Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 56, Number 3 (2015), 417-428.
Regular Ultrapowers at Regular Cardinals
In earlier work by the first and second authors, the equivalence of a finite square principle with various model-theoretic properties of structures of size and regular ultrafilters was established. In this paper we investigate the principle —and thereby the above model-theoretic properties—at a regular cardinal. By Chang’s two-cardinal theorem, holds at regular cardinals for all regular filters if we assume the generalized continuum hypothesis (GCH). In this paper we prove in ZFC that, for certain regular filters that we call regular, 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.
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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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