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 Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α<ω CK 1 let hα be the weak degree of 0 (α),theαth Turing jump of 0. If p is the weak degree of any mass problem P,letp ∗ be the weak degree

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Abstract. Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated. §1. Introduction. This article concerns the algebraic study of the upper semilattice of Turing degrees. Upper semilattices of interest in this regard tend to have a least element, hence for convenience the following definition is made. Definition 1.1. A unital semilattice (usl) is a structure L = (L, ∗, e) satisfying the following equalities for all a, b, c ∈ L.

Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ω-c.e. approximation. 1.

We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular--- the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals

Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order theoretic properties of a degree and its complexity of definition in arithmetic as expressed by the Turing jump operator which embodies a single step in the hierarchy of quantification. For example, there is a long history of results showing that 0 ′

xperience, but simply as irreducible points comparable, epistemologically, to the gods of Homer.") Of course, the theory itself does indicate di#culties in substantiating the Turing model, but, if not overstretched (viz. the ubiquitous Godel's [15], [16] Theorem) such asymptotic representations can be useful and productive adjuncts to subjective intuition. For instance, unlike in mathematics where small variations in axioms can lead to fundamentally di#erent theories, Turing nonrigidity and known countable automorphism bases indicate that although diverse basic assumptions about the real world, related to culture or religion, for example, are inevitable (perhaps even necessary), relative to the Turing model there is a convergence at higher levels of the informational structure suggested by relative rigidity of substructures. The purpose of this note is to describe how, at a more basic level, the material Universe can be modelled according to the underlying structure of

Abstract. It is shown that for any computably enumerable (c.e.) degree w, if w � = 0, then there is a c.e. degree a such that (a ∨ w) ′ = a ′ ′ = 0 ′′ (so a is low2 and a ∨ w is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low2 c.e. degrees are not elementarily equivalent as partial orderings. 1.

We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.

We exhibit a structural difference between the truth-table degrees of the sets which are truth-table above 0 # and the PTIME-Turing degrees of all sets.