Scheme representation for first-order logic

Spencer Breiner

Published 2014-02-11Version 1

Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes, geometric entities which relate to first-order logical theories in much the same way that algebraic schemes relate to commutative rings. The construction relies on a Grothendieck-style representation theorem which associates every coherent or classical first-order theory with an affine scheme: a topological groupoid (the spectrum of the theory) together with a sheaf of (local) syntactic categories. The groupoid is constructed from the semantics of the theory (models and isomorphisms) and topologized using a Stone-type construction. The sheaf of categories can be regarded as a logical theory varying over the spectrum, and its global sections recover the theory up to semantic equivalence. These affine pieces can be glued together to give more general logical schemes and these are studied using methods from algebraic geometry. The final chapter also presents some connections between schemes and other areas of logic such as model theory, type theory and topos theory.