On the possible values of the Rearrangement Number

2026/07/21 by Vinicius de Oliveira Rodrigues
#math.LO #math.CO

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Abstract

The rearrangement number \mathfrakrr 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\cov(\mathcal N),\mathfrak b\≤\mathfrakrr\leqnon(\mathcal M) and asked whether \mathfrakrr<non(\mathcal M) is consistent. We prove that \mathfrakrr<non(\mathcal M) is consistent with ZFC. We also prove, in a different forcing extension, that max\cov(\mathcal N),\mathfrak b\<\mathfrakrr. We further derive consequences for the subseries number \mathfraks_sub and the splitting number \mathfrak s.

Related

• The Rearrangement Number
• Density cardinals
• The subseries number
• On Riemann's Theorem About Conditionally Convergent Series
• Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces