Abstract.
We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of L κ ω and L κ κ : For weakly compact κ there is no strongest extension of L κ ω with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of L κ κ with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to <κ.
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References
Barwise, J.: Axioms for abstract model theory. Ann. Math. Logic 7, 221–265 (1974)
Barwise, J., Feferman, S.: Model-theoretic logics. Perspectives in Mathematical Logic, Springer-Verlag, New York, 1985, xviii+893
Fuhrken, G.: Skolem-type normal forms for first-order languages with a generalized quantifier. Fund. Math. 54, 291–302 (1964)
Lindström, P.: On extensions of elementary logic. Theoria 35, 1–11 (1969)
Lipparini, P.: Limit ultraproducts and abstract logics. J. Symb. Logic 52, 437–454 (1987)
Luosto, K.: Filters in abstract model theory. Ph.D. Thesis, University of Helsinki, 1992, 81 p
Makowsky, J., Shelah, S.: The theorems of Beth and Craig in abstract model theory. II. Compact logics, Archiv für Mathematische Logik und Grundlagenforschung 21, 13–35 (1981)
Mostowski, A.: On a generalization of quantifiers. Fund. Math. 44, 12–36 (1957)
Shelah, S.: Generalized quantifiers and compact logic. Trans. Am. Math. Soc. 204, 342–364 (1975)
Shelah, S.: Models with second-order properties. I. Boolean algebras with no definable automorphisms. Ann. Math. Logic, Ann. Math. Logic 14 (1), 57–72 (1978)
Wacławek, M.: On ordering of the family of logics with Skolem-Löwenheim property and countable compactness property. In: Quantifiers: Logics, Models and Computation, Vol. 2, Michał Krynicki, Marcin Mostowski, Lesław Szczerba (eds.), Kluwer Academic Publishers, Dordrecht, Boston, London, 1995, pp. 229–236
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We are indebted to Lauri Hella, Tapani Hyttinen and Kerkko Luosto for useful suggestions.
Research partially supported by the United States-Israel Binational Science Foundation. Publication number [ShVa:726]
Research partially supported by grant 40734 of the Academy of Finland.
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Shelah, S., Väänänen, J. A note on extensions of infinitary logic. Arch. Math. Logic 44, 63–69 (2005). https://doi.org/10.1007/s00153-004-0212-8
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DOI: https://doi.org/10.1007/s00153-004-0212-8