Creativity and effective inseparability

Abstract

Introduction. Our terminology is from [1]. We consider the collection of all r.e. (recursively enumerable) sets arranged in an effective sequence 01, (02, ***, (Oi... as in the Post-Kleene enumeration, i.e., coi is the set of all numbers satisfying the condition (]y)T1(i,x, y), where 7&(z,x,y) is the Kleene predicate (cf. [2]). A number set a is called productive iff there exists a recursive function f(x) (called a productivefunction for x) such that for every number i, if w.i c oc thenf(i) E ac-coi. A number set a is called creative iff a is r.e. and the complement oc of oL is productive. A disjoint pair (cL, ,B) of number sets is called recursively inseparable (abbreviated R. I.) iff there exists no recursive superset of x which is disjoint from ,B. The pair (cL, ,B) is effectively inseparable (abbreviated E.I.)iff there is a recursive function $(x, y)-called an E.I. function for the pair CL, /3-such that for every disjoint pair (oi, oj) of respective r.e. supersets of (oL, /3), the number 8(i,j) is outside both coi and co,. It has been independently shown by Trahtenbrat and by Tennenbaum that there exists an R.I. pair (o,,B) of r.e. sets which is not E.I. In this paper we obtain some new characterizations of creativity and effective inseparability (as well as some closely allied notions) which seem surprisingly weak. These results are in some respects stronger than those obtained in [1]. The characterizations arose from the consideration of the following metamathematical problem. Consider a first order theory (T) with standard formalization (cf. Tarski [3]) and an effective G6del numbering g of the expressions of (T). We let To, Ro, respectively, be the set of Godel numbers of the provable and refutable sentences of (T) [by a refutable sentence we mean one whose negation is provable]. We refer to these 2 sets as the nuclei of (T). For the time being we shall consider only those theories (T) which are axiomatizable-i.e., such that To (and hence also RO) is an r.e. set. A formula F(x) (with just one free variable x) is said to represent in (T) a number set A if F(x) is provable for every n E A but for no n ? A-i.e., for every n, n e A iff F(An) is provable. Myhill [4] showed that if every r.e. set is representable in (T) then (T) is undecidable-i.e., To is not recursive. We have shown [5; 1], that a sufficient condition for (T) to be undecidable is that every recursive set be representable in (T). However, under Myhill's stronger hypothesis that every recursively enumerable set is representable in (T), the set To is not