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A downward Löwenheim-Skolem theorem for infinitary theories which have the unsuperstability property

  • Rami Grossberg (a1)

Abstract

We present a downward Löwenheim-Skolem theorem which transfers downward formulas from L∞,ω to , ω. The simplest instance is:

Theorem 1. Let λ > κ be infinite cardinals, and let L be a similarity type of cardinality κ at most. For every L-structure M of cardinality λ and every XM there exists a model N ≺ M containing the set X of powerX∣ · κ such that for every pair of finite sequences a, b ∈ N

The following theorem is an application:

Theorem 2. Let λ<κ, T, ω, and suppose χ is a Ramsey cardinal greater than λ. If T has the (χ, , ω)-unsuperstability property, then T has the (χ, , ω)-unsuperstability property.

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[Ch] Chang, C. C., Some remarks on the model theory of infinitary languages, The syntax and semantics of infinitary languages (Barwise, J., editor), Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin, 1968, pp. 3663.
[Di] Dickmann, M. A., larger infinitary languages, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 317363.
[GS] Gale, D. and Stewart, F. M., Infinite games with perfect information, Contributions to the theory of games. II (Kuhn, H. W. and Tucker, A. W., editors), Princeton University Press, Princeton, New Jersey, 1953, pp. 245266.
[GrSh] Grossberg, R. and Shelah, S., A non structure theorem for an infinitary theory which has the unsuperstability property, Illinois Journal of Mathematics, vol. 30 (1986), pp. 364390.
[He] Henkin, L., Some remarks on infinitely long formulas, Infinitistic methods: proceedings of the symposium on foundations of mathematics, PWN, Warsaw, and Pergamon Press, Oxford, 1961, pp. 167183.
[Je] Jech, T., Set theory, Academic Press, New York, 1978.
[Kar] Karp, C., Finite quantifier equivalence, The theory of models (Addison, J. W.et al., editors), North-Holland, Amsterdam, 1965, pp. 405412.
[Kel] Keisler, H. J., Formulas with linearly ordered quantifiers, The syntax and semantics of infinitary languages (Barwise, J., editor), Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin, 1968, pp. 96130.
[Ke2] Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.
[Ko] Kolaitis, PH. G., Game quantification, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, 1985, pp. 365421.
[Ma] Malitz, J., Problems in the model theory of infinite languages, Ph.D. thesis, University of California, Berkeley, California, 1966.
[Sh1] Shelah, S., On the number of non-almost isomorphic models of Tin a power, Pacific Journal Mathematics, vol. 36 (1971), pp. 811818.
[Sh2] Shelah, S., A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific Journal of Mathematics, vol. 41 (1972), pp. 247267.
[Sh3] Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.

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