On the possible values of the Rearrangement Number

Abstract

The rearrangement number $\mathfrak{rr}$ is the least cardinality of a collection of permutations of $ω$ such that every conditionally convergent real series is disrupted by some permutation in the collection. Blass, Brendle, Brian, Hamkins, Hardy, and Larson proved that $\max\{\operatorname{cov}(\mathcal N),\mathfrak b\}\leq\mathfrak{rr}\leq\operatorname{non}(\mathcal M)$ and asked whether $\mathfrak{rr}<\mathrm{non}(\mathcal M)$ is consistent. We prove that $\mathfrak{rr}<\operatorname{non}(\mathcal M)$ is consistent with ZFC. We also prove, in a different forcing extension, that $\max\{\operatorname{cov}(\mathcal N),\mathfrak b\}<\mathfrak{rr}$. We further derive consequences for the subseries number $\mathfrak{s}_{\mathrm{sub}}$ and the splitting number $\mathfrak s$.

Disclosure

“as financed, in part, by the São Paulo Research Foundation (FAPESP), Brazil. Process Number 2025/07302-0. Declaration on the use of generative artificial intelligence. During the research and preparation of this manuscript, the author used OpenAI Codex, powered by GPT-5.6 Sol, as for exploratory proof development, literature discovery, revision, grammar, and LATEX editing. The author critically evaluated and edited all AI-assisted material used to write the manuscript. The author t”

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2026-08-23T19:01:14.728302Z

Structural counts