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

Published online by Cambridge University Press:  12 March 2014

Lauri Hella  and
Kerkko Luosto
Lauri Hella
Affiliation:
Department of Mathematics, University of Helsinki, 00100 Helsinki, Finland, E-mail: hella@cc.helsinki.fi
Kerkko Luosto
Affiliation:
Department of Mathematics, University of Helsinki, 00100 Helsinki, Finland, E-maii: kluosto@cc.helsinki.fi
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Abstract

We prove that if ℵα is uncountable and regular, then the Beth-closure of ℒωω(Qα) is not a sublogic of ℒαω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(ℒωω(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(ℒωω(Qα)) ≤ ℒωω(Q).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 2 , June 1992 , pp. 442 - 448
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[Eb]Ebbinghaus, H-D., Extended logics: the general framework, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 25–76.Google Scholar
[Fr]Friedman, H., Beth's theorem in cardinality logic, Israel Journal of Mathematics, vol. 14 (1973), pp. 205–212.CrossRefGoogle Scholar
[He]Hella, L., Definability hierarchies of generalized quantifiers, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 235–271.CrossRefGoogle Scholar
[Lu]Luosto, K., On interpolation and preservation in abstract model theory, Report of the Department of Mathematics, University of Helsinki, Helsinki, 1989. (17 pp.)Google Scholar
[Ma]Makowsky, J., Compactness, embeddings and definability, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 645–716.Google Scholar
[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. 215–239.Google Scholar
[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. 13–35.CrossRefGoogle Scholar