Codable sets and orbits of computably enumerable sets

Journal of Symbolic Logic 63 (1):1-28 (1998)
A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let ε denote the structure of the computably enumerable sets under inclusion, $\varepsilon = (\{W_e\}_{e\in \omega}, \subseteq)$ . We previously exhibited a first order ε-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has a certain "slowness" property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A ∈ ε there exists B in the orbit of A such that X ≤ T B under relative Turing computability (≤ T ). We produce B using the Δ 0 3 -automorphism method we introduced earlier
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DOI 10.2307/2586583
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References found in this work BETA
Donald A. Martin (1966). Classes of Recursively Enumerable Sets and Degrees of Unsolvability. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 12 (1):295-310.

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Citations of this work BETA
Peter Cholak, Rod Downey & Eberhard Herrmann (2001). Some Orbits for E. Annals of Pure and Applied Logic 107 (1-3):193-226.
Peter A. Cholak, Peter M. Gerdes & Karen Lange (2012). On N-Tardy Sets. Annals of Pure and Applied Logic 163 (9):1252-1270.

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