Double Generativity and Complete Effective

Abstract

Abstract §1. Complete Effective Inseparability. A disjoint pair (A1,A2) is by definition recursively inseparable if no recursive superset of A1 is disjoint from A2. This is equivalent to saying that for any disjoint r.e. supersets ωi and ωj of A1 and A2, the set ωi is not the complement of ωj —in other words, there is a number n outside both ωi and ωj. The disjoint pair (A1,A2) is called effectively inseparable—abbreviated E.I.—if there is a recursive function δ(x, y)—called an E.I. function for (A1, A2)—such that for any numbers i and j such that A1⊆ ωi and A2Í ωj. with ωi being disjoint from ωj, the number d(i , j) is outside both a;,- and ωj. We shall call a disjoint pair (A1, A2) completely E.I. if there is a recursive function δ(x, y)—which we call a complete E.I. function for (A1, A2)—such that for any numbers i and j, if A1⊆ ωi and A2Í ωj, then δ(i , j) Í ωi ↔ d(i , j)Í ωj (in other words, d(i, j) is either inside or outside both sets ωi and ωj.). [If ωi and ωj happen to be disjoint, then, of course, d(i, j) is outside both ωi and ωj, so any complete E.I. function for (A1,A2) is also an E.I. function for (A1,A2) In a later chapter, we will prove the non-trivial fact that if (A1, A2) is E.I. and A1 and A2 are both r.e., then (A1,A2) is completely E.I. [The proof of this uses the result known as the double recursion theorem, which we will study in Chapter 9.] Effective inseparability has been well studied in the literature. Complete effective inseparability will play a more prominent role in this volume—especially in the next few chapters. Proposition 1. (1) If (A1,A2) is completely E.I., then so is (A2,A1) —in fact, if d(x,y) is a complete E.I. function for (A1,A2), then d(y,x) is a complete E.I. function for (A2, A1).