Game sentences and ultrapowers

Annals of Pure and Applied Logic 60 (3):261-274 (1993)
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Abstract

We prove that if is a model of size at most [kappa], λ[kappa] = λ, and a game sentence of length 2λ is true in a 2λ-saturated model ≡ , then player has a winning strategy for a related game in some ultrapower ΠD of . The moves in the new game are taken in the cartesian power λA, and the ultrafilter D over λ must be chosen after the game is played. By taking advantage of the expressive power of game sentences, we obtain several applications showing the existence of ultrapowers with certain properties. In each case, it was previously known that such ultrapowers exist under the assumption of the GCH, and we get them without the GCH. Article O

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Citations of this work

The strength of the isomorphism property.Renling Jin & Saharon Shelah - 1994 - Journal of Symbolic Logic 59 (1):292-301.
Type two cuts, bad cuts and very bad cuts.Renling Jin - 1997 - Journal of Symbolic Logic 62 (4):1241-1252.

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References found in this work

Some remarks on infinitely long formulas.L. Henkin - 1961 - Journal of Symbolic Logic 30 (1):167--183.
Some applications of infinitely long formulas.H. Jerome Keisler - 1965 - Journal of Symbolic Logic 30 (3):339-349.
Countable ultraproducts without CH.Michael Canjar - 1988 - Annals of Pure and Applied Logic 37 (1):1-79.
Ultraproducts and Elementary Classes.H. Jerome Keisler - 1962 - Journal of Symbolic Logic 27 (3):357-358.

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