A Paradox Arising from the Elimination of a Paradox. Issue 4 (21st April 2019)

• Record Type: Journal Article
• Authors: Taylor, Alan D.; Wagon, Stan
• Is Part Of: American Mathematical Monthly. Volume 126:Issue 4(2019)
• Page Start: 306
• Page End: 318
• Publication Date: 2019-04-21
• Subjects: MSC: Primary 03E25 -- Secondary 03E30 -- 28A05 -- 03E15; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2019.1559416
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: We present a result of Mycielski and Sierpiński—remarkable and underappreciated in our view—showing that the natural way of eliminating the Banach–Tarski paradox by assuming all sets of reals to be Lebesgue measurable leads to another paradox about division of sets that is just as unsettling as the paradox being eliminated. The division paradox asserts that the reals can be divided into nonempty classes so that there are strictly more classes than there are reals.