On the characterization of geometric logic
Summary
In a letter to Michael Makkai of 1989, Ieke Moerdijk proved a classification theorem for finitary first-order theories which are axiomatized by coherent sequents and he asked for a generalization of his result to infinitary logic for geometric theories.
It was only in 2009 that Olivia Caramello came with a more general characterization theorem which answered Moerdijks question positively.
It states the following:
Let L be a possibly infinitary first-order language and let S be a class of L-structures in Grothendieck topoi, closed under isomorphism of L-structures. Then S is the class of all models in Grothendieck topoi of a geometric L-theory T if and only if the following two conditions are
satisfied:
1. Elements of S are sent to elements of S by all inverse image parts of all geometric morphisms between Grothendieck topoi.
2. Given any set-indexed jointly surjective family of geometric morphisms with codomain E and any L-structure in E, then if all the images of the structure under the inverse image functors are elements of S, then so is the structure itself.
In the presentation of my masterthesis we shall understand and investigate the results of Moerdijk and Caramello and witness the beauty of the underlying topos theory which mixes geometry and logic to perfection. It is my goal to make this theory available to a much wider audience then is currently the case, as it can be a powerful asset in affiliated mathematical theory.