Productivity and Double Productivity

Abstract

Abstract Now we will give the promised applications of the (strong) recursion and double recursion theorems to the theory of productivity and effective inseparability (and also to double productivity—a double analogue of productivity—which we will define). §1. Weak Productivity. We recall that a set α is said to be co-productive under a recursive function g(x) if for every number i, such that ωi is disjoint from α , the number g(i) is outside both a and ωi. This, of course, implies the folloωing weaker condition: C1: For every i, such that ωi is disjoint from α and such that ωi contains at most one element, the number g(i) is outside both α and ωi