CODABLE SETS AND ORBITS OF COMPUTABLY ENUMERABLE SETS

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 epsilon denote the structure of the computab ly enumerable sets under inclusion, epsilon = ({W-e}(e is an element o f omega), subset of or equal to). We previously exhibited a first orde r epsilon-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 thoug h X may have high information content. second we prove that every X wi th this slowness property is computable in some member of any nontrivi al orbit, namely for any noncomputable A is an element of epsilon ther e exists B in the orbit of A such that X less than or equal to(T) B un der relative Turing computability (less than or equal to(T)). We produ ce B using the Delta(3)(0)-automorphism method we introduced earlier.