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On Σ1-Structural Differences Among Finite Levels of the Ershov Hierarchy

Published online by Cambridge University Press:  12 March 2014

Yue Yang  and
Liang Yu
Yue Yang
Affiliation:
Department of Mathematics Faculty of Science, National University of Singapore, 2 Science Drive 2, Singapore 117543. Singapore, E-mail: matyangy@nus.edu.sg
Liang Yu
Affiliation:
Institute of Mathematical Sciences, Nanjing University, Nanjing, Jiangsu Province, 210093, China, E-mail: yuliang.nju@gmail.com
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Abstract

We show that the structure of recursively enumerable degrees is not a Σ1-elementary substructure of , where (n > 1) is the structure of n-r.e. degrees in the Ershov hierarchy.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1223 - 1236
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Arslanov, Marat M., Structural properties of the degrees below 0', Doklady Akademiia Nauk SSSR, (1985). pp. 270–273.Google Scholar
[2]Arslanov, Marat M., Kalimullin, Iskander Sh., and Lempp, Steffen, On Downey's conjecture, preprint, 2005.Google Scholar
[3]Cooper, S. Barry, Harrington, Leo, Lachlan, Alistair H., Lempp, Steffen, and Soare, Robert I., The d.r.e. degrees are not dense, Annals of Pure and Applied Logic, vol. 55 (1991), no. 2, pp. 125–151.CrossRefGoogle Scholar
[4]Downey, Rodney G., Intervals and sublattices of the recursively enumerable T- and w-degrees, part I: Density, Annals of Pure and Applied Logic, vol. 49 (1989), pp. 1–27.CrossRefGoogle Scholar
[5]Ershov, Ju. L., A certain hierarchy of sets. I, Algebra i Logika, vol. 7 (1968), no. 1, pp. 47–74.Google Scholar
[6]Ershov, Ju. L., A certain hierarchy of sets. II, Algebra i Logika, vol. 7 (1968), no. 4, pp. 15–47.Google Scholar
[7]Ershov, Ju. L., A certain hierarchy of sets. III, Algebra i Logika, vol. 9 (1970), pp. 34–51.Google Scholar
[8]Gold, E. Mark, Limiting recursion, this Journal, vol. 30 (1965), no. 1, pp. 28–48.Google Scholar
[9]Putnam, Hillary, Trial and error predicates and the solution to a problem of Mostowski, this Journal, vol. 30 (1965), no. 1, pp. 49–57.Google Scholar
[10]Sacks, Gerald E., A minimal degree below 0′, Bulletin of the American Mathematical Society, vol. 67 (1961), pp. 416–419.CrossRefGoogle Scholar
[11]Sacks, Gerald E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[12]Shore, Richard A. and Slaman, Theodore A., Working below a high recursively enumerable degree, this Journal, vol. 58 (1993), no. 3, pp. 824–859.Google Scholar
[13]Slaman, Theodore A., The recursively enumerable degrees as a substructure of the Δ20, hand-written notes, 1983.Google Scholar
[14]Soare, Robert I., Recursively enumerable sets and degrees, Springer–Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[15]Yang, Yue and Yu, Liang, Elementary differences among finite levels of the Ershov hierarchy, Proceedings of TAMC2006: Theory and Applications of Models of Computation (Cai, Jin-Yi, Cooper, S. Barry, and Li, Angsheng, editors), Lecture Notes in Computer Science, vol. 3959, Springer–Verlag, 2006, pp. 765–771.CrossRefGoogle Scholar