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The Beth-closure of (Qα) is not finitely generated

  • Lauri Hella (a1) and Kerkko Luosto (a2)
Abstract

We prove that if ℵα is uncountable and regular, then the Beth-closure of ωω(Qα) is not a sublogic of αω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(ωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindström quantifiers such that B(ωω(Qα)) ≤ ωω(Q).

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[Eb]Ebbinghaus, H-D., Extended logics: the general framework, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 2576.
[Fr]Friedman, H., Beth's theorem in cardinality logic, Israel Journal of Mathematics, vol. 14 (1973), pp. 205212.
[He]Hella, L., Definability hierarchies of generalized quantifiers, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 235271.
[Lu]Luosto, K., On interpolation and preservation in abstract model theory, Report of the Department of Mathematics, University of Helsinki, Helsinki, 1989. (17 pp.)
[Ma]Makowsky, J., Compactness, embeddings and definability, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 645716.
[MS1]Makowsky, J. and Shelah, S., The theorems of Beth and Craig in abstract model theory. I: The abstract setting, Transactions of the American Mathematical Society, vol. 256 (1979), pp. 215239.
[MS2]Makowsky, J. and Shelah, S., The theorems of Beth and Craig in abstract model theory. II: Compact logics, Archiv für Mathematische Logik und Grundlagenforschung, vol. 21 (1981), pp. 1335.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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