Mechanised Reverse Mathematics in the Calculus of Inductive Constructions

Description du projet

Résoudre le théorème de complétude

Pendant des siècles, les mathématiciens ont cherché à définir les fondements logiques précis qui sous-tendent les vérités mathématiques et leurs applications. Le théorème de complétude de Gödel (un résultat fondamental en logique mathématique), parmi d’autres résultats fondamentaux, constitue un tournant dans cette entreprise. Ses implications informatiques et logiques demeurent toutefois mal comprises. Soutenu par le programme Actions Marie Skłodowska-Curie (MSCA), le projet MeReMath se propose d’exploiter la puissance de la mécanisation informatique et le calcul des constructions inductives pour explorer les mathématiques à rebours. Il entend rigoureusement analyser le théorème de complétude et les théorèmes connexes, ouvrant la voie à une nouvelle bibliothèque Coq susceptible de remodeler la façon dont nous comprenons et appliquons les principes mathématiques fondamentaux.

Objectif

“Foundations of mathematics” labels the centuries-old interdisciplinary vision to secure the logical basis for mathematics and its applications. Gödel's celebrated completeness theorem for first-order logic, identifying semantic truth with syntactic deduction, is a key result heralding the formal phase of that vision. Yet, this and other foundational results have not been fully characterised regarding their logical strength and computational content, limiting their understanding and applicability.

The MeReMath project aims at closing this gap by systematically employing computer mechanisation to the programme of reverse mathematics, the ongoing effort to identify the exact logical principles underlying completeness and other results. For this analysis, the project will use the calculus of inductive constructions (CIC) as a logical base system, embodying an agnostic intuitionistic base system unveiling fine logical structure, and the Coq proof assistant, an interactive software tool for modelling logical reasoning and its computational content.

Specifically, continuing previous research of the experienced researcher (ER) and the supervisor, the MeReMath project will contribute the first comprehensive constructive and computational analysis of the completeness theorem, taking into account all dimensions relevant to its logical strength and implementing modularly mechanised proofs as executable Coq code. By further accommodating a similar analysis of the related Löwenheim-Skolem theorems and other results in the canon, the main outcome of the MeReMath project will be a well-designed, collaboratively developed Coq library for (constructive) reverse mathematics, with novel logical observations and mechanisation techniques developed on the way.

Hosted at the IRIF lab in Paris, the project will be developed in the centre of the original creation of CIC and Coq, providing a world-class environment for the project’s aims and the ER’s academic career prospects.