Guessing genericity -- looking at parametrized diamonds from a different perspective
Ziemowit Kostana
TL;DR
The work develops a unified, model-theoretic perspective on parametrized diamond principles by introducing invariants $ (A,B,E) $ and associated guessing statements, and by formulating game- and forcing-based frameworks (STR and EL diamonds) that relate forcing, elementary submodels, and genericity. It shows that strategic and elementary diamonds coincide in key contexts and proves preservation results under both ccc and proper forcing, including a measure-algebra forcing that strengthens older results on $\diamondsuit(\mathfrak{d})$. The framework yields concrete combinatorial consequences, such as the existence of Suslin trees from $\diamondsuit_{EL}(\mathfrak{non}(\mathcal{M}),(H_{\omega_1};\in))$, and derives various $\clubsuit$-type principles and P-point constructions via $\diamondsuit_{EL}$-instances. Overall, the paper connects forcing-theoretic, model-theoretic, and combinatorial facets of guessing principles, offering a versatile toolkit for deriving continuum invariants and applications in topology and set theory.
Abstract
We introduce and study a family of axioms that closely follows the pattern of parametrized diamonds, studied by Moore, Hrušák, and Džamonja in [13]. However, our approach appeals to model theoretic / forcing theoretic notions, rather than pure combinatorics. The main goal of the paper is to exhibit a surprising, close connection between seemingly very distinct principles. As an application, we show that forcing with a measure algebra preserves (a variant of) $\diamondsuit(\mathfrak{d})$, improving an old result of M. Hrušák.