Recursive and Recursively Enumerable

I. Basic concepts 1. Recursive enumerability and recursiveness. We consider infinite sequences of non-negative integers, free from repetitions. A familiar equivalence relation between such sequences is that based on sets: two sequences are equivalent when they enumerate the same set. The equivalence classes under this relation, with the necessary operations introduced, form a system isomorphic to the algebra of all infinite sets of non-negative integers, with the set of all such integers, e, as a distinguished element. In this algebra we can introduce the concepts of recursive enumerability and recursiveness as follows. If an equivalence class a contains a sequence produced by a recursive function, we say that a corresponds to a recursively enumerable (r.e.) set (or, informally, that a is an r.e. set). In each equivalence class there is a sequence which is in order of size; we call this the principal sequence of its class. If the principal sequence of a is produced by a recursive function, we say that a is a recursive set [7, p. 291 ]. This development is interesting chiefly because it can be generalized. For any equivalence relation between sequences, we call recursively enumerable any element of the equivalence class algebra which contains a sequence produced by a recursive function. If a nontrivial principal sequence can be defined for each element, we call recursive those elements whose principal sequences are produced by recursive functions. (A trivial principal sequence would be one which was never produced by a recursive function, or which was produced by a recursive function whenever any sequence of its equivalence class was.) In this paper we study the system arising from a particular equivalence relation based on order. The set equivalence relation ignores differences in order between sequences; the order equivalence relation will ignore differences in sets. However, the introduction of recursion theory produces strong interconnections between the two systems, which we will use constantly in attempting to gain information about them both. 2. Orders. Definition. Two sequences aa, ci, a2, • • • and bo, bi, b2, ■ • ■ are equivalent if the same permutation of their terms arranges each in order of size.