The Descriptive Complexity of Series Rearrangements

Abstract

We consider the descriptive complexity of some subsets of the infinite permutation group \(S_\infty\) which arise naturally from the classical series rearrangement theorems of Riemann, Levy, and Steinitz. In particular, given some fixed conditionally convergent series of vectors in Euclidean space \(\mathbb{R}^d\), we study the set of permutations which make the series diverge, as well as the set of permutations which make the series diverge properly. We show that both collections are \(\boldsymbol\Sigma^0_3\)-complete in \(S_\infty\), regardless of the particular choice of series.