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 ∑𝑛𝑎𝑛 is absolutely convergent if and only if the value of the sum is invariant under all rearrangements ∑𝑛𝑎𝑝⁡(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 ∑𝑛𝑎𝑝⁡(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 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 ∑𝑛𝑎𝑛 remains convergent and with the same sum after all rearrangements ∑𝑛𝑎𝑝⁡(𝑛) by a permutation 𝑝 ∈𝑃, then it is absolutely convergent. The corresponding rearrangement number for sums, denoted 𝔯𝔯Σ, is the smallest cardinality of a family 𝑃 of permutations, such that if a series ∑𝑛𝑎𝑛 is conditionally convergent, then there is a rearrangement ∑𝑛𝑎𝑝⁡(𝑛), by some permutation 𝑝 ∈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 𝔟 ≤𝔯𝔯 ≤𝐧⁢𝐨⁢𝐧⁡(M), that 𝔡 ≤𝔯𝔯Σ, and that 𝔟 <𝔯𝔯 is relatively consistent.

MathOverflow question | CUNY Set Theory Seminar