The Banach-Tarski Paradox

ISBN-13: 9781107617315 / Angielski / Miękka / 2019 / 366 str.

Grzegorz Tomkowicz; Stan Wagon

Opis

The Banach–Tarski Paradox seems patently false. The authors explain it and its implications in terms appropriate for an undergraduate.

Recenzje

'For some people the book will be over by page 36, because by then one has seen full treatments of the results of Hausdorff and of Banach and Tarski. These people are short-sighted; there is much fascinating mathematics to be learned from the further developments. As the recent result of Marks and Unger shows, there is probably still much to discover. Indeed, the book contains some very interesting questions that still await solution.' Klaas Pieter Hart, Mathematical Reviews

Spis treści

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.