The subseries number

Abstract

Every conditionally convergent series of real numbers has a divergent subseries. How many subsets of the natural numbers are needed so that every conditionally convergent series diverges on the subseries corresponding to one of these sets? The answer to this question is defined to be the subseries number, a new cardinal characteristic of the continuum. This cardinal is bounded below by ℵ 1 and above by the cardinality of the continuum, but it is not provably equal to either. We define three natural variants of the subseries number, and compare them with each other, with their corresponding rearrangement numbers, and with several well-studied cardinal characteristics of the continuum. Many consistency results are obtained from these comparisons, and we obtain another by computing the value of the subseries number in the Laver model.

Three conditionally convergent series, but not four

“…These two pictures represent different subsets of Fn (3,2). But these two sets of functions are equivalent (in the sense defined above), precisely because their pictures can be changed from one to the other simply by relabelling.…”

“…The idea here is that a member of Fn (3,2) can represent the effect of a tame set on our three series. More precisely:…”

“…Definition 2.9. Two sets F, G ⊆ Fn (3,2) are equivalent if there is a permutation π of {1, 2, 3} and permutations τ 1 , τ 2 , and τ 3 of {p, n} such that…”

“…The intuition here is that two subsets of Fn (3,2) are equivalent if they have the same 'shape.' For example, consider the following sets of functions:…”

“…Thus, the main theorem cannot be improved by replacing 'three with 'four. ' The ideas in this paper emerged from set-theoretic investigations into cardinal characteristics of the continuum in [2,3]. In the course of these investigations, the question arose: How small can a collection C of conditionally convergent series be with the property that every A ⊆ N fails to send some member of C to infinity?…”