Regular ultrapowers at regular cardinals. (English) Zbl 1334.03043

Kennedy, Juliette; Shelah, Saharon; Väänänen, Jouko

Notre Dame J. Formal Logic 56, No. 3, 417-428 (2015).

Summary: 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 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 [C. C. Chang and H. J. Keisler, Model theory. 3rd rev. ed. Amsterdam etc.: North-Holland (1990; Zbl 0697.03022)].

MSC:

03E05	Other combinatorial set theory
03C55	Set-theoretic model theory
03C20	Ultraproducts and related constructions

Keywords:

regular filter; square principle; good ultrafilter; ultraproduct; reduced product

Citations:

Zbl 0697.03022