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Elementary differences between the (2p)-C. E. and the (2p +1)-c. e. enumeration degrees

  • I. S. Kalimullin (a1)

Abstract

It is proved that the (2p)-c. e. e-degrees are not elementarily equivalent to the (2p + 1)-c. e. e-degrees for each nonzero p ∈ ω. It follows that m-c. e. e-degrees are not elementarily equivalent to the n-c e. e-degrees if 1 <m < n.

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[1]Cooper, S. B., Enumeration reducibility, nondeterministic computations and relative computability of partial functions, Recursion Theory Week, Oberwolfach 1989 (Ambos-Spies, K., Müller, G., and Sacks, G. E., editors), Lecture Notes in Mathematics, vol. 1432, Springer, 1990, pp. 57–110.
[2]Kalimullin, I. S., Elementary theories of semilattices of n-recursive enumerable degrees with respect to enumerability, Russian Mathematics, vol. 45 (2001), no. 4, pp. 22–25.
[3]Kalimullin, I. S., Splitting properties of n-c. e. enumeration degrees, this Journal, vol. 67 (2002), pp. 537–546.
[4]Kalimullin, I. S., Definability of the jump operator in the enumeration degrees, Journal of Mathematical Logic, vol. 3 (2003), no. 2, pp. 257–267.
[5]Lachlan, A. H., Lower bounds for pairs of recursively enumerable degrees, Proceedings of the London Mathematical Society, vol. 16 (1966), pp. 537–569.
[6]Rogers, H. Jr, Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967.
[7]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer, 1987.

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