Determining the Rearrangement Number

Abstract

The rearrangement number $\mathfrak{rr}$ is the least cardinality of a collection of permutations of $\omega$ such that every conditionally convergent real series is disrupted by some permutation in the collection. Blass, Brendle, Brian, Hamkins, Hardy, and Larson proved that $\max \{\mathrm{cov}(\mathcal{N}),\mathfrak{b}\}\le \mathfrak{rr}\le \mathrm{non}(\mathcal{M})$ and asked whether $\mathfrak{rr}=\mathrm{non}(\mathcal{M})$. We prove in ZFC that $\mathfrak{rr}=\mathrm{non}(\mathcal{M})$.

9 pages, 3 figures. New title; substantially rewritten. A serious error in the use of the Laver property invalidated the v1 claim that rr < non(M) is consistent. This version replaces that claim with a ZFC proof of the opposite conclusion: rr = non(M)