Definable Encodings in the Computably Enumerable Sets

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets. We will work in the structure of the computably enumerable sets. The language is just inclusion, ⊆. This structure is called e. All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3. We will also consider the quotient structure e modulo the ideal of finite sets, e*. e* is a definable quotient structure of e since “Χ is finite” is definable in e; “Χ is finite” iff all subsets of Χ are computable (it takes a little computability theory to show if Χ is infinite then Χ has an infinite non-computable subset). We use A* to denote the equivalent class of A under the ideal of finite sets.