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On Downey's conjecture

Published online by Cambridge University Press:  12 March 2014

Marat M. Arslanov ,
Iskander Sh. Kalimullin  and
Steffen Lempp
Marat M. Arslanov
Affiliation:
Department of Mathematics, Kazan State University, Ul. Kremlevskaya 18, 420008 Kazan, Russia. E-mail: Marat.Arslanov@ksu.ru
Iskander Sh. Kalimullin
Affiliation:
Department of Mathematics, Kazan State University, Ul. Kremlevskaya 18, 420008 Kazan, Russia. E-mail: Iskander.Kalimullin@ksu.ru
Steffen Lempp
Affiliation:
Department of Mathematics, University of Wisconsin, Madison. WI 53706-1388, USA. E-mail: lempp@math.wisc.edu, URL: http://www.math.wisc.edu/~lempp
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Abstract

We prove that the degree structures of the d.c.e. and the 3-c.e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey. More specifically, we show that the following statement fails in the former but holds in the latter structure: There are degrees f > e > d > 0 such that any degree u ≤ f is either comparable with both e and d, or incomparable with both.

Keywords

d.c.e. degreesn-c.e. degreesDowney's conjecture
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 2 , June 2010 , pp. 401 - 441
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[Ar88]Arslanov, Marat M., The lattice of the degrees below 0′, Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, (1988), no. 7, pp. 27–33.Google Scholar
[ArOO]Arslanov, Marat M., Open questions about the n-c.e. degrees, Computability theory and its applications (Boulder, CO, 1999), Contemporary Mathematics, no. 257, American Mathematical Society, Providence. RI, 2000.Google Scholar
[ALS96]Arslanov, Marat M., Lempp, Steffen, and Shore, Richard A., On isolating r.e. and isolated d-r.e. degrees, Computability, enumerability, unsolvability (Cooper, S. B., Slaman, T. A., and Wainer, S. S., editors), London Mathematical Society, Cambridge University Press, Cambridge, 1996, pp. 61–80.CrossRefGoogle Scholar
[Co71]Cooper, S. Barry, Degrees of unsolvability, Ph.D. thesis. Leicester University, Leicester, England, 1971,Google Scholar
[CHLLS91]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), pp. 125–151.CrossRefGoogle Scholar
[CLW89]Cooper, S. Barry, Lempp, Steffen, and Watson, Philip, Weak density and cupping in the d-r.e. degrees, Israel Journal of Mathematics, vol. 67 (1989), pp. 137–152.CrossRefGoogle Scholar
[Do89]Downey, Rodney G., D.r.e. degrees and the nondiamond theorem, Bulletin of the London Mathematical Society, vol. 21 (1989), pp. 43–50.CrossRefGoogle Scholar
[Er68a]Ershov, Yuri L., A certain hierarchy of sets I, Algebra and Logic, vol. 7 (1968), no. 1, pp. 47–74, (English translation pp. 25–43).Google Scholar
[Er68b]Ershov, Yuri L., A certain hierarchy of sets II, Algebra and Logic, vol. 7 (1968), no. 4, pp. 15–47. (English translation pp. 212–232).Google Scholar
[Er70]Ershov, Yuri L., A certain hierarchy of sets III, Algebra and Logic, vol. 9 (1970), no. 1, pp. 34–51. (English translation pp. 20–31).CrossRefGoogle Scholar
[La66]Lachlan, Alistair H., Lower bounds for pairs of recursively enumerable degrees, Proceedings of the London Mathematical Society, vol. 16 (1966), pp. 537–569.CrossRefGoogle Scholar
[Mi81]Miller, David P., High recursively enumerable degrees and the anticupping property, Logic Year 1979–80 (Proceedings of the Seminars and Conference on Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80), Lecture Notes in Mathematics, no. 859. Springer, Berlin, 1981, pp. 230–245.CrossRefGoogle Scholar
[Pu65]Putnam, Hilary, Trial and error predicates and the solution to a problem of Mostowski, this Journal, vol. 30 (1965), pp. 49–57.Google Scholar
[Sa61]Sacks, Gerald E., A minimal degree less than 0′, Bulletin of the American Mathematical Society, vol. 67 (1961), pp. 416–419.CrossRefGoogle Scholar
[Sa64]Sacks, Gerald E., The recursively enumerable degrees are dense, Annals of Mathematics (2), vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[So87]Soare, Robert I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar