The Rearrangement Number

TL;DR

The paper introduces the rearrangement number, a new cardinal characteristic of the continuum, measuring how many permutations are needed to alter the convergence of conditionally convergent series, and explores its properties and consistency results.

Contribution

It defines the rearrangement number, compares it with existing cardinal characteristics, and develops forcing constructions to analyze its set-theoretic properties.

Findings

Rearrangement number is a new cardinal characteristic of the continuum.

Forcing constructions demonstrate the consistency of various properties of the rearrangement number.

Comparisons with known cardinal characteristics reveal its relative size and behavior.

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 \emph{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.