Game sentences and ultrapowers |
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Authors: | Renling JinH Jerome Keisler |
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Institution: | Department of Mathematics, University of California, Berkeley, CA 94720, USA Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA |
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Abstract: | We prove that if
is a model of size at most kappa], λkappa] = 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. |
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Keywords: | |
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