this thing is all about. en.wikipedia.org /wiki/Talk:Banach-Tarski_paradox   (2668 words)

Math Forum - Ask Dr. Math Date: 07/20/2002 at 02:38:00 From: Tony Zhu Subject: Axiom of choice and Banach-Tarski paradox The following is an excerpt from _The Mystery of the Aleph_ by Amir Aczel. Banach-Tarski in fact splits the sphere into 5 pieces, one of which is the point at the center of the sphere. The reason it seems paradoxical is because the sphere itself is measurable - it has a volume given by the formula V = (4/3)pi*r^3, where r is the radius. www.bonus.com /contour/askdrmath/http@@/mathforum.org/library/drmath/view/61014.html   (416 words)

PlanetMath: proof of Banach-Tarski paradox This is version 3 of proof of Banach-Tarski paradox, born on 2005-05-28, modified 2005-05-29. "proof of Banach-Tarski paradox" is owned by GrafZahl. Cross-references: vectors, countable, Hausdorff paradox, sphere, infinite, sequence, inclusion, image, preimage, intersection, theorem, proper subset, disjoint union, bijective, map, integer, unions, isometry, congruent, transitivity, Euclidean space, subsets, equivalence relation, disjoint, equi-decomposable, origin, balls, unit, argument, line, clear, properties planetmath.org /encyclopedia/ProofofBanachTarskiParadox.html   (274 words)

Banach-Tarski paradox The fact that the Banach-Tarski paradox depends on the axiom of choice (AC), yet is so strongly counterintuitive, has been used by some mathematics to suggest that AC must be wrong; however, the benefits of adopting AC are so great that such black sheep of the mathematical family as the paradox are generally tolerated. The Banach-Tarski paradox, which mathematicians often refer to as the Banach-Tarski decomposition because it's really a proof not a paradox, highlights the fact that among the infinite set of points that make up a mathematical ball, the concept of volume and of measure can't be defined for all possible subsets. In any case, the Banach-Tarski paradox doesn't give a prescription for how to produce the subsets: it only proves their existence and that there must be at least five of them to produce a second copy of the original ball. www.daviddarling.info /encyclopedia/B/Banach-Tarski_paradox.html   (475 words)

Cutting a sphere into pieces of larger volume This construction is known as the Banach-Tarski paradox or the Banach-Tarski-Hausdorff paradox (Hausdorff did an early version of it). The full Banach-Tarski paradox is stronger than just doubling the ball. ``Banach and Tarski had hoped that the physical absurdity of this theorem would encourage mathematicians to discard AC. www.cs.uwaterloo.ca /~alopez-o/math-faq/node70.html   (753 words)

sciforums.com - Banach-Tarski paradox holds key to universe? In fact, what the Banach-Tarski paradox shows is that no matter how you try to define "volume" so that it corresponds with our usual definition for nice sets, there will always be "bad" sets for which it is impossible to define a "volume"! Many great thinkers and math geniuses have tried to debunk te Tarski paradox, but it profed to be mathematically sound, that is, if you assume that the basic believes of our math axiomas are sound. Rejecting the Tarski paradox would mean you reject todays way of doing math and you know what, maybe you are right!!! www.sciforums.com /printthread.php?t=13655   (601 words)

paradox for Books Great Deals on paradox for Books Justification And Variegated Nomism: The Paradoxes Of Paul (Justification and Variegated Nomism) Kingdom, Grace, Judgment: Paradox, Outrage, and Vindication in the Parables of Jesus Paradox of Plenty: A Social History of Eating in Modern America, Revised Edition (California Studies in Food and Culture) www.negative-procreative.biz /all-about-paradox.php   (937 words)

The Banach-Tarski Paradox - Cambridge University Press 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. 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. www.cup.cam.ac.uk /uk/catalogue/catalogue.asp?isbn=0521302447   (235 words)

k17. the infinite choice axiom and so... the impossible Banach-Tarski paradox 26 On page 16 you can find a part for the exact proof of the Banach-Tarski paradox. Look at the proof of the Banach-Tarski paradox. This is also a link about that real paradox. nnw.berlios.de /docs.php/introftk17/noflash   (427 words)

PlanetMath: Banach-Tarski paradox This is version 7 of Banach-Tarski paradox, born on 2003-07-17, modified 2005-11-02. See Also: pseudoparadox in measure theory, Hausdorff paradox, proof of Hausdorff paradox So some of the pieces in which the ball is divided are not constructable. planetmath.org /encyclopedia/BanachTarskiParadox.html   (355 words)

iqexpand.com The Banach-Tarski paradox claims that you can take a ball of radius 1, dissect it into 5 parts, move and rotate the parts and get two balls of radius 1. Dewdney published a letter from his friend Arlo Lipof in the computer recreations column of the Scientific American where he describes an underground operation "in a South American country" of doubling gold balls using the Banach-Tarski paradox. For a technical description see measure (mathematics) and the various constructions of non-measurable sets, Vitali set, Hausdorff paradox, Banach-Tarski paradox. non-measurable_set.iqexpand.com   (828 words)

BAN from FOLDOC This paradox is a consequence of the Axiom of Choice. In a Banach space the inverse to a continuous linear mapping is continuous. Hilbert spaces, spaces of integrable functions and spaces of absolutely convergent series are examples of infinite-dimensional Banach spaces. www.instantweb.com /d/dictionary/foldoc.cgi?query=BAN   (958 words)

Paradox - Layman's Guide to the Banach-Tarski Paradox The Bertrand's Paradox is one such discovery that made mathematicians wary of the whole notion of probability. If you have any comments about Paradox or their songs, then please post them below: Paradox are awesome, very unique singing style but also very cool. A Paradox of Logic by Lewis Carroll, from the Platonic Realms Interactive Math Encyclopedia. webinfosites.com /q/paradox.htm   (191 words)

thesamis.net » Blog Archive » The Banach-Tarski Paradox A pretty good redux of the Banach-Tarski Paradox has been posted over at K5. thesamis.net » Blog Archive » The Banach-Tarski Paradox WP-Theme by Oliver Roick and modified by Bryan Samis. www.thesamis.net /2003/05/26/the-banach-tarski-paradox   (310 words)

Banach - Tarski Paradox The topic I chose was an interesting dilemma known as the Banach - Tarski Paradox. The idea behind this paradox can be easily explained by the following. Any pea sized ball can be cut up into a finite number of pieces, the pieces can then be reassembled, after rotation, into a ball the size of the earth. www.math.uga.edu /~chadm/balls.html   (117 words)

The Banach-Tarski Paradox The cool way to phrase the Banach-Tarski Paradox is this: "You can take a ball, break it into ten pieces, and rearrange them to form two balls each as big the original one." To my mind, that phrasing makes the Paradox transcend gee-whiz-math-puzzle-neatness and attain the geomantic allure of the Occult section of the bookstore. In fact, the case of the actual Paradox is worse, because neither you nor any mathematician can specify the infinite sets of points you would like to separate. [In fact, this weaker version is called the Hausdorff Paradox.] Think about this fact for a minute: Our proof failed because it did not really specify a unique color for each point, some being accessible via multiple distinct journeys from their capitals. www.thebandarlog.com /math/BTParadox.html   (4288 words)

< FeroxLog > » Blog Archive » The Banach-Tarski Paradox The Banach-Tarski Paradox is well-known among mathematicians, particularly among set theorists.1 The paradox states that it is possible to take a solid sphere (a “ball”), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere. No matter how many times I read it or come back to it (sometimes years later) I still can’t get my head completely around The Banach-Tarski Paradox In other words, you’ve doubled the volume of the original sphere. www.ferox.haxial.net /weblog?p=445   (182 words)

Citations: The Banach-Tarski Paradox - Wagon (ResearchIndex) The Banach-Tarski Paradox, Cambridge University Press, Cambridge, 1986. ....equidecompositions are physically forbidden 14 Suppose a group G acts on A X. Then Tarski s theorem states that there exists a finitely additive, G invariant measure : P(x) 0, with (A) 1 if and only if A is not G paradoxical. Augenstein [43] and Pitowsky [44] have given two possible applications of paradoxical equidecomposibility in physics. citeseer.ist.psu.edu /context/124486/0   (825 words)

Topological Curiosities The down side is that, for example, the sets whose existence is claimed by Banach and Tarski could not be visualized, let alone built. Two theorems I am going to state are mind boggling results associated with the names of F. Hausdorff, A. Tarski, S. Banach, J. von Neumann and R. Robinson. All the more so because some results (Tarski-Banach Decompositions is one example) are only obtainable with the help of this axiom or equivalent statements. www.cut-the-knot.org /do_you_know/banach.shtml   (579 words)

The Banach-Tarski paradox We shall use the axiom of choice to prove an extremely wimpy version of the Banach Tarski paradox, to wit: It is possible to take a subset of the interval [0,2], cut it up into a countable number of disjoint pieces, and then translate each of these pieces so that their union is the entire real line. www.math.ucla.edu /~tao/resource/general/121.1.00s/tarski.html   (356 words)

Banach-Tarski Paradox made from the mathematical properties of the banach-tarski paradox All graphics here are free for personal use only. www.graphicsforums.com /public/list.asp?id=292   (59 words)

Amazon.ca: The Banach-Tarski Paradox: Books Look for books like The Banach-Tarski Paradox by subject: Write an online review and share your thoughts with other shoppers! www.amazon.ca /exec/obidos/ASIN/0521457041   (294 words)

paradoxo de Banach-Tarski 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)

Banach-Tarski_paradox - OneLook Dictionary Search Other places to try your search for Banach-Tarski paradox: General Web searches for dictionaries containing Banach-Tarski paradox: www.onelook.com /?w=Banach-Tarski_paradox&other=1   (72 words)

Universal Book of Mathematics: list of entries The fact that the Banach-Tarski paradox depends on the axiom of choice (AC), yet is so strongly counterintuitive, has been used by some mathematics to suggest that AC must be wrong; however, the benefits of adopting AC are so great that such black sheep of the mathematical family as the paradox are generally tolerated. The Banach-Tarski paradox, which mathematicians often refer to as the Banach-Tarski decomposition because it's really a proof not a paradox, highlights the fact that among the infinite set of points that make up a mathematical ball, the concept of volume and of measure can't be defined for all possible subsets. In any case, the Banach-Tarski paradox doesn't give a prescription for how to produce the subsets: it only proves their existence and that there must be at least five of them to produce a second copy of the original ball. www.daviddarling.info /works/Mathematics/mathematics_samples.html   (5705 words)

PlanetMath: Hausdorff paradox The theorem itself is a cruical ingredient to the proof of the so-called Banach-Tarski paradox. See Also: choice function, Banach-Tarski paradox, proof of Banach-Tarski paradox This is version 4 of Hausdorff paradox, born on 2005-05-15, modified 2005-06-25. planetmath.org /encyclopedia/HausdorffParadox.html   (215 words)

Axiom of Choice Thus, the Banach-Tarski Paradox does not violate the Law of Conservation of Mass; it merely tells us that the notion of "volume" is more complicated than we might have expected. One or more of the sets in the decomposition must be Lebesgue unmeasurable; thus a corollary of the Banach-Tarski Theorem is the fact that there exist sets that are not Lebesgue measurable. Here is an on-line survey article, The Hahn-Banach Theorem: The Life and Times, by Lawrence Narici and Edward Beckenstein. www.math.vanderbilt.edu /~schectex/ccc/choice.html   (3626 words)

Banach–Tarski paradox - Wikipedia, the free encyclopedia The Banach–Tarski "paradox": A ball can be decomposed and reassembled into two balls the same size as the original. Banach and Tarski intended for this proof to demonstrate that the axiom of choice was incorrect, but the nature of the proof is such that most mathematicians take it to mean that the axiom of choice merely results in bizarre and unintuitive consequences. "Sur la décomposition des ensembles de points en parties respectivement congruentes", Fundamenta Mathematicae, 6, (1924), 244-277, Banach and Tarski's original paper (in French). en.wikipedia.org /wiki/Banach-Tarski_Paradox   (1267 words)

Try your search on: Qwika (all wikis)

About us   |   Why use us?   |   Reviews   |   Press   |   Contact us   Copyright © 2005-2007 www.factbites.com Usage implies agreement with terms.