Notre Dame Journal of Formal Logic

Regular Ultrapowers at Regular Cardinals

Juliette Kennedy, Saharon Shelah, and Jouko Väänänen

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Abstract

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

Article information

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

Dates
Received: 18 July 2012
Accepted: 7 February 2013
First available in Project Euclid: 22 July 2015

Permanent link to this document
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

Keywords
regular filter square principle good ultrafilter ultraproduct reduced product

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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References

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