Forcing Completeness: A Model-Theoretic Recasting of First-Order Deduction

This paper explores a novel perspective on the completeness of first-order deduction by recasting it through a model-theoretic lens, drawing inspiration from the forcing technique in set theory. We propose a framework where the existence of models for consistent first-order theories is established via a "forcing-like" construction. Rather than relying on traditional Henkin constructions or direct proof-theoretic methods, our approach defines a notion of "forcing condition" for first-order logic sentences and uses these conditions to build "generic" models. We demonstrate how this method yields a proof of Gödel's completeness theorem, providing an alternative conceptual and formal understanding of the fundamental link between syntax and semantics. The implications of this recasting are discussed, including potential avenues for exploring non-classical logics, enriching the foundations of automated reasoning, and fostering new insights into the interplay between proof theory and model theory.