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A computably categorical structure whose expansion by a constant has infinite computable dimension

Published online by Cambridge University Press:  12 March 2014

Denis R. Hirschfeldt ,
Bakhadyr Khoussainov  and
Richard A. Shore
Denis R. Hirschfeldt
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA
Bakhadyr Khoussainov
Affiliation:
Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand
Richard A. Shore
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
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Abstract

Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1199 - 1241
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Cholak, P., Goncharov, S. S.. Khoussainov, B.. and Shore, R. A., Computably categorical structures and expansions by constants, this Journal, vol. 64 (1999), pp. 13–37.Google Scholar
[2] Ershov, Y. L., Goncharov, S. S., Nerode, A., and Remmel, J. B. (editors). Handbook of recursive mathematics. Studies in Logic and the Foundations of Mathematics, vol, 138-139, Elsevier Science. Amsterdam, 1998.Google Scholar
[3] Goncharov, S. S., Computable single-valued numerations. Algebra and Logic, vol. 19 (1980), pp. 325–356.Google Scholar
[4] Goncharov, S. S., Problem of the number of non-self-equivalent constructivizations. Algebra and Logic, vol. 19 (1980), pp. 401–414.CrossRefGoogle Scholar
[5] Harizanov, V. S., Pure computable model theory, in Ershov, et al. [2], pp. 3–114.Google Scholar
[6] Hirschfeldt, D. R., Degree spectra of intrinsically c. e. relations, this Journal, vol. 66 (2001), pp. 441–469.Google Scholar
[7] Hirschfeldt, D. R., Degree spectra of relations on structures of finite computable dimension. Annals of Pure and Applied Logic, vol. 115 (2002), pp. 233–277.Google Scholar
[8] Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimension in algebraic structures. Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71–113.Google Scholar
[9] Khoussainov, B. and Shore, R. A., Computable isomorphisms, degree spectra of relations, and Scott families. Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153–193.CrossRefGoogle Scholar
[10] Khoussainov, B. and Shore, R. A., Effective model theory: the number of models and their complexity, Models and computability (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Note Series, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193–239.CrossRefGoogle Scholar
[11] Millar, T., Recursive categoricity and persistence, this Journal, vol. 51 (1986), pp. 430–434.Google Scholar
[12] Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg, 1987.Google Scholar