A general theory of iterated forcing using finitely additive measures

Abstract

Based on the work of Shelah, Kellner, and Tănasie (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 $μ$-$\mathrm{FAM}$-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 Cichoń'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 $F_σ$ measure zero set. Finally, we apply our theory to prove a new separation of the left-hand side of Cichoń's diagram where $\mathrm{cov}(\mathcal{N})$ is possibly singular, even with countable cofinality.