Journal of Symbolic Logic

Codable Sets and Orbits of Computably Enumerable Sets

Leo Harrington and Robert I. Soare

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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 $\varepsilon$ denote the structure of the computably enumerable sets under inclusion, $\varepsilon = (\{W_e\}_{e\in \omega}, \subseteq)$. We previously exhibited a first order $\varepsilon$-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 $\in \varepsilon$ there exists B in the orbit of A such that X $\leq_T$ B under relative Turing computability ($\leq_T$). We produce B using the $\Delta^0_3$-automorphism method we introduced earlier.

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J. Symbolic Logic, Volume 63, Issue 1 (1998), 1-28.

First available in Project Euclid: 6 July 2007

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Harrington, Leo; Soare, Robert I. Codable Sets and Orbits of Computably Enumerable Sets. J. Symbolic Logic 63 (1998), no. 1, 1--28.

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