Forcing with Language Fragments, Extending Namba Forcing, and Models of Theories with Constraints in Interpretation
Desmond Lau
TL;DR
The paper develops a forcing framework based on language fragments and a canonical term model, enabling translation of extension requirements into fragment-level constraints and connecting forcing with theories that include constraints in interpretation (TCIs).It introduces a meta-language and a robust apparatus for forcing with language fragments, then demonstrates how this framework yields generic witnesses to sets of $\\mathcal{L}^{*}_{\\mathfrak{A}}$-$\\Pi_2$ sentences, grounding applications to variants of the extended Namba problem and to TCIs.A key contribution is the conditional Nb1 Nb'_1(\\lambda) result, achieved via a side-condition–enhanced construction that, under a precipitous ${\\mathrm{NS}}_{\\omega_1}$ assumption, forces $cof(\\alpha)=\\omega$ for a broad class of regular cardinals and $cof(\\lambda)=\\omega_1$ at a target index.The work culminates by showing how to weave in the Asperó–Schindler construction to realize MM$^{++}$-style outcomes within this modular forcing framework, illustrating the framework’s potential to generate rich generic objects with controlled structural properties.
Abstract
We develop a forcing framework based on the idea of amalgamating language fragments into a theory with a canonical term model. We then demonstrate the usefulness of this framework by applying it to variants of the extended Namba problem, as well as to the analysis of models of certain theories with constraints in interpretation (TCIs). The foundations for a theory of TCIs and their models are laid in parallel to the development of our framework, and are of independent interest.