Forcing with Language Fragments, Extending Namba Forcing, and Models of Theories with Constraints in Interpretation

We develop a forcing framework based on the idea of amalgamating language fragments into a theory with a canonical Henkin model. We then demonstrate the usefulness of this framework by applying it to both the extended Namba problem and 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.