On Properties of Regressive Sets

Abstract

It was recently demonstrated by R. Mansfield (unpublished) that complementary retraceable sets must be recursive. Our main result, proved in §3, is that at least one member of any complementary pair of regresssive sets is recursively enumerable. This is a generalization of Mansfield's theorem, but the method of proof, in §3, is quite different. In §4, one of the two principal lemmas used by Mansfield is generalized, and some related material is developed, including an alternative derivation of the main theorem. In §5, we show that if the intersection of a pair of regressive sets is infinite then it has an infinite regressive subset. As a corollary to this last result, we prove that a coregressive hypersimple set is many-one incomparable with any hyperhypersimple set. 2. Definitions and notations. An infinite set a of numbers is regressive (see [l]) iff there exists a partial recursive function g, and a nonrepetitive listing bo, bx, ■■■ of the elements of a, such that (1) aEog, and (2) g(bo) = b0, (\/n)(g(bn+x) = bJ. (The notation 'b',as well as V,' + ','•','*', and '^', is used as in [1].) An infinite set a of numbers is said to be retraceable iff a is regressive, in the above sense, with 60,bx, •■■ specified to be the listing of a in order of magnitude. If a is regressive (retraceable) with respect to the listing b0, bx, ••• of its elements and the partial recursive function g, we say that g regresses (retraces) a, or is a regressing (retracing) function for a, and that b0 is the root of a under g. We shall freely use the results of [l] and [4]. In particular, we make use of the fact that if g regresses a with respect to the listing b0, bx, ■ ■ • of a, then there is a partial recursive subfunction g0 of g which also regresses a with respect to the listing b0, bx, ■ ■ • and is "special" in the sense that (a) pgo C ago, and (b) x G ôgo ==> (3 re) OfQ(x) = bQ). With the exception of our discussion of Theorem 3 in §4, whenever we make use of regressing functions they will be special in the above sense. Given a partial number-theoretic function g, we set g(x)=df[y\(ln)(gn(x)=y)].

Co-immune retraceable sets

“…This is closely related to the following general assertion : 1 We are indebted to Paul Young for a conversation which took place in August, 1963. At that time he made a suggestion which has proved to be susceptible of elaboration into proofs of Propositions A and B.…”

On the intersection of regressive sets

“…This verifies properties (1) and (3). For (2), assume deg/S ^ degδ. Then there would exist a total function F, partial recursive in the characteristic function of d, such that meβ<=*F(m) = 1.…”

The family of all recursively enumerable classes of finite sets

“…Some use will be made of the notation y*(x) (y an arbitrary one-place partial number-theoretic function); the meaning of this notation is that prescribed in [2, p. 81]. For some of the basic properties of special retracing functions and of the mapping y -> y*, the reader may consult [1], [2], and [7]. For any partial recursive function y such that x e 8y => y(x) £ x, we denote by KY the disjoint r.e.…”

“…Our first two lemmas serve as technical lubrication for the proof of Lemma C. Lemma A. There exists a recursive function £ such that (1) We^Fin => (pf(e) is a row-disjoint enumeration of a subfamily of Fin*, and (2) Wf e We n FR => the family enumerated by cp%(e) contains the class Wf.…”