Abstract
We prove that a sequence of sets containing representatives of cupping partners for every nonzero \({\Delta^0_2}\) enumeration degree cannot have a \({\Delta^0_2}\) enumeration. We also prove that no subclass of the \({\Sigma^0_2}\) enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to \({\mathbf{0}_e'}\) by a single incomplete \({\Sigma^0_2}\) enumeration degree.
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References
Arslanov, M.M., Sorbi, A.: Relative splittings of \({0'_e}\) in the Δ2 enumeration degrees. In: Buss, S., Pudlak, P. (eds.) Logic Colloquium ’98, pp. 44–56. Lecture Notes in Logic 13 (1999)
Cooper S.B.: Partial Degrees and the density problem. Part 2: The enumeration degrees of the Σ 2 sets are dense. J Symb Log 49, 503–513 (1984)
Cooper, S.B.: Enumeration reducibility, nondeterminitsic computations and relative computability of partial functions. In: Ambos-Spies, K., Muller, G.H., Sacks G.E. (eds.) Recursion Theory Week, Proceedings Oberwolfach 1989, pp. 57–110. Lecture Notes in Mathematics 1432 (1990)
Cooper S.B.: Computability Theory. Chapman & Hall/CRC Mathematics, Boca Raton, FL (2004)
Cooper, S.B.: On a theorem of C.E.M. Yates. Handwritten notes (1973)
Cooper S.B., Sorbi A., Yi X.: Cupping and noncupping in the enumeration degrees of \({\Sigma_2^0}\) sets. Ann. Pure Appl. Log. 82, 317–342 (1996)
Ershov Y.L.: A hierarchy of sets. I. Algebra i Logika 7.1, 47–74 (1968)
Ershov Y.L.: On a hierarchy of sets, II. Algebra i Logika 7.4, 15–47 (1968)
Jockusch C.G. Jr: Semirecursive sets and positive reducibility. Trans. Am. Math. Soc. 131, 420–436 (1968)
Kent, T.F.: Decidability and definability in the \({\Sigma^0_2}\)-enumeration degrees. PhD thesis, University of Wisconsin - Madison (2005)
Lachlan A.H., Shore R.A.: The n-rea enumeration degrees are dense. Arch. Math. Log. 31, 277–285 (1992)
Lewis A.E.M.: Finite cupping sets. Arch. Math. Log. 43, 845–858 (2004)
Nies A., Sorbi A.: Branching in the Σ 2 enumeration degrees. Isr. J. Math. 110, 29–59 (1999)
Posner D., Robinson R.: Degrees joining to 0′. J. Symb. Log. 46, 714–722 (1981)
Soare R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)
Soskova, M.I.: Cupping classes of \({\Sigma^0_2}\) enumeration degrees. In: Beckmann, Costas Dimitracopoulos, Benedikt Löwe (eds.) Logic and Theory of Algorithms, Fourth Conference on Computability in Europe, CiE 2008, Athens, Greece, June 2008, Proceedings, pp. 554–566. Lecture Notes in Computer Science 5028 (2008)
Soskova, M.I., Wu, G.: Cupping \({\Delta^0_2}\) enumeration degrees to 0′. In: Cooper, S., Löwe, B., Sorbi, A. (eds.) Computation and Logic in the Real World, pp. 727–738. Lecture Notes in Computer Science 4497 (2007)
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Soskova, M.I. The limitations of cupping in the local structure of the enumeration degrees. Arch. Math. Logic 49, 169–193 (2010). https://doi.org/10.1007/s00153-009-0171-1
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DOI: https://doi.org/10.1007/s00153-009-0171-1