# The subseries number

• J. Brendle, W. Brian, and J. D. Hamkins, “The subseries number,” Fund. Math., vol. 247, iss. 1, pp. 49-85, 2019.
@ARTICLE{BrendleBrianHamkins2019:The-subseries-number,
AUTHOR = {Brendle, J\"{o}rg and Brian, Will and Hamkins, Joel David},
TITLE = {The subseries number},
JOURNAL = {Fund. Math.},
FJOURNAL = {Fundamenta Mathematicae},
VOLUME = {247},
YEAR = {2019},
NUMBER = {1},
PAGES = {49--85},
ISSN = {0016-2736},
MRCLASS = {03E17 (03E35 40A05)},
MRNUMBER = {3984279},
DOI = {10.4064/fm667-11-2018},
url = {http://jdh.hamkins.org/the-subseries-number},
eprint = {1801.06206},
archivePrefix = {arXiv},
primaryClass = {math.LO},
}

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.

This paper grew naturally out of our previous paper, The rearrangement number, which considered the minimal number of permutations of $\mathbb{N}$ which suffice to reveal the conditional convergence of all conditionally convergent series. I had defined the subseries number in my answer to a MathOverflow question, On Hamkins’s answer to a question of Michael Hardy’s, asked by M. Rahman in response to the earlier MO questions on the rearrangement number.

In the paper, we situation the subseries number  (German sharp s) with respect to other cardinal characteristics, including the rearrangement numbers.