[Encyclopedia of Mathematics and its Applications 163] • The Banach–Tarski Paradox by Tomkowicz, Grzegorz -- Read -- Imperial Library of Trantor

Index

Cover Half title Series Title Copyright Dedication Table of 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

5.1 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 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

Appendices

A Euclidean Transformation Groups B Jordan Measure C Graph Theory

Bibliography List of Symbols Index