Regular Ultrapowers at Regular Cardinals

Abstract

In earlier work by the first and second authors, the equivalence of a finite square principle □ finλD with various model-theoretic properties of structures of size λ and regular ultrafilters was established. In this paper we investigate the principle □ finλD -and thereby the above model-theoretic properties-at a regular cardinal. By Chang's two-cardinal theorem, □ finλD holds at regular cardinals for all regular filters D if we assume the generalized continuum hypothesis (GCH). In this paper we prove in ZFC that, for certain regular filters that we call doubly+ regular, □finD holds at regular cardinals, with no assumption about GCH. Thus we get new positive answers in ZFC to Open Problems 18 and 19 in Chang and Keisler's book Model Theory.