Syntax and Semantics of the logic L_omega omega^lambda

Abstract

In this paper we study the logic L_omega omega^lambda , which is first order logic<br /> extended by quantification over functions (but not over relations).<br > We give the syntax of the logic, as well as the semantics in Heyting<br /> categories with exponentials. Embedding the generic model of a theory<br /> into a Grothendieck topos yields completeness of L_omega omega^lambda with respect<br /> to models in Grothendieck toposes, which can be sharpened to completeness<br />with respect to Heyting valued models. The logic L_omega omega^lambda is the<br />strongest for which Heyting valued completeness is known. Finally,<br />we relate the logic to locally connected geometric morphisms between toposes.