Syntax and Semantics of the Logic Lλωω

In this paper we study the logic Lωω, which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of Lωω with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting valued models. The logic Lωω is the strongest for which Heyting valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes. In this paper we study aspects of completeness of the logic Lωω, which is intuitionistic first order logic extended by quantification over functions. This logic may be seen as well as λ–calculus enriched with first order logic. The details of the syntax are given in section 1. The logic Lωω is of interest for many reasons: it is reasonably powerful and (therefore) incomplete with respect to models in Sets. But the logic Lωω is complete with respect to Heyting valued models. In fact, the infinitary variants Lκω are the strongest logics we know that are complete with respect to Heyting valued models. Secondly, the logic Lωω characterises a class of geometric morphisms between Grothendieck toposes, namely the locally connected ones: We show that a geometric morphisms f :F → E between ∗BRICS, Basic Research in Computer Science, Centre of the Danish National Research Foundation, Computer Science Department, Aarhus University, Ny Munkegade, DK–8000 Århus C, Denmark, [email protected].