The subseries number

TL;DR

This paper introduces the subseries number, a new cardinal characteristic of the continuum, measuring how many subsets of natural numbers are needed to ensure divergence of subseries for all conditionally convergent series.

Contribution

It defines the subseries number and its variants, compares them with existing cardinal characteristics, and explores their properties and values in different models of set theory.

Findings

The subseries number is bounded between  and the continuum.

The paper establishes inequalities and relationships with other cardinal characteristics.

The value of the subseries number is computed in the Laver model.

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 $\aleph_1$ 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.