The rearrangement number
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- by Andreas Blass, Jörg Brendle, Will Brian, Joel David Hamkins, Michael Hardy and Paul B. Larson PDF
- Trans. Amer. Math. Soc. 373 (2020), 41-69 Request permission
Abstract:
How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the rearrangement number, a new cardinal characteristic of the continuum, as the answer to this question. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied cardinal characteristics of the continuum. We present some new forcing constructions designed to add permutations that rearrange series from the ground model in particular ways, thereby obtaining consistency results going beyond those that follow from comparisons with familiar cardinal characteristics. Finally, we deal briefly with some variants concerning rearrangements by a special sort of permutation and with rearranging some divergent series to become (conditionally) convergent.References
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Additional Information
- Andreas Blass
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043
- MR Author ID: 37805
- Email: ablass@umich.edu
- Jörg Brendle
- Affiliation: Graduate School of System Informatics, Kobe University, 1–1 Rokkodai, Nada-ku, 657-8501 Kobe, Japan
- Email: brendle@kobe-u.ac.jp
- Will Brian
- Affiliation: Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223
- MR Author ID: 1003948
- Email: wbrian.math@gmail.com
- Joel David Hamkins
- Affiliation: Department of Mathematics, The Graduate Center of the City University of New York, 365 Fifth Avenue, New York, New York 10016; and Department of Mathematics, College of Staten Island of CUNY, Staten Island, New York 10314
- MR Author ID: 347679
- Email: jhamkins@gc.cuny.edu
- Michael Hardy
- Affiliation: Department of Mathematics, Hamline University, Saint Paul, Minnesota 55104
- MR Author ID: 702080
- Email: drmichaelhardy@gmail.com
- Paul B. Larson
- Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
- MR Author ID: 646854
- Email: larsonpb@miamioh.edu
- Received by editor(s): November 22, 2016
- Received by editor(s) in revised form: May 10, 2018, and March 20, 2019
- Published electronically: October 1, 2019
- Additional Notes: The research of the second author was partially supported by Grant-in-Aid for Scientific Research (C) 15K04977, Japan Society for the Promotion of Science.
The research of the fourth author was supported in part by Simons Foundation grant 209252.
The research of the sixth author was supported in part by NSF grant DMS-1201494.
Commentary concerning this article can be made on the fourth author’s blog at http://jdh.hamkins.org/the-rearrangment-number. An extended version of the paper, designed to make the first four sections more accessible to students at the graduate level and those who are not experts in set theory, can be accessed at https://arxiv.org/abs/1612.07830/v3. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 41-69
- MSC (2010): Primary 03E17; Secondary 03E35, 40A05
- DOI: https://doi.org/10.1090/tran/7881
- MathSciNet review: 4042868