UID:
almafu_9960117364302883
Format:
1 online resource (xviii, 348 pages) :
,
digital, PDF file(s).
Edition:
Second edition.
ISBN:
1-316-57056-8
,
1-316-57287-0
,
1-107-33714-3
Series Statement:
Encyclopedia of mathematics and its applications ; 163
Content:
The Banach-Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theory, and logic. This new edition of a classic book unifies contemporary research on the paradox. It has been updated with many new proofs and results, and discussions of the many problems that remain unsolved. Among the new results presented are several unusual paradoxes in the hyperbolic plane, one of which involves the shapes of Escher's famous 'Angel and Devils' woodcut. A new chapter is devoted to a complete proof of the remarkable result that the circle can be squared using set theory, a problem that had been open for over sixty years.
Note:
Title from publisher's bibliographic system (viewed on 06 Jun 2016).
,
Cover -- Half title -- Series -- Title -- Copyright -- Dedication -- Contents -- Foreword -- Addendum to the Foreword -- Preface -- Part One Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures -- 1 Introduction -- 1.1 Examples of Paradoxical Actions -- 1.2 Geometrical Paradoxes -- 2 The Hausdorff Paradox -- 3 The Banach-Tarski Paradox: Duplicating Spheres and Balls -- 4 Hyperbolic Paradoxes -- 4.1 The Hyperbolic Plane -- 4.2 A Hyperbolic Hausdorff Paradox -- 4.3 A Banach-Tarski Paradox of the Whole Hyperbolic Plane -- 4.4 Paradoxes in an Escher Design -- 4.5 The Disappearing Hyperbolic Squares -- 4.6 A Bounded Hyperbolic Paradox -- 5 Locally Commutative Actions: Minimizing the Number of Pieces in a Paradoxical Decomposition -- 5.1 A Minimal Decomposition of a Sphere -- 5.2 A Minimal Decomposition of a Solid Ball -- 5.3 General Systems of Congruences -- 6 Higher Dimensions -- 6.1 Euclidean Spaces -- 6.2 Non-Euclidean Spaces -- 6.3 Tetrahedral Chains -- 7 Free Groups of Large Rank: Getting a Continuum of Spheres from One -- 7.1 Large Free Groups of Isometries -- 7.2 Large Free Semigroups of Isometries -- 7.3 Sets Congruent to Proper Subsets -- 8 Paradoxes in Low Dimensions -- 8.1 Paradoxes in the Plane -- 8.2 Paradoxes of the Real Line -- 9 Squaring the Circle -- 9.1 Changing the Group -- 9.2 The Squaring of the Circle -- 9.3 Generalizations and Open Problems -- 10 The Semigroup of Equidecomposability Types -- 10.1 The Semigroup of Equidecomposability Types -- 10.2 A Cancellation Law -- 10.3 Restrictions on the Pieces -- Part Two Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions -- 11 Transition -- 11.1 Tarski's Theorem -- 11.2 The Marczewski Problem: A Paradox Using Baire Sets -- 11.3 Equidecomposability with Countably Many Pieces -- 12 Measures in Groups -- 12.1 Amenable Groups.
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12.2 Classes of Groups -- 12.3 Invariant Measures -- 12.4 Characterizations of Amenability -- 12.5 Topological Amenability -- 13 Applications of Amenability -- 13.1 Exotic Measures -- 13.2 Paradoxes modulo an Ideal -- 13.3 How to Eliminate Exotic Measures in R[sup(2)] -- 13.4 Paradoxes Using Measurable Pieces -- 13.5 Characterizing Isometry Groups That Yield Paradoxes -- 14 Growth Conditions in Groups and Supramenability -- 14.1 Supramenable Groups -- 14.2 Bounded Paradoxical Sets -- 14.3 Group Growth -- 14.4 Cogrowth and Amenability -- 15 The Role of the Axiom of Choice -- 15.1 The Axiom of Choice Is Essential -- 15.2 The Axiom of Choice Can Sometimes Be Eliminated -- 15.3 Foundational Implications of the Banach-Tarski Paradox -- Appendices -- A Euclidean Transformation Groups -- B Jordan Measure -- C Graph Theory -- Bibliography -- List of Symbols -- Index.
,
English
Additional Edition:
ISBN 1-107-04259-3
Additional Edition:
ISBN 1-107-61731-6
Language:
English
URL:
https://doi.org/10.1017/CBO9781107337145
