Abstract.
We show that there is a limit lemma for enumeration reducibility to 0 e ', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X< e W such that W≤ e X' there is a set A such that X≤ e A< e W and A'=W'. (All jumps are enumeration degree jumps.) The degrees of sets with good approximations include the Σ0 2 degrees and the n-CEA degrees.
Article PDF
Similar content being viewed by others
References
Calhoun, W.C., Slaman, T.A.: The Π0 2 enumeration degrees are not dense. Journal of Symbolic Logic 61, 1364–1379 (1996)
Cooper, S.B.: Partial degrees and the density problem, part ii: the enumeration degrees of σ2 sets are dense. Journal of Symbolic Logic 49, 503–513 (1984)
Cooper, S.B.: Enumeration reducibility, nondeterministic computations and relative computability of partial functions. In Ambos-Spies K., Müller G.H., and Sacks G.E. (editors), Recursion Theory Week, Proceedings Oberwolfach, Lecture Notes in Mathemathics, 1432, 57–110. Springer-Verlag, Berlin Heidelberg New York, 1989
Cooper, S.B., Copestake, C.S.: Properly §igma 2 enumeration degrees. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 34, 491–522 (1988)
Lachlan, A.H., Shore, R.A.: The n-rea enumeration degrees are dense. Archive for Mathematical Logic 31, 277–285 (1992)
McEvoy, K.: Jumps of quasi-minimal enumeration degrees. Journal of Symbolic Logic 50, 839–848 (1985)
McEvoy, K., Cooper, S.B.: On minimal pairs of enumeration degrees. Journal of Symbolic Logic 50, 983–1001 (1985)
Rogers, H.: Theory of recursive functions and effective computability. New York: McGraw-Hill, 1967
Sacks, G.E.: Recursive enumerability and the jump operator. Transactions of the American Mathematical Society 108, 223–239 (1963)
Sorbi, A.: The enumeration degrees of the §igma 0 2 sets. In Lecture Notes in Pure and Applied Math., 187, 303–330. Dekker, New York, 1997
Author information
Authors and Affiliations
Corresponding author
Additional information
The results in this paper form part of the author's doctoral dissertation written under the supervision of Prof. Steffen Lempp at the University of Wisconsin Madison. The author is grateful to an anonymous referee for helpful comments and suggestions.
Rights and permissions
About this article
Cite this article
Griffiths, E. Limit lemmas and jump inversion in the enumeration degrees. Arch. Math. Logic 42, 553–562 (2003). https://doi.org/10.1007/s00153-002-0161-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-002-0161-z