The Co-Simple Isols

Abstract

Let E {0, 1, .. }, E* {O, ? 1, ...I}, P(E) = the class of all subsets of E, XkE the Cartesian product of k copies of E. A partial isomorphism is a 1-1 partial recursive function defined on a subset of E. If a, Se P(E), a and , are recursively equivalent (a 2? S) if there is a partial isomorphism p with a ' domain p and p(a) = 6; the equivalence class of a under this relation is its recursive equivalence type (r.e.t.) and the collection of all r.e.t.'s is denoted by t7. The r.e.t. of an isolated (i.e., finite or immune) set is an isol; A denotes the collection of isols, and A* the isolic integers (ring of differences of isols). These systems were introduced in [3], where various subsystems were also considered. Among these are i from this it follows that the results of ?? 2-3 of [7] remain valid when A* is replaced by A*. We are able to prove only a weak