The Banach–Tarski Paradox: 163 (Encyclopedia of Mathematics and its Applications, Series Number 163)
Cambridge University Press
Edition: 2, 6/14/2016
EAN 9781107042599, ISBN10: 1107042593
Hardcover, 360 pages, 16 x 14.2 x 2.3 cm
Language: English
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.
Part I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures
1. Introduction
2. The Hausdorff paradox
3. The Banach–Tarski paradox
duplicating spheres and balls
4. Hyperbolic paradoxes
5. Locally commutative actions
minimizing the number of pieces in a paradoxical decomposition
6. Higher dimensions
7. Free groups of large rank
getting a continuum of spheres from one
8. Paradoxes in low dimensions
9. Squaring the circle
10. The semigroup of equidecomposability types
Part II
Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions
11. Transition
12. Measures in groups
13. Applications of amenability
14. Growth conditions in groups and supramenability
15. The role of the axiom of choice.