The Δ₃⁰-automorphism method and noninvariant classes of degrees

A set A 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 E denote the structure of the computably enumerable sets under inclusion, E = ({We}e∈ω ,⊆). Most previously known automorphisms Φ of the structure E of sets were effective (computable) in the sense that Φ has an effective presentation. We introduce here a new method for generating noneffective automorphisms whose presentation is ∆3, and we apply the method to answer a number of long open questions about the orbits of c.e. sets under automorphisms of E. For example, we show that the orbit of every noncomputable (i.e., nonrecursive) c.e. set contains a set of high degree, and hence that for all n > 0 the well-known degree classes Ln (the lown c.e. degrees) and Hn = R −Hn (the complement of the highn c.e. degrees) are noninvariant classes. Department of Mathematics, University of California at Berkeley, Berkeley, California 94720 E-mail address: leo@math.berkeley.edu Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637-1538 E-mail address: soare@math.uchicago.edu World Wide Web address: http://www.Cs.uchicago.edu/∼soare