This will be a talk for the CUNY Set Theory Seminar on November 6, 2015.

The Riemann rearrangement theorem states that a convergent real series $\sum _n{a}_n$ is absolutely convergent if and only if the value of the sum is invariant under all rearrangements $\sum _n{a}_{p(n)}$ by any permutation $p$ on the natural numbers; furthermore, if the series is merely conditionally convergent, then one may find rearrangements for which the new sum $\sum _n{a}_{p(n)}$ has any desired (extended) real value or which becomes non-convergent.  In recent joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson, based on an exchange in reply to a Hardy’s MathOverflow question on the topic, we investigate the minimal size of a family of permutations that can be used in this manner to test an arbitrary convergent series for absolute convergence.

Specifically, we define the rearrangement number $\mathfrak{rr}$ (“double-r”), a new cardinal characteristic of the continuum, to be the smallest cardinality of a set $P$ of permutations of the natural numbers, such that if a convergent real series $\sum _n{a}_n$ remains convergent and with the same sum after all rearrangements $\sum _n{a}_{p(n)}$ by a permutation $p\in P$, then it is absolutely convergent. The corresponding rearrangement number for sums, denoted ${\mathfrak{rr}}_{\mathrm{\Sigma }}$, is the smallest cardinality of a family $P$ of permutations, such that if a series $\sum _n{a}_n$ is conditionally convergent, then there is a rearrangement $\sum _n{a}_{p(n)}$, by some permutation $p\in P$, which converges to a different sum. We investigate the basic properties of these numbers, and explore their relations with other cardinal characteristics of the continuum. Our main results are that $\mathfrak{b}\le \mathfrak{rr}\le \mathbf{n}\mathbf{o}\mathbf{n}(\mathcal{M})$, that $\mathfrak{d}\le {\mathfrak{rr}}_{\mathrm{\Sigma }}$, and that $\mathfrak{b}<\mathfrak{rr}$ is relatively consistent.

MathOverflow question | CUNY Set Theory Seminar