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An algebraic result about soft model theoretical equivalence relations with an application to H. Friedman's fourth problem

Published online by Cambridge University Press:  12 March 2014

Daniele Mundici
Daniele Mundici*
Affiliation:
Loc. Romola N. 76, 50060 Donnini, Florence, Italy
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Abstract

We prove the following algebraic characterization of elementary equivalence: ≡ restricted to countable structures of finite type is minimal among the equivalence relations, other than isomorphism, which are preserved under reduct and renaming and which have the Robinson property; the latter is a faithful adaptation for equivalence relations of the familiar model theoretical notion. We apply this result to Friedman's fourth problem by proving that if is an (ω1, ω)-compact logic satisfying both the Robinson consistency theorem on countable structures of finite type and the Löwenheim-Skolem theorem for some λ < ωω for theories having ω1 many sentences, then ≡L = ≡ on such structures.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 3 , September 1981 , pp. 523 - 530
Copyright
Copyright © Association for Symbolic Logic 1981

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References

REFERENCES

[Ba]Barwise, K.J., Axioms for abstract model theory, Annals of Mathematical Logic, vol. 7 (1974), pp. 221–265.CrossRefGoogle Scholar
[BKM]Barwise, K.J., Kaufmann, M. and Makkai, M., Stationary logic, Annals of Mathematical Logic, vol. 13 (1978), pp. 171–224.CrossRefGoogle Scholar
[Fe]Feferman, S., Two notes on abstract model theory. I, Fundament a Mathematicae, vol. 82 (1974), pp. 153–165 and II, Fundament a Mathematicae, vol. 89 (1975), pp. 111–130.CrossRefGoogle Scholar
[Fl]Flum, J., First order logic and its extensions, Lecture Notes in Mathematics, no. 499, Springer, Berlin, 1976, pp. 248–310.Google Scholar
[Fr]Friedman, H., One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113–129.Google Scholar
[Ke]Keisler, H.J., Constructions in model theory, CIME II Ciclo, Coord. Mangani, Model theory and applications, Cremonese, Rome, 1975.Google Scholar
[MSI]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 (to appear).Google Scholar
[MSS]Makowsky, J. A., Shelah, S. and Stavi, J., ⊿-logics and generalized quantifiers, Annals of Mathematical Logic, vol. 10(1976),pp. 155–192.CrossRefGoogle Scholar
[Mul]Mundici, D., Compactness + Craig interpolation = Robinson consistency in any logic (preprint).Google Scholar
[Mu2]Mundici, D., Compactness = JEP in any logic, Fundamenta Mathematicae (to appear).Google Scholar
[Mu3]Mundici, D., Compactness, interpolation and Friedman's third problem (to appear).Google Scholar
[Mu4]Mundici, D., Applications of many-sorted Robinson consistency theorem, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (to appear).Google Scholar
[Mu5]Mundici, D., Robinson's consistency theorem in soft model theory, Transactions of the American Mathematical Society (to appear).Google Scholar