A Paradox Arising from the Elimination of a Paradox

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.

• MSC: Primary 03E25
• Secondary 03E30
• 28A05
• 03E15

Acknowledgments

We are grateful to Andreas Blass for drawing our attention to the division paradox, and to Randall Dougherty, Matt Foreman, Asaf Karagila, Menachem Magidor, Andrew Marks, Greg Moore, and Dan Velleman for much helpful correspondence.

Additional information

Notes on contributors

Alan D. Taylor

ALAN TAYLOR is the Marie Louise Bailey Professor of Mathematics at Union College. He spent the first 15 years of his career and the last 15 years of his career as a set theorist, with the middle portion devoted to fair division and voting theory. His latest book (with C. Hardin) is The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems. In his spare time he enjoys the three Rs: running, renovating (houses), and returning (to Maine). [CrossRef]

Stan Wagon

STAN WAGON is recently retired from Macalester College. His main interest is in using the power of modern software to visualize abstract mathematical concepts. His latest book (with G. Tomkowicz) is a second edition of The Banach–Tarski Paradox. Other interests include ski mountaineering, climbing, mushroom hunting, and playing the piano. He is one of the founding editors of Ultrarunning magazine, but now finds that covering long distances is much easier on skis than in running shoes.