Separating cardinal characteristics of the strong measure zero ideal

TL;DR

This paper develops forcing techniques to separate the four cardinal characteristics of the strong measure zero ideal, demonstrating their pairwise inequalities, and combines this with Cichoń's maximum in a single model.

Contribution

It introduces general properties of forcing notions to control the additivity of the strong measure zero ideal and constructs a model where all four associated cardinal characteristics are pairwise different.

Findings

The four cardinal characteristics of the strong measure zero ideal are pairwise different.

A forcing extension satisfying both the separation of these characteristics and Cichoń's maximum is constructed.

The paper provides new forcing methods to manipulate the additivity of strong measure zero sets.

Abstract

Let $\mathcal{SN}$ be the $\sigma$-ideal of the strong measure zero sets of reals. We present general properties of forcing notions that allow to control of the additivity of $\mathcal{SN}$ after finite support iterations. This is applied to force that the four cardinal characteristics associated with $\mathcal{SN}$ are pairwise different: \[\mathrm{add}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN}).\] Furthermore, we construct a forcing extension satisfying the above and Cicho\'n's maximum (i.e.\ that the non-dependent values in Cicho\'n's diagram are pairwise different).