More separations of cardinal characteristics of the strong measure zero ideal

TL;DR

The paper advances the study of cardinal characteristics of the strong measure zero ideal $\mathcal{SN}$ by developing a new forcing framework, $\sigma$-$\bar{\rho}$-linked, which helps keep $\mathsf{add}(\mathcal{N})$ small under finite-support iterations. It combines this with uf-extendable matrix iterations to realize sophisticated left-hand side separations in Cichon’s diagram, producing a constellation of Tukey-relations and cardinal equalities such as $\mathsf{add}(\mathcal{N})=\theta_1$, $\mathsf{add}(\mathcal{SN})=\theta_2$, $\mathsf{cov}(\mathcal{N})=\theta_3$, $\mathsf{cov}(\mathcal{SN})=\mathsf{supcov}=\theta_4$, $\mathfrak{b}=\theta_5$, $\mathsf{non}(\mathcal{M})=\theta_6$, and $\mathsf{cov}(\mathcal{M})=\mathsf{non}(\mathcal{SN})=\mathfrak{d}=\mathfrak{c}=\theta_7$, with additional refinements under further cardinals. By refining preservation theory and exploiting anti-localization and Tukey order techniques, the authors provide a robust toolkit for constructing models with prescribed SN-related features. These results contribute to a deeper structural understanding of Cichon’s diagram and demonstrate new ways to separate left-hand side cardinals while controlling $\mathcal{SN}$.

Abstract

Let $\mathcal{N}$ be the $σ$-ideal of the null sets of reals. We introduce a new property of forcing notions that enable control of the additivity of $\mathcal{N}$ after finite support iterations. This is applied to answer some open questions from the work of Brendle, the first author, and Mejía~\cite{BCM2}.

Figures (6)

• Figure 1: Diagram of the cardinal characteristics associated with $\mathcal{I}$. An arrow $\mathfrak x\rightarrow\mathfrak y$ means that (provably in ZFC) $\mathfrak x\le\mathfrak y$.
• Figure 2: An arrow $\mathfrak{x}\rightarrow\mathfrak{y}$ means that (provably in ZFC) $\mathfrak{x}\le\mathfrak{y}$. Moreover, $\mathsf{add}(\mathcal{M})=\min\{\mathfrak{b},\mathsf{non}(\mathcal{SN})\}$ and $\mathsf{cof}(\mathcal{M})=\max\{\mathfrak{d},\mathsf{supcov}\}$. The ideals $\mathcal{I}_f$ and $\mathcal{I}_g$ are illustrated for arbitrary $f$ and $g$ to emphasize that the covering of any Yorioka ideal is an upper bound of the additivities of all Yorioka ideals, likewise for the cofinality and uniformity.
• Figure 3: Constellation forced in \ref{['Thm:a0']}.
• Figure 4: Constellation forced in \ref{['Thm:a2']}.
• Figure 5: Constellation forced in \ref{['Thm:a1']}.

Theorems & Definitions (70)

• Definition 1.1
• Lemma 1.2
• Definition 1.3: Yorioka Yorioka
• Theorem 1: \ref{['d3']}
• Theorem 2: \ref{['appl:I']}
• Theorem 3
• Theorem 4: \ref{['appl:III']}
• Definition 2.1
• Example 2.2
• Definition 2.3