Notre Dame Journal of Formal Logic

Regular Ultrapowers at Regular Cardinals

Juliette Kennedy, Saharon Shelah, and Jouko Väänänen

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

regular filter square principle good ultrafilter ultraproduct reduced product


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.

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