# The rearrangement number

• A. Blass, J. Brendle, W. Brian, J. D. Hamkins, M. Hardy, and P. B. Larson, “The rearrangement number,” Trans. Amer. Math. Soc., vol. 373, iss. 1, pp. 41-69, 2020.
@ARTICLE{BlassBrendleBrianHamkinsHardyLarson2020:TheRearrangementNumber,
author = {Andreas Blass and Jörg Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. Larson},
title = {The rearrangement number},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {373},
YEAR = {2020},
NUMBER = {1},
PAGES = {41--69},
ISSN = {0002-9947},
MRCLASS = {03E17 (03E35 40A05)},
MRNUMBER = {4042868},
DOI = {10.1090/tran/7881},
note = {},
url = {http://jdh.hamkins.org/the-rearrangement-number},
eprint = {1612.07830},
archivePrefix = {arXiv},
primaryClass = {math.LO},
abstract = {},
keywords = {},
source = {},
}

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 show that the minimum number of permutations needed for this purpose, which we call the rearrangement number, is uncountable, but whether it equals the cardinal of the continuum is independent of the usual axioms of
set theory. 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 permutations and with rearranging some divergent series to become (conditionally) convergent.

This project started with Michael Hardy’s question on MathOverflow, How many rearrangements must fail to alter the value of a sum before you conclude that none do? I had proposed in my answer that we should think of the cardinal in question as a cardinal characteristic of the continuum, the rearrangement number, since we could prove that it was uncountable and that it was the continuum under MA, and had begun to separate it from other familiar cardinal characteristics. Eventually, the research effort grew into the collaboration of this paper. What a lot of fun!

Here are the lecture notes for an introductory talk on the topic I had given at the Vassar College Mathematics Colloquium