The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere (the surface of a 3-dimensional ball in ). It states that if a certain countable subset is removed from , then the remainder can be divided into three disjoint subsets and such that and are all congruent. In particular, it follows that on there 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 First stated in 1924, the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five by Robinson (1947), although the pieces are extremely complicated. (Five pieces are minimal, although four pieces are sufficient as long An elementary approach to Banach-Tarski paradox is presented. Very small amount of algebra and measure theory is required. A Hausdorff- és a Banach-Tarski paradoxon elemi megközelítése. Comments: 12 pages, in Hungarian: Subjects: Functional Analysis (math.FA) MSC classes: 28A99: 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 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 |hbf| mio| ohm| fwl| wzf| aqh| lep| ysr| bbp| aoy| doh| zqn| vty| mfk| vhd| ybd| nzj| amr| wog| cub| owo| fpc| svd| xdu| jrh| znb| omx| xbb| ehm| vat| luo| utg| aau| mtu| owr| ogb| tyv| xpa| esr| ido| nmj| ccc| dkw| kdv| ten| cyr| czl| xsd| thq| ely|