Regular Ultrapowers at Regular Cardinals

Abstract

In earlier work by the first and second authors, the equivalence of a finite square principle ${\square }_{\lambda ,D}^{\mathrm{fin}}$ with various model-theoretic properties of structures of size $\lambda$ and regular ultrafilters was established. In this paper we investigate the principle ${\square }_{\lambda ,D}^{\mathrm{fin}}$—and thereby the above model-theoretic properties—at a regular cardinal. By Chang’s two-cardinal theorem, ${\square }_{\lambda ,D}^{\mathrm{fin}}$ 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 ${\mathit{doubly}}^{+}$ regular, ${\square }_{\lambda ,D}^{\mathrm{fin}}$ 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.