Model-Theoretic Forcing for Impredicative Type Theories

This paper develops a framework for model-theoretic forcing in the context of impredicative type theories, such as the Calculus of Constructions. Drawing inspiration from Cohen's set-theoretic forcing and its categorical generalizations in topos theory, we adapt these techniques to the semantics of higher-order constructive logic. The methodology involves defining forcing notions as internal partial orders or locales within a suitable base model, typically a realizability topos or a category of assemblies. The generic extension is constructed as a category of sheaves over this forcing notion. We establish the fundamental soundness theorem for this construction, ensuring that the generic model validates the rules of the type theory. The primary result of this work is an application of the framework to demonstrate a new independence result. Specifically, we construct a forcing notion that adds a 'generic' object violating a weak form of Markov's Principle, thereby proving its independence from the Calculus of Constructions. This demonstrates the power of forcing as a tool for fine-grained metatheoretic analysis of impredicative systems, opening new avenues for exploring the consistency and relative strength of various type-theoretic axioms without resorting to purely syntactic methods.