Open Access
2015 Regular Ultrapowers at Regular Cardinals
Juliette Kennedy, Saharon Shelah, Jouko Väänänen
Notre Dame J. Formal Logic 56(3): 417-428 (2015). DOI: 10.1215/00294527-3132788


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.


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Juliette Kennedy. Saharon Shelah. Jouko Väänänen. "Regular Ultrapowers at Regular Cardinals." Notre Dame J. Formal Logic 56 (3) 417 - 428, 2015.


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

zbMATH: 1334.03043
MathSciNet: MR3373611
Digital Object Identifier: 10.1215/00294527-3132788

Primary: 03C20
Secondary: 03E05

Keywords: good ultrafilter , reduced product , regular filter , square principle , ultraproduct

Rights: Copyright © 2015 University of Notre Dame

Vol.56 • No. 3 • 2015
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