An analysis of the constructive content of Henkin's proof of G\"odel's completeness theorem

Abstract

G{\"o}del's completeness theorem for classical first-order logic is one of the most basic theorems of logic. Central to any foundational course in logic, it connects the notion of valid formula to the notion of provable formula.We survey a few standard formulations and proofs of the completeness theorem before focusing on the formal description of a slight modification of Henkin's proof within intuitionistic second-order arithmetic.It is standard in the context of the completeness of intuitionistic logic with respect to various semantics such as Kripke or Beth semantics to follow the Curry-Howard correspondence and to interpret the proofs of completeness as programs which turn proofs of validity for these semantics into proofs of derivability.We apply this approach to Henkin's proof to phrase it as a program which transforms any proof of validity with respect to Tarski semantics into a proof of derivability.By doing so, we hope to shed an effective light on the relation between Tarski semantics and syntax: proofs of validity are syntactic objects with which we can compute.