## Notre Dame Journal of Formal Logic

### Regular Ultrapowers at Regular Cardinals

#### Abstract

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

#### Article information

Source
Notre Dame J. Formal Logic, Volume 56, Number 3 (2015), 417-428.

Dates
Accepted: 7 February 2013
First available in Project Euclid: 22 July 2015

https://projecteuclid.org/euclid.ndjfl/1437570853

Digital Object Identifier
doi:10.1215/00294527-3132788

Mathematical Reviews number (MathSciNet)
MR3373611

Zentralblatt MATH identifier
1334.03043

Subjects
Primary: 03C20: Ultraproducts and related constructions
Secondary: 03E05: Other combinatorial set theory

#### Citation

Kennedy, Juliette; Shelah, Saharon; Väänänen, Jouko. Regular Ultrapowers at Regular Cardinals. Notre Dame J. Formal Logic 56 (2015), no. 3, 417--428. doi:10.1215/00294527-3132788. https://projecteuclid.org/euclid.ndjfl/1437570853

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