Minimal Covers and Hyperdegrees

Abstract

ABSTRACT. Every hyperdegree at or above that of Kleene's O is the hyperjump and the supremum of two minimal hyperdegrees (Theorem 3.1). There is a nonempty ~ZX class of number-theoretic predicates each of which has minimal hyperdegree (Theorem 4.7). If V = L or a generic extension of L, then there are arbitrarily large hyperdegrees which are not minimal over any hyperdegree (Theorems 5.1, 5.2). If <?# exists, then there is a hyperdegree such that every larger hyperdegree is minimal over some hyperdegree (Theorem 5.4). Several other theorems on hyperdegrees and degrees of nonconstructibility are presented.1. Introduction. In this paper are proved several results concerning hyperdegrees. The methods of proof in § §3 and 4 are based on the methods of Gandy and Sacks [9]. In § §5 and 7 some ideas of modern set theory are applied. The significance of the results is discussed in §6.In this introductory section, the contents of the paper are summarized and compared to some of the recent literature on Turing degrees. A Turing degree m is said to be minimal if m > 0 and there is no Turing degree strictly between m and 0. S. B. Cooper [3], [4] has investigated minimal Turing degrees and has proved the following two theorems. Let 0' be the complete r.e. Turing degree.(1) 0' is the supremum of two minimal Turing degrees. (2) Every Turing degree > 0' is the jump of a minimal Turing degree. The proofs of (1) and (2) involve delicate applications of the priority method. In §3 below is proved a theorem for hyperdegrees which has as corollaries the hyperdegree analogs of (1) and (2). It is found that, once the basic techniques from Gandy-Sacks [9] have been mastered, the proof of the hyperdegree analogs is much easier than the proofs of (1) and (2). In particular, the priority method is not used for the hyperdegree analogs. Pursuant to Cooper's theorem (2), L. P. Sasso [28](2) has

Jumps of Minimal Degrees Below 0 ′

“…Simultaneously build two sets A 0 , A 1 of minimal degree as above but use the narrow subtrees in each construction to code the final result of the other in an interleaved way as in Simpson [1975]. Suppose the sequences of trees for A 0 and A 1 at stage s are T 0 0,s , .…”

Forcing and reductibilities. II. Forcing in fragments of analysis

“…• Similarly, Simpson [28] has proved that if a > 0' then there are b, c minimal hyperdegrees such that b' ~ c' = b [j c = a.…”

Basis Theorems for -Sets

“…Then for any x ∈ L, every Σ 1 2 (x) set is Σ 1 2 (x ) for some . We use an idea in Simpson [11] as presented in Chong and Yu [2] to construct A. For any perfect tree T , we use f T : 2 < → T to denote the canonical homeomorphism from 2 to [T ].…”