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Countable structures, Ehrenfeucht strategies, and Wadge reductions

Published online by Cambridge University Press:  12 March 2014

Tom Linton
Tom Linton*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
*
Department of Mathematics, California Institute of Technology, Pasadena, California 91125.
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Abstract

For countable structures and , let abbreviate the statement that every sentence true in also holds in . One can define a back and forth game between the structures and that determines whether . We verify that if θ is an Lω,ω sentence that is not equivalent to any Lω,ω sentence, then there are countably infinite models and such that ⊨ θ, ⊨ ¬θ, and . For countable languages ℒ there is a natural way to view ℒ structúres with universe ω as a topological space, Xℒ. Let [] = { ∊ Xℒ∣ ≅ } denote the isomorphism class of . Let and be countably infinite nonisomorphic ℒ structures, and let C ⊆ ωω be any subset. Our main result states that if , then there is a continuous function f: ωω → Xℒ with the property that x ∊ C ⇒ f(x) ∊ [] and x ∉ C ⇒ f(x) ∊ f(x) ∈ []. In fact, for α ≤ 3, the continuous function f can be defined from the relation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 4 , December 1991 , pp. 1325 - 1348
Copyright
Copyright © Association for Symbolic Logic 1991

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References

REFERENCES

[1]Addison, J. W., The method of alternating chains, The theory of models (proceedings of the 1963 international symposium at Berkeley), North-Holland, Amsterdam, 1965, pp. 1–16.Google Scholar
[2]Addison, J. W., Some problems in hierarchy theory, Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Providence, Rhode Island, 1962, pp. 123–130.CrossRefGoogle Scholar
[3]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[4]Ehrenfeucht, A., An application of games to the completeness problem for formalized theories, Fundamenta Mathematicae, vol. 49 (1961), pp. 129–141.CrossRefGoogle Scholar
[5]Friedman, H. and Stanley, L., A Borel reducibility theory for classes of countable structures, this Journal, vol. 54 (1989), pp. 894–914.Google Scholar
[6]John, T., Strategies for Borel games, Ph.D. Thesis, University of California, Berkeley, California, 1981.Google Scholar
[7]Keisler, H. J., Theory of models with generalized atomic formulas, this Journal, vol. 25 (1960), pp. 1–26.Google Scholar
[8]Keisler, H. J., Finite approximations of infinitely long formulas, The theory of models (proceedings of the 1963 international symposium at Berkeley), North-Holland, Amsterdam, 1965, pp. 158–169.Google Scholar
[9]Linton, T. J., Partial isomorphisms and continuous reductions with games, Ph.D. Thesis, University of Wisconsin, Madison, Wisconsin, 1991.Google Scholar
[10]López-Escobar, E. G. K., An interpolation theorem for denumerably long formulas, Fundamenta Mathematicae, vol. 57 (1965), pp. 253–272.CrossRefGoogle Scholar
[11]Louveau, A., Some results in the Wadge hierarchy of Borel sets, Cabel seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, New York, 1983, pp. 28–55.CrossRefGoogle Scholar
[12]Louveau, A. and Saint-Raymond, J., The strength of Borel Wadge determinacy, Cabal seminar 81–85, Lecture Notes in Mathematics, vol. 1333, Springer-Verlag, New York, 1988, pp. 1–30.CrossRefGoogle Scholar
[13]Louveau, A. and Saint-Raymond, J., Borel classes and closed games: Wadge-type and Hurewicz-type results, Transactions of the American Mathematical Society, vol. 304 (1987), pp. 431–467.CrossRefGoogle Scholar
[14]Martin, D., Borel determinancy, Annals of Mathematics, ser. 2, vol. 102 (1975), pp. 363–371.CrossRefGoogle Scholar
[15]Miller, A., On the Borel classification of the isomorphism class of a countable model, Notre Dame Journal of Formal Logic, vol. 24(1983), pp: 22–34.CrossRefGoogle Scholar
[16]Miller, D. E., The invariant separation principle, Transactions of the American Mathematical Society, vol. 242 (1978), pp. 185–204.Google Scholar
[17]Moschovakis, Y., Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
[18]Mostowski, A., On absolute properties of relations, this Journal, vol. 12 (1947), pp. 33–42.Google Scholar
[19]Mostowski, A. and Ehrenfeucht, A., A compact space of models of first order theories, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 9 (1961), pp. 369–373.Google Scholar
[20]Scott, D., Invariant Borel sets, Fundamenta Mathematicae, vol. 56 (1964), pp. 117–128.CrossRefGoogle Scholar
[21]Van Wesep, R., Wadge degrees and descriptive set theory, Cabal seminar 76–77, Lecture Notes in Mathematics, vol. 689, Springer-Verlag, New York, 1978, pp. 151–170.CrossRefGoogle Scholar
[22]Vaught, R., Invariant sets in topology and logic, Fundamenta Mathematicae, vol. 82 (1974), pp. 269–294.CrossRefGoogle Scholar
[23]Wadge, W., Reducibility and determinateness on the Boire space, Ph.D. thesis, University of California, Berkeley, California, 1984.Google Scholar
[24]Wadge, W., Degrees of complexity of subsets of the Baire space, Notices of the American Mathematical Society, vol. 19 (1972), pp. A–714-A-715 (abstract 72T-E91).Google Scholar