The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...

The primary notion of effective computability is that provided by Turing machines (or equivalently any of the other common models of computation). We denote the partial function computed by the eth Turing machine in some standard list by # e . When these machines are equipped with an "oracle" for a subset A of the natural numbers #, i.e. an external procedure that answers questions of the form "is n in A", they define the basic notion of relative computability or Turing reducibility (from Turing (1939)). We say that A is computable from (or recursive in) B if there is a Turing machine which, when equipped with an oracle for B, computes (the characteristic function of) A, i.e. for some e, # B e = A. We denote this relation by A # T<F10

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. A major focus of research in computability theory in recent years has involved denability issues in degree structures. There has been much success in getting general results by coding methods that translate rst or second order arithmetic into the structures. In this paper we concentrate on the issues of getting denitions of interesting, apparently external, relations on degrees that are order-theoretically natural in the structures D and R of all the Turing degrees and of the r.e. Turing degrees, respectively. Of course, we have no formal denition of natural but we oer some guidelines, examples and suggestions for further research. 1. Introduction A major focus of research in computability theory in recent years has involved denability issues in degree structures. The basic question is, which interesting apparently external relations on degrees can actually be dened in the structures themselves, that is, in the rst order language with the single fundamental relation...

$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.

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

We prove definability results for the structure R T of computably enumerable Turing degrees. Some of the results can be viewed as approximations to an affirmative answer for the biinterpretability conjecture in parameters for. For instance, all uniformly computably enumerable sets of nonzero c.e. Turing degrees can be defined from parameters by a fixed formula. This implies that the finite subsets are uniformly definable. As a consequence we obtain a new ;-definable ideal, and all arithmetical ideals are parameter definable. 1 Introduction Let R T denote the upper semilattice of computably enumerable (c.e.) Turing degrees. We are concerned with definability results in R T which can be viewed as approximations to the biinterpretability conjecture for R T . The biinterpretability conjecture in parameters for an arithmetical structure A (in brief, BI-conjecture) states that there is a parameter defined copy M of (N; +; \Theta) and a parameter definable 1-1 map f : M 7! A. This h...

Slaman and Woodin have developed and used set-theoretic methods to prove some remarkable theorems about automorphisms of, and de…nability in, the Turing degrees. Their methods apply to other coarser degree structures as well and, as they point out, give even stronger results for some of them. In particular, their methods can be used to show that the hyperarithmetic degrees are rigid and biinterpretable with second order arithmetic. We give a direct proof using only older coding style arguments to prove these results without any appeal to set-theoretic or metamathematical considerations. Our methods also apply to various coarser reducibilities. 1

Gödel's work [Gö34] on undecidable theories and the subsequent formalisations of the notion of a recursive function ([Tu36], [K136] etc.) have led to an ever deepening understanding of the nature of the non-computable universe (which as Gödel himself showed, includes sets and functions of everyday significance). The nontrivial aspect of Church's Thesis (any function not contained within one of the equivalent definitions of recursive/Turing computable, cannot be considered to be effectively computable) still provides a basis not only for classical and generalised recursion theory, but also for contemporary theoretical computer science. Recent years, in parallel with the massive increase in interest in the computable universe and the development of much subtler concepts of 'practically computable', have seen remarkable progress with some of the most basic and challenging questions concerning the non-computable universe, results both of philosophical significance and of potentially wider technical importance. Relativising Church's Thesis, Kleene and Post [KP54] proposed the now

Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK