Journal of Symbolic Logic 67 (3):1169-1177 (2002)

Authors
Juliette Cara Kennedy
University of Helsinki
Abstract
Assume $\langle \aleph_0, \aleph_1 \rangle \rightarrow \langle \lambda, \lambda^+ \rangle$ . Assume M is a model of a first order theory T of cardinality at most λ+ in a language L(T) of cardinality $\leq \lambda$ . Let N be a model with the same language. Let Δ be a set of first order formulas in L(T) and let D be a regular filter on λ. Then M is $\Delta-embeddable$ into the reduced power $N^\lambda/D$ , provided that every $\Delta-existential$ formula true in M is true also in N. We obtain the following corollary: for M as above and D a regular ultrafilter over $\lambda, M^\lambda/D$ is $\lambda^{++}-universal$ . Our second result is as follows: For $i < \mu$ let Mi and Ni be elementarily equivalent models of a language which has cardinality $\leq \lambda$ . Suppose D is a regular filter on λ and $\langle \aleph_0, \aleph_1 \rangle \rightarrow \langle \lambda, \lambda^+ \rangle$ holds. We show that then the second player has a winning strategy in the $Ehrenfeucht-Fra\ddot{i}ss\acute{e}$ game of length λ+ on $\prod_i M_i/D$ and $\prod_i N_i/D$ . This yields the following corollary: Assume GCH and λ regular (or just $\langle \aleph_0, \aleph_1 \rangle \rightarrow \langle \lambda, \lambda^+ \rangle$ and 2λ = λ+). For L, Mi and Ni be as above, if D is a regular filter on λ, then $\prod_i M_i/D \cong \prod_i N_i/D$
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DOI 10.2178/jsl/1190150156
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References found in this work BETA

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
The Fine Structure of the Constructible Hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229.
Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.
Onκ-Complete Reduced Products.Tapani Hyttinen - 1992 - Archive for Mathematical Logic 31 (3):193-199.

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Citations of this work BETA

A Gap 1 Cardinal Transfer Theorem.Luis M. Villegas-Silva - 2006 - Mathematical Logic Quarterly 52 (4):340-350.
Regular Ultrapowers at Regular Cardinals.Juliette Kennedy, Saharon Shelah & Jouko Väänänen - 2015 - Notre Dame Journal of Formal Logic 56 (3):417-428.
Atomic Saturation of Reduced Powers.Saharon Shelah - forthcoming - Mathematical Logic Quarterly.

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