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η-representation of sets and degrees

Published online by Cambridge University Press:  12 March 2014

Kenneth Harris
Kenneth Harris*
Affiliation:
Department of Computer Science, University of Chicago1100 E. 5TH Street, Chicago, IL 60637, USA, E-mail: [email protected]
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Abstract

We show that a set has an η-representation in a linear order if and only if it is the range of a 0′-computable limitwise monotonic function. We also construct a Δ3 Turing degree for which no set in that degree has a strong η-representation, answering a question posed by Downey.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1097 - 1121
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[CDK98]Coles, R. J., Downey, R., and Khoussainov, B. M., On initial segments of computable linear orders, Order, vol. 14 (1998), pp. 107–24.CrossRefGoogle Scholar
[CHKS04]Csima, B., Hirschfeldt, D., Knight, J. F., and Soare, R. I., Bounding prime models, this Journal, vol. 69 (2004), no. 4, pp. 1117–1142.Google Scholar
[Dow98]Downey, Rod, Computability theory and linear orders, Handbook of recursive mathematics (Ershov, Y. L., Goncharov, S. S., Nerode, A., and Remmel, J. B., editors), Studies in Logic and the Foundations of Mathematics, vol. 138, Elsevier, 1998.Google Scholar
[HMP07]Hirschfeldt, D., Miller, R., and Podzorov, S., Order-computable sets, The Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 3, pp. 317–347.CrossRefGoogle Scholar
[Khi98]Khisamiev, N. G., Constructive abelian groups, Handbook of recursive mathematics (Ershov, Yu. L., Goncharov, S. S., Nerode, A., and Remmel, J. B., editors), Studies in Logic and the Foundations of Mathematics, vol. 138-139, Elsevier Science, 1998, pp. 1177–1231.Google Scholar
[KNS97]Khoussainov, B. M., Nies, A., and Shore, R. A., Computable models of theories with few models, Notre Dame Journal of Formal Logic, vol. 38 (1997),pp. 165–178.CrossRefGoogle Scholar
[Ler81]Lerman, Manuel, On recursive linear orders, Logic year 1979-80 (Lerman, M., Schmerl, J. H., and Soare, R. I., editors), Lecture Notes in Mathematics, vol. 859, 1981, pp. 132–142.CrossRefGoogle Scholar
[Soa87]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematics, Springer-Verlag, 1987.CrossRefGoogle Scholar