The Rearrangement Number

• Published in 2016
• Added on 2018-05-16

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.

Links

• http://arxiv.org/abs/1612.07830v1
• http://arxiv.org/pdf/1612.07830v1

Other information

BibTeX entry

@article{TheRearrangementNumber,
	key = {TheRearrangementNumber},
	type = {article},
	title = {The Rearrangement Number},
	author = {Andreas Blass and J{\"{o}}rg Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. Larson},
	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.},
	comment = {},
	date_added = {2018-05-16},
	date_published = {2016-10-09},
	urls = {http://arxiv.org/abs/1612.07830v1,http://arxiv.org/pdf/1612.07830v1},
	collections = {Combinatorics},
	url = {http://arxiv.org/abs/1612.07830v1 http://arxiv.org/pdf/1612.07830v1},
	year = 2016,
	urldate = {2018-05-16},
	archivePrefix = {arXiv},
	eprint = {1612.07830},
	primaryClass = {math.LO}
}