The Banach-Tarski Paradox

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Cambridge University Press, Sep 24, 1993 - Mathematics - 253 pages
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This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, and logic. It unifies the results of contemporary research on the paradox and presents several new results including some unusual paradoxes in hyperbolic space. It also provides up to date proofs and discusses many unsolved problems.
  

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This is one of the most enjoyable mathematics books I have seen. The collection of results is both unified and fascinating, and the bibliographic information very helpful.

Contents

Introduction
3
The Hausdorff Paradox
15
The BanachTarski Paradox Duplicating Spheres and Balls
21
Locally Commutative Actions Minimizing the Number of Pieces in a Paradoxical Decomposition
34
Higher Dimensions and NonEuclidean Spaces
52
Free Groups of Large Rank Getting a Continuum of Spheres from One
73
Paradoxes in Low Dimensions
96
The Semigroup of Equidecomposability Types
109
Applications of Amenability Marczewski Measures and Exotic Measures
165
Growth Conditions in Groups and Supramenability
188
The Role of the Axiom of Choice
207
Euclidean Transformation Groups
222
Jordan Measure
227
Unsolved Problems
229
Addendum to Second Printing
234
References
236

Finitely Additive Measures or the Nonexistence of Paradoxical Decompositions
123
Transition
125
Measures in Groups
146
List of Symbols
248
Index
250
Copyright

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About the author (1993)

Wagon is Professor of Mathematics and Computer Science at Macalester College, St. Paul, Minnesota.

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