A general theory of iterated forcing using finitely additive measures

TL;DR

This paper develops a comprehensive iterated forcing framework using finitely additive measures, introducing new concepts and generalizations that preserve key set-theoretic properties and lead to novel results in Cichoń's diagram.

Contribution

It introduces the $ ext{FAM}$-linked property and generalizes intersection numbers, extending iterated forcing techniques with applications to Cichoń's diagram and measure theory.

Findings

Preserves strong unbounded families and anti-Bendixson families during iteration.

Does not increase the minimal size of non-measure-zero sets.

Provides a new separation result in Cichoń's diagram with possibly singular $ ext{cov}( ext{N})$.

Abstract

Based on the work of Shelah, Kellner, and T\u{a}nasie (Fund. Math., 166(1-2):109-136, 2000 and Comment. Math. Univ. Carolin., 60(1):61-95, 2019), and the recent developments in the third author's master's thesis, we develop a general theory of iterated forcing using finitely additive measures. For this purpose, we introduce two new notions: on the one hand, we define a new linkedness property, called $\mu$-$\mathrm{FAM}$-linked and, on the other hand, we generalize the notion of intersection number to forcing notions, which justifies the limit steps of our iteration theory. Our theory also generalizes iterations with ultrafilters, which have played an important role in the proof of the consistency of Cicho\'n's maximum. We further show that any iteration constructed with our theory preserves strong unbounded families and what we call anti-Bendixson families, which play a central role in preserving witnesses of $\mathrm{cov}(\mathcal{N})$ of singular size (even of countable cofinality). We also show that our iteration method does not increase $\mathrm{non}(\mathcal{E})$, the smallest size of a set of reals that cannot be covered by an $F_\sigma$ measure zero set. Finally, we apply our theory to prove a new separation of the left-hand side of Cicho\'n's diagram where $\mathrm{cov}(\mathcal{N})$ is possibly singular, even with countable cofinality.