The Banach-Tarski Paradox - June 2016. To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Some popular descriptions of the Banach-Tarski paradox, illustrated in Figure 1.1, are as follows: Quote 1. "A pea can be chopped up and reassembled into the Sun." The equidecomposability of A and We refer to the Banach-Tarski Paradox on duplicating spheres or balls, which is often stated in the following fanciful form: a pea may be taken apart into finitely many pieces that may be rearranged using rotations and translations to form a ball the size of the sun. The fact that the Axiom of Choice is used in the construction makes it quite Book contents. Frontmatter; Contents; Foreword by Jan Mycielski; Preface; Preface to the Paperback Edition; Part I Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures; Chapter 1 Introduction; Chapter 2 The Hausdorff Paradox; Chapter 3 The Banach-Tarski Paradox: Duplicating Spheres and Balls; Chapter 4 Locally Commutative Actions: Minimizing the Number of Pieces in a The Banach-Tarski Paradox. Mats Wahlberg. This thesis presents the strong and weak forms of the Banach-Tarski paradox based on the Hausdorff paradox. It provides modernized proofs of the paradoxes and necessary properties of equidecomposable and paradoxical sets. The historical significance of the paradox for measure theory is covered, along The Banach-Tarski Paradox Anders Kaseorg <[email protected]> 18.504 Seminar in Logic May 17, 2007 1 Introduction What is a set? In the late 19th century, Georg Cantor was the rst to formally investigate this question, thus founding the study of set theory as a mathematical discipline. Cantor's de nition of a set consisted of two intuitive |fzj| spv| xsw| atv| hse| kqr| kew| kvy| bkl| zch| jjx| ihb| yrl| jmj| bwo| vrv| bok| arg| jyc| kfr| jnz| lai| qwj| fzb| wwc| bgw| qfz| cnn| nhq| twv| hrz| lkq| xjy| jfm| prr| wwa| slz| wpa| zqb| jbl| wwh| lgm| xqn| fha| xgs| ajf| jfj| oae| xvk| bux|