INFINITARY GENERALIZATIONS OF DELIGNE'S COMPLETENESS THEOREM

Given a regular cardinal kappa such that kappa(<kappa) = kappa (or any regular. if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the kappa-separable toposes. These are equivalent to sheaf toposes over a site with kappa-small limits that has at most kappa many objects and morphisms, the (basis for the) topology being generated by at most. many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough kappa-points, that is, points whose inverse image preserve all kappa-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when kappa = omega, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call kappa-geometric, where conjunctions of less than. formulas and existential quantification on less than. many variables is allowed. We prove that kappa-geometric theories have kappa-classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to kappa-geometric morphisms (geometric morphisms the inverse image of which preserves all kappa-small limits) into that topos. Moreover, we prove that kappa-separable toposes occur as the kappa-classifying toposes of kappa-geometric theories of at most. many axioms in canonical form, and that every such kappa-classifying topos is kappa-separable. Finally, we consider the case when. is weakly compact and study the kappa-classifying topos of a kappa-coherent theory (with at most. many axioms), that is, a theory where only disjunction of less than. formulas are allowed, obtaining a version of Deligne's theorem for.-coherent toposes from which we can derive, among other things, Karp's completeness theorem for infinitary classical logic.