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On inverse γ-systems and the number of L∞λ-equivalent, non-isomorphic models for λ singular

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel Rutgers University, Hill Ctr-Busch, New Brunswick, New Jersey 08903, USA, E-mail: shelah@math.rutgers.edu
Pauli Väisänen
Affiliation:
Department of Mathematics, P. O. Box 4, 00014, University of Helsinki, Finland, E-mail: pauli.vaisanen@helsinki.fi
Corresponding

Abstract

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model of cardinality λ, let No() denote the number of isomorphism types of models of cardinality λ which are L∞λ-equivalent to . In [7] Shelah considered inverse κ-systems of abelian groups and their certain kind of quotient limits Gr()/ Fact(). In particular Shelah proved in [7, Fact 3.10] that for every cardinal Μ there exists an inverse κ-system such that consists of abelian groups having cardinality at most Μκ and card(Gr()/ Fact()) = Μ. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that θκ < λ for every θ < λ): if is an inverse κ-system of abelian groups having cardinality < λ, then there is a model such that card() = λ and No() = card(Gr()/ Fact()). The following was an immediate consequence (when θκ < λ for every θ < λ): for every nonzero Μ < λ or Μ = λκ there is a model , of cardinality λ with No() = Μ. In this paper we show: for every nonzero Μ ≤ λκ there is an inverse κ-system of abelian groups having cardinality < λ such that card(Gr()/ Fact()) = Μ (under the assumptions 2κ < λ and θ < λ for all θ < λ when Μ > λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No() = λ is possible when θκ > λ for every λ < λ.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2000

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References

[1]Chang, C. C., Some remarks on the model theory ofinfinitary languages, The syntax and semantics of inflnitary languages (Barwise, J., editor), Lecture Notes in Mathematics, no. 72, Springer-Verlag, Berlin, 1968, pp. 3663.CrossRefGoogle Scholar
[2]Palyutin, E. A., Number of models in L ∞,ω theories, II, Algebra i Logika, vol. 16 (1977), no. 4, pp. 443456, English translation in [3].Google Scholar
[3]Palyutin, E. A., Number of models in theories, II, Algebra and Logic, vol. 16 (1977), no. 4, pp. 299309.CrossRefGoogle Scholar
[4]Scott, Dana, Logic with denumerably long formulas and finite strings of quantifiers, Theory of models (Proceedings of the 1963 International Symposium, Berkeley) (Addison, J. W., Henkin, Leon, and Tarski, Alfred, editors), North-Holland, Amsterdam, 1965, pp. 329334.Google Scholar
[5]Shelah, Saharon, On the number of nonisomorphic models of cardinality λ L∞,λ-equivalent to a fixed model, Notre Dame Journal of Formed Logic, vol. 22 (1981), no. 1, pp. 510.CrossRefGoogle Scholar
[6]Shelah, Saharon, On the number of nonisomorphic models in L∞,λ when κ is weakly compact, Notre Dame Journal of Formal Logic, vol. 23 (1982), no. 1, pp. 2126.CrossRefGoogle Scholar
[7]Shelah, Saharon, On the possible number no(M) = the number of nonisomorphic models L∞,λ-equivalent to M of power λ, for λ singular, Notre Dame Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 3650.CrossRefGoogle Scholar
[8]Shelah, Saharon, On the no(M) for M of singular power, Around classification theory of models, Lecture Notes in Mathematics, no. 1182, Springer-Verlag, Berlin, 1986, pp. 120134.CrossRefGoogle Scholar
[9]Shelah, Saharon, The number ofpairwisenon-elementarily-embeddable models, this Journal, vol. 54 (1989), no. 4, pp. 14311455.Google Scholar
[10]Shelah, Saharon, Cardinal arithmetic, Oxford Logic Guides, no. 29, The Clarendon Press Oxford University Press, New York, 1994, Oxford Science Publications.Google Scholar
[11]Shelah, Saharon and VÄisÄnen, Pauli, On the number of L∞,λ-equivalent, non-isomorphic models, to appear in Transactions of the American Mathematical Society.Google Scholar

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