Game sentences and ultrapowers

Abstract

We prove that if $\text{A}$ is a model of size at most [kappa], λ^[kappa] = λ, and a game sentence of length 2^λ is true in a 2^λ-saturated model $\text{M}$ ≡ $\text{A}$, then player ∃ has a winning strategy for a related game in some ultrapower Π_D $\text{A}$ of $\text{A}$. 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.