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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The rearrangement number


Authors: Andreas Blass, Jörg Brendle, Will Brian, Joel David Hamkins, Michael Hardy and Paul B. Larson
Journal: Trans. Amer. Math. Soc. 373 (2020), 41-69
MSC (2010): Primary 03E17; Secondary 03E35, 40A05
DOI: https://doi.org/10.1090/tran/7881
Published electronically: October 1, 2019
MathSciNet review: 4042868
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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.


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Additional Information

Andreas Blass
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043
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
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
Email: jhamkins@gc.cuny.edu

Michael Hardy
Affiliation: Department of Mathematics, Hamline University, Saint Paul, Minnesota 55104
Email: drmichaelhardy@gmail.com

Paul B. Larson
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
Email: larsonpb@miamioh.edu

DOI: https://doi.org/10.1090/tran/7881
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 \url{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 \url{https://arxiv.org/abs/1612.07830/v3}.
Article copyright: © Copyright 2019 American Mathematical Society