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On the T-degrees of partial functions

Published online by Cambridge University Press:  12 March 2014

Paolo Casalegno
Paolo Casalegno*
Affiliation:
Dipartimento di F1Losofia, Università di Pisa, 56100 Pisa, Italy
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Abstract

Let 〈, ≤ 〉 be the usual structure of the degrees of unsolvability and 〈, ≤ 〉 the structure of the T-degrees of partial functions defined in [7]. We prove that every countable distributive lattice with a least element can be isomorphically embedded as an initial segment of 〈, ≤ 〉: as a corollary, the first order theory of 〈, ≤ 〉 is recursively isomorphic to that of 〈, ≤ 〉. We also show that 〈, ≤ 〉 and 〈, ≤ 〉 are not elementarily equivalent.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 3 , September 1985 , pp. 580 - 588
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

[1]Cooper, S. B., Partial degrees and the density problem. I, II, this Journal, vol. 47 (1982), pp. 854–859; vol. 49 (1984), pp. 503-513.Google Scholar
[2]Epstein, R. L., Degrees of unsolvability: structure and theory, Lecture Notes in Mathematics, vol. 759, Springer-Verlag, Berlin, 1979.CrossRefGoogle Scholar
[3]Grätzer, G., Lattice theory, Freeman, San Francisco, California, 1971.Google Scholar
[4]Nerode, A. and Shore, R. A., Second order logic and first order theories of reducibility orderings, The Kleene symposium (Barwise, J., Keisler, J and Kunen, K., editors), North-Holland, Amsterdam, 1980, pp. 181–200.CrossRefGoogle Scholar
[5]Sasso, L., Degrees of unsolvability of partial functions, Ph.D. Thesis, University of California, Berkeley, California, 1971.Google Scholar
[6]Sasso, L., A minimal partial degree ≤ 0′, Proceedings of the American Mathematical Society, vol. 38 (1973), pp. 388–392.Google Scholar
[7]Sasso, L., A survey of partial degrees, this Journal, vol. 40 (1975), pp. 130–140.Google Scholar