The Banach Tarski Paradox 2nd Edition by Grzegorz Tomkowicz, Stan Wagon ISBN 1107042593 9781107042599

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ISBN 10: 1107042593 
ISBN 13: 9781107042599
Author: Grzegorz Tomkowicz, Stan Wagon

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.

The Banach Tarski Paradox 2nd Table of contents:

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
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

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Tags: Grzegorz Tomkowicz, Stan Wagon, Banach