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Published online by Cambridge University Press:  12 March 2014

Juliette Cara Kennedy
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hällströmin Katu 2B), FI-00014, University of Helsinki, Finland, E-mail: juliette.kennedy@helsinki.fi
Saharon Shelah
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem, Israel, E-mail: shelah@math.huji.ac.il
Corresponding

Abstract.

The authors show, by means of a finitary version of the combinatorial principle of [7]. the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal λ. if Mi and Ni are elementarily equivalent models of a language of size ≤ λ, then the second player has a winning strategy in the Ehrenfeucht-Fraïssé game of length λ+ on ΠiMi/D and ΠiNi/D. If in addition 2λ = λ+ and i < λ implies |Mi| + |Ni| ≤ λ+ this means that the ultrapowers are isomorphic. This settles negatively conjecture 18 in [2].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

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References

[1]Ben-David, S., On Shelah's compactness of cardinals, Israel Journal of Mathematics, vol. 31 (1978), pp. 34–56.CrossRefGoogle Scholar
[2]Chang, C. C. and Keisler, J., Model theory, North-Holland, 1990.Google Scholar
[3]Keisler, J., Ultraproducts and saturated models, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings Series A, vol. 67 (1964), pp. 178–186, (= Indagationes Mathematicae. New Series, vol. 26).Google Scholar
[4]Kennedy, J. and Shelah, S., On regular reduced products, this Journal, vol. 67 (2002), pp. 1169–1177.Google Scholar
[5]Kennedy, J. and Shelah, S., On embedding models of arithmetic of cardinality ℵ1 into reduced powers, Fundamenta Mathematicae, vol. 176 (2003), no. 1, pp. 17–24.CrossRefGoogle Scholar
[6]Shelah, S., Every two elementarily equivalent models have isomorphic ultrapowers, Israel Journal of Mathematics, vol. 10 (1971), pp. 224–233.CrossRefGoogle Scholar
[7]Shelah, S., “Gap 1” two-cardinal principles and the omitting types theorem for L(Q), Israel Journal of Mathematics, vol. 65 (1989), no. 2, pp. 133–152.CrossRefGoogle Scholar

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