The Banach-Tarski paradox:
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...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York
Cambridge University Press
2016
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Encyclopedia of mathematics and its applications
163 |
| Schlagworte: | |
| Online-Zugang: | BSB01 FHN01 Volltext Inhaltsverzeichnis |
| Zusammenfassung: | 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 |
| Beschreibung: | Title from publisher's bibliographic system (viewed on 06 Jun 2016) |
| Beschreibung: | 1 online resource (xviii, 348 pages) |
| ISBN: | 9781107337145 |
| DOI: | 10.1017/CBO9781107337145 |
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Datensatz im Suchindex
| _version_ | 1804176880257990656 |
|---|---|
| adam_text | Titel: The Banach-Tarski paradox
Autor: Tomkowicz, Grzegorz
Jahr: 2016
Contents
Foreword page xi
Jan Mycielski
Addendum to the Foreword xiii
Preface xv
Part One Paradoxical Decompositions, or the Nonexistence
of Finitely Additive Measures
1 Introduction 3
1.1 Examples of Paradoxical Actions 4
1.2 Geometrical Paradoxes 8
2 The Hausdorff Paradox 14
3 The Banach-Tarski Paradox: Duplicating Spheres and Balls 23
4 Hyperbolic Paradoxes 36
4.1 The Hyperbolic Plane 36
4.2 A Hyperbolic Hausdorff Paradox 38
4.3 A Banach-Tarski Paradox of the Whole Hyperbolic Plane 42
4.4 Paradoxes in an Escher Design 48
4.5 The Disappearing Hyperbolic Squares 51
4.6 A Bounded Hyperbolic Paradox 53
5 Locally Commutative Actions: Minimizing the Number
of Pieces in a Paradoxical Decomposition 62
5.1 A Minimal Decomposition of a Sphere 62
5.2 A Minimal Decomposition of a Solid Ball 66
5.3 General Systems of Congruences 69
6 Higher Dimensions 78
6.1 Euclidean Spaces 78
6.2 Non-Euclidean Spaces 86
6.3 Tetrahedral Chains 88
viii Contents
7 Free Groups of Large Rank: Getting a Continuum of Spheres
from One 93
7.1 Large Free Groups of Isometries 93
7.2 Large Free Semigroups of Isometries 107
7.3 Sets Congruent to Proper Subsets 109
8 Paradoxes in Low Dimensions 116
8.1 Paradoxes in the Plane 116
8.2 Paradoxes of the Real Line 126
9 Squaring the Circle 133
9.1 Changing the Group 133
9.2 The Squaring of the Circle 135
9.3 Generalizations and Open Problems 162
10 The Semigroup of Equidecomposability Types 168
10.1 The Semigroup of Equidecomposability Types 168
10.2 A Cancellation Law 175
10.3 Restrictions on the Pieces 178
Part Two Finitely Additive Measures, or the Nonexistence
of Paradoxical Decompositions
11 Transition 193
11.1 Tarski s Theorem 193
11.2 The Marczewski Problem: A Paradox Using Baire Sets 199
11.3 Equidecomposability with Countably Many Pieces 207
12 Measures in Groups 219
12.1 Amenable Groups 219
12.2 Classes of Groups 223
12.3 Invariant Measures 225
12.4 Characterizations of Amenability 230
12.5 Topological Amenability 243
13 Applications of Amenability 247
13.1 Exotic Measures 247
13.2 Paradoxes modulo an Ideal 257
13.3 How to Eliminate Exotic Measures in R2 259
13.4 Paradoxes Using Measurable Pieces 261
13.5 Characterizing Isometry Groups That Yield Paradoxes 262
14 Growth Conditions in Groups and Supramenability 270
14.1 Supramenable Groups 270
14.2 Bounded Paradoxical Sets 273
14.3 Group Growth 282
14.4 Cogrowth and Amenability 291
Contents ix
15 The Role of the Axiom of Choice 296
15.1 The Axiom of Choice Is Essential 296
15.2 The Axiom of Choice Can Sometimes Be Eliminated 306
15.3 Foundational Implications of the Banach-Tarski Paradox 308
Appendices
A Euclidean Transformation Groups 315
B Jordan Measure 320
C Graph Theory 322
Bibliography 325
List of Symbols 339
Index 343
|
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| spelling | Tomkowicz, Grzegorz Verfasser aut The Banach-Tarski paradox Grzegorz Tomkowicz, Stan Wagon Second edition New York Cambridge University Press 2016 1 online resource (xviii, 348 pages) txt rdacontent c rdamedia cr rdacarrier Encyclopedia of mathematics and its applications 163 Title from publisher's bibliographic system (viewed on 06 Jun 2016) 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 Banach-Tarski paradox Measure theory Decomposition (Mathematics) Maßtheorie (DE-588)4074626-4 gnd rswk-swf Banach-Tarskisches Paradoxon (DE-588)4143975-2 gnd rswk-swf Zerlegung Mathematik (DE-588)4190746-2 gnd rswk-swf Banach-Tarskisches Paradoxon (DE-588)4143975-2 s Maßtheorie (DE-588)4074626-4 s Zerlegung Mathematik (DE-588)4190746-2 s 1\p DE-604 Wagon, S. Sonstige oth Erscheint auch als Druckausgabe 978-1-107-04259-9 Erscheint auch als Druckausgabe 978-1-107-61731-5 https://doi.org/10.1017/CBO9781107337145 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029349131&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Tomkowicz, Grzegorz The Banach-Tarski paradox Banach-Tarski paradox Measure theory Decomposition (Mathematics) Maßtheorie (DE-588)4074626-4 gnd Banach-Tarskisches Paradoxon (DE-588)4143975-2 gnd Zerlegung Mathematik (DE-588)4190746-2 gnd |
| subject_GND | (DE-588)4074626-4 (DE-588)4143975-2 (DE-588)4190746-2 |
| title | The Banach-Tarski paradox |
| title_auth | The Banach-Tarski paradox |
| title_exact_search | The Banach-Tarski paradox |
| title_full | The Banach-Tarski paradox Grzegorz Tomkowicz, Stan Wagon |
| title_fullStr | The Banach-Tarski paradox Grzegorz Tomkowicz, Stan Wagon |
| title_full_unstemmed | The Banach-Tarski paradox Grzegorz Tomkowicz, Stan Wagon |
| title_short | The Banach-Tarski paradox |
| title_sort | the banach tarski paradox |
| topic | Banach-Tarski paradox Measure theory Decomposition (Mathematics) Maßtheorie (DE-588)4074626-4 gnd Banach-Tarskisches Paradoxon (DE-588)4143975-2 gnd Zerlegung Mathematik (DE-588)4190746-2 gnd |
| topic_facet | Banach-Tarski paradox Measure theory Decomposition (Mathematics) Maßtheorie Banach-Tarskisches Paradoxon Zerlegung Mathematik |
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