The Hahn–Banach theorem and a six-piece paradoxical decomposition of a ball
HTML articles powered by AMS MathViewer
- by Haruka Sato and Masahiro Shioya PDF
- Proc. Amer. Math. Soc. 150 (2022), 365-369 Request permission
Abstract:
The Banach–Tarski paradox holds using six pieces in Zermelo–Fraenkel set theory with the Hahn–Banach theorem.References
- S. Banach and A. Tarski, Sur la décomposition des ensembles de points en parties respectivement congruentes, Fund. Math. 6 (1924), 244–277.
- F. Hausdorff, Bemerkung über den Inhalt von Punktmengen, Math. Ann. 75 (1914), no. 3, 428–433 (German). MR 1511802, DOI 10.1007/BF01563735
- Sabine Koppelberg, Handbook of Boolean algebras. Vol. 1, North-Holland Publishing Co., Amsterdam, 1989. Edited by J. Donald Monk and Robert Bonnet. MR 991565
- W. A. J. Luxemburg, Reduced powers of the real number system and equivalents of the Hahn-Banach extension theorem, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 123–137. MR 0237327
- J. von Neumann, Zur allgemeinen Theorie des Masses, Fund. Math. 13 (1929), 73–116.
- Janusz Pawlikowski, The Hahn-Banach theorem implies the Banach-Tarski paradox, Fund. Math. 138 (1991), no. 1, 21–22. MR 1122274, DOI 10.4064/fm-138-1-21-22
- Raphael M. Robinson, On the decomposition of spheres, Fund. Math. 34 (1947), 246–260. MR 26093, DOI 10.4064/fm-34-1-246-260
- Waclaw Sierpinski, Sur le paradoxe de MM. Banach et Tarski, Fund. Math. 33 (1945), 229–234 (French). MR 16441, DOI 10.4064/fm-33-1-229-234
- Grzegorz Tomkowicz and Stan Wagon, The Banach-Tarski paradox, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 163, Cambridge University Press, New York, 2016. With a foreword by Jan Mycielski. MR 3616119
Additional Information
- Haruka Sato
- Email: harupie_529@yahoo.co.jp
- Masahiro Shioya
- Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan
- MR Author ID: 336423
- Email: shioya@math.tsukuba.ac.jp
- Received by editor(s): September 14, 2020
- Received by editor(s) in revised form: December 30, 2020
- Published electronically: October 25, 2021
- Communicated by: Heike Mildenberger
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 365-369
- MSC (2020): Primary 03E25, 46A22
- DOI: https://doi.org/10.1090/proc/15550
- MathSciNet review: 4335883