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Limit ultrapowers and abstract logics

Published online by Cambridge University Press:  12 March 2014

Paolo Lipparini
Paolo Lipparini*
Affiliation:
Department of Mathematics, University of Tor Vergata, Rome, Italy
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Abstract

We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L.

For every countably generated [ω, ω]-compact logic L, our main applications are:

(i) Elementary classes of L can be characterized in terms of ≡L only.

(ii) If and are countable models of a countable superstable theory without the finite cover property, then .

(iii) There exists the “largest” logic M such that complete extensions in the sense of M and L are the same; moreover M is still [ω, ω]-compact and satisfies an interpolation property stronger than unrelativized ⊿-closure.

(iv) If L = Lωω(Qx), then cf(ωx) > ω and λω < ωx, for all λ < ωx.

We also prove that no proper extension of Lωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning Lκλ-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 2 , June 1987 , pp. 437 - 454
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[BF]Barwise, J. and Feferman, S. (editors), Model-theoretic logics, Perspectives in Mathematical Logic, vol. 6, Springer-Verlag, Berlin, 1985.Google Scholar
[Ca]Caicedo, X., Failure of interpolation for quantifiers of monadic type, Methods in mathematical logic (proceedings of the sixth Latin American symposium in mathematical logic; Di Prisco, C. A., editor), Lecture Notes in Mathematics, Vol. 1130, Springer-Verlag, Berlin, 1985, pp. 1–12.Google Scholar
[CK]Chang, C. C. and Keisler, J., Model theory, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam, 1977.Google Scholar
[Eb]Ebbinghaus, H. D., Extended logics: the general framework, in [BF], pp. 25–76.CrossRefGoogle Scholar
[Ke]Keisler, J., On cardinalities of ultraproducts, Bulletin of the American Mathematical Society, Vol. 70 (1964), pp. 644–647.CrossRefGoogle Scholar
[Ko]Kopperman, R., Model theory and its applications, Allyn and Bacon, Boston, Massachusetts, 1972.Google Scholar
[Li]Lindström, P., On extensions of elementary logic, Theoria, vol. 35 (1969), pp. 1–11.CrossRefGoogle Scholar
[Lp1]Lipparini, P., Duality for compact logics and substitution in abstract model theory, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 31 (1985), pp. 517–532.CrossRefGoogle Scholar
[Lp2]Lipparini, P., Logics closed under first order unary operations (abstract), this Journal, vol. 51 (1986), p. 491.Google Scholar
[Lp3]Lipparini, P., Robinson equivalence relations through limit ultrapowers, Bollettino dell'Unione Matematica Italiana, vol. 4-B (1985), pp. 569–583.Google Scholar
[Ma]Makowsky, J. A., Vopěnka's principle and compact logics, this Journal, vol. 50 (1985), pp. 42–48.Google Scholar
[MS1]Makowsky, J. A. 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. A. and Shelah, S., Positive results in abstract model theory: a theory of compact logics, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 263–299.CrossRefGoogle Scholar
[Mu1]Mundici, D., Interpolation, compactness and JEP in soft model theory, Archiv für Mathematische Logik und Grundlagenforschung, vol. 22 (1982), pp. 61–67.CrossRefGoogle Scholar
[Mu2]Mundici, D., Duality between logics and equivalence relations, Transactions of the American Mathematical Society, vol. 270 (1982), pp. 111–129.CrossRefGoogle Scholar
[Sh]Shelah, S., Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North Holland, Amsterdam, 1978.Google Scholar