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## Fundamenta Mathematicae

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## The subseries number

### Volume 247 / 2019

Fundamenta Mathematicae 247 (2019), 49-85 MSC: 03E17, 03E35, 40A05. DOI: 10.4064/fm667-11-2018 Published online: 15 March 2019

#### Abstract

Every conditionally convergent series of real numbers has a divergent subseries. How many subsets of the natural numbers are needed so that every conditionally convergent series diverges on the subseries corresponding to one of these sets? The answer to this question is defined to be the subseries number, a new cardinal characteristic of the continuum. This cardinal is bounded below by $\aleph _1$ and above by the cardinality of the continuum, but it is not provably equal to either. We define three natural variants of the subseries number, and compare them with each other, with their corresponding rearrangement numbers, and with several well-studied cardinal characteristics of the continuum. Many consistency results are obtained from these comparisons, and we obtain another by computing the value of the subseries number in the Laver model.

#### Authors

• Jörg BrendleGraduate School of System Informatics
Kobe University
657-8501 Kobe, Japan
e-mail
• Will BrianDepartment of Mathematics and Statistics
University of North Carolina at Charlotte
9201 University City Blvd.
Charlotte, NC 28223-0001, U.S.A.
wbrian.math@gmail.com
e-mail
• Joel David HamkinsMathematics
The Graduate Center of the City University of New York
365 Fifth Avenue
New York, NY 10016, U.S.A.
and
Mathematics
College of Staten Island of CUNY
Staten Island, NY 10314, U.S.A.
jdh.hamkins.org
e-mail

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