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Expansions of models and turing degrees

Published online by Cambridge University Press:  12 March 2014

Julia Knight  and
Mark Nadel
Julia Knight
Affiliation:
University of Notre Dame, Notre Dame, Indiana 46556
Mark Nadel
Affiliation:
University of Notre Dame, Notre Dame, Indiana 46556
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Extract

If is a countable recursively saturated structure and T is a recursively axiomatizable theory that is consistent with Th(), then it is well known that can be expanded to a recursively saturated model of T [7, p. 186]. This is what has made recursively saturated models useful in model theory. Recursive saturation is the weakest notion of saturation for which this expandability result holds. In fact, if is a countable model of Pr = Th(ω, +), then can be expanded to a model of first order Peano arithmetic P just in case is recursively saturated (see [3]).

In this paper we investigate two natural sets of Turing degrees that tell a good deal about the expandability of a given structure. If is a recursively saturated structure, I() consists of the degrees of sets that are recursive in complete types realized in . The second set of degrees, D(), consists of the degrees of sets S such that is recursive in S-saturated. In general, I() ⊆ D(). Moreover, I() is obviously an “ideal” of degrees. For countable structures , D() is “closed” in the following sense: For any class C ⊆ 2ω, if C is co-r.e. in S for some set S such that , then there is some σ ∈ C such that . For uncountable structures , we do not know whether D() must be closed.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 3 , September 1982 , pp. 587 - 604
Copyright
Copyright © Association for Symbolic Logic 1982

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References

REFERENCES

[1]Jockusch, C. G. Jr. and Soare, R. I., Π10-classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[2]Kunen, K., Combinatorics, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 371–401.CrossRefGoogle Scholar
[3]Lipshitz, L. and Nadel, M., The additive structure of models of arithmetic, Proceedings of the American Mathematical Society, vol. 68 (1978), pp. 331–336.CrossRefGoogle Scholar
[4]MacDowell, R. and Specker, E., Modelle der Arithmetik, Infinitistic methods, Pergamon, London, 1961, pp. 257–263.Google Scholar
[5]Nadel, M., On a problem of MacDowell and Specker, this Journal, vol. 45 (1980), pp. 612–622.Google Scholar
[6]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[7]Schlipf, J. S., Model theory and recursive saturation, this Journal, vol. 43 (1978), pp. 183–206.Google Scholar
[8]Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Providence, R.I., 1962, pp. 117–121.CrossRefGoogle Scholar
[9]Simpson, S., Degrees of unsolvability, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 631–665.CrossRefGoogle Scholar
[10]Simpson, S., Sets which do not have subsets of every higher degree, this Journal, vol. 43 (1978), pp. 135–138.Google Scholar
[11]Szmielew, W., Elementary properties of Abelian groups, Fundamenta Mathematicae, vol. 41 (1954), pp. 203–271.CrossRefGoogle Scholar