## Journal of Symbolic Logic

### Codable Sets and Orbits of Computably Enumerable Sets

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745453

Mathematical Reviews number (MathSciNet)
MR1610758

Zentralblatt MATH identifier
0906.03041

JSTOR