MASS PROBLEMS AND HYPERARITHMETICITY

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let [Formula: see text] be the lattice of weak degrees of mass problems associated with nonempty [Formula: see text] subsets of the Cantor space. The lattice [Formula: see text] has been studied in previous publications. The purpose of this paper is to show that [Formula: see text] partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in [Formula: see text] which are indexed by the ordinal numbers less than [Formula: see text] and which correspond to the hyperarithmetical hierarchy. Namely, for each [Formula: see text], let hα be the weak degree of 0(α), the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p* be the weak degree of the mass problem P* = {Y | ∃X (X ∈ P and BLR (X) ⊆ BLR (Y))} where BLR (X) is the set of functions which are boundedly limit recursive in X. Let 1 be the top degree in [Formula: see text]. We prove that all of the weak degrees [Formula: see text], [Formula: see text], are distinct and belong to [Formula: see text]. In addition, we prove that certain index sets associated with [Formula: see text] are [Formula: see text] complete.

The isomorphism problem for computable Abelian p-groups of bounded length

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples.We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian p-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.

A hierarchy for the 1-section of any type two object

Shoenfield [6] constructed a hierarchy for any type two object F in which 2E is recursive by generalizing the hyperarithmetical hierarchy, using a jump operation jF defined by(A similar hierarchy was constructed independently by Hinman [3].) In order to construct a hierarchy for an arbitrary type two F we must first associate with F an operator which, on one hand will always be recursive in F, and on the other hand will generate all the functions recursive in F when iterated over a simultaneously generated set of ordinal notations OF. Clearly the use of 2E in the above definition of jF(f) can be avoided if, instead of diagonalizing F over all functions recursive in f, we only diagonalize F over, say, the functions primitive recursive in f. If furthermore we code in a function which enumerates all functions primitive recursive in f then the resulting operation will certainly generate a primitive recursively expanding hierarchy of functions recursive in F. The problem that remains is whether this hierarchy will exhaust the 1-section of F. But this reduces to an effectiyized version of the following problem: If g is recursive in some level of the hierarchy, is g primitive recursive in some higher level ? An affirmative answer is suggested by the completeness results of Feferman [2], and our main theorem below will be proved by combining his ideas with those of Shoenfield [6]. The result is a hierarchy which applies to all type two objects, and which replaces the notion of recursion in F by the simpler notion of primitive recursion in certain functions generated by F. Unfortunately, in contrast with the Shoenfield hierarchy, the hierarchy developed here cannot always be expected to have the uniqueness property (even w.r.t. ≤T), and for this reason the proof of our main theorem is rather more complicated than the corresponding proof in [6].

A note on the hyperarithmetical hierarchy

The hyperarithmetical hierarchy assigns a degree of unsolvability hγ to each constructive ordinal γ. This assignment has the properties that h0 is the recursive degree and hγ+1 is the jump h′γ of hγ. For a limit ordinal λ < ω1 it is not so easy to define hγ. The original definitions used systems of notations for ordinals, see Spector [6]. There are also later notation-free definitions due to Shoenfield (unpublished) and to Hensel and Putnam [4].

Recursive well-orderings

Cantor's second ordinal number class is perhaps the simplest example of a set of mathematical objects which cannot all be named in one language. In this paper we shall investigate a system of names for a segment of the first and second number classes in relation to decision problems. The system, except for one minor difference, is the one studied by Markwald in [12]. In our system ordinals are named by natural numbers from a set W via recursive well-orderings of subsets of the natural numbers.The decision problems will be related to the hyperarithmetical hierarchy of Davis [2], [3] and Kleene [8]. This hierarchy is indexed by ordinal notations from Kleene's system S3 [4], [6], [9], in which ordinals are named by natural numbers from a set O, partially well-ordered ([12] p. 138) by a relation a≤Ob; O and ≤O are defined inductively by applications of the successor and limit operations. As results of this investigation, we shall (i) answer negatively Markwald's question [12] Theorem 12 whether his set “W” is arithmetical by showing that it is not even hyperarithmetical, (ii) obtain a new proof of the main result of Kleene [10] that every predicate expressible in both the one-function-quantifier forms of [8] is recursive in Hα for some aεO, (iii) answer affirmatively the question raised by Davis [2], [3] whether all the Church-Kleene constructive ordinals are uniqueness ordinals, and (iv) solve the function-quantifier analog of Post's problem [15]. Strong use will be made of the well-orderings that can be constructed from one-function-quantifier predicates as in [9].