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The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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Global Properties of the Turing Degrees and the Turing Jump
"... We present a summary of the lectures delivered to the Institute for Mathematical Sciences, Singapore, during the 2005 Summer School in Mathematical Logic. The lectures covered topics on the global structure of the Turing degrees D, the countability of its automorphism group, and the definability of ..."
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We present a summary of the lectures delivered to the Institute for Mathematical Sciences, Singapore, during the 2005 Summer School in Mathematical Logic. The lectures covered topics on the global structure of the Turing degrees D, the countability of its automorphism group, and the definability of the Turing jump within D.
Conjectures and Questions from Gerald Sacks’s Degrees of Unsolvability
 Archive for Mathematical Logic
, 1993
"... 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, parti ..."
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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
Turing Incomparability in Scott Sets
 Proceedings of the American Mathematical Society
"... Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1. ..."
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Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1.
Double Jump Inversions and Strong Minimal Covers in the Turing Degrees
, 2004
"... Decidability problems for (fragments of) the theory of the structure D of Turing degrees, form a wide and interesting class, much of which is yet unsolved. Lachlan showed in 1968 that the first order theory of D with the Turing reducibility relation is undecidable. Later results concerned the decida ..."
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Decidability problems for (fragments of) the theory of the structure D of Turing degrees, form a wide and interesting class, much of which is yet unsolved. Lachlan showed in 1968 that the first order theory of D with the Turing reducibility relation is undecidable. Later results concerned the decidability (or undecidability) of fragments of this theory, and of other theories obtained by extending the language (e.g. with 0 or with the Turing jump operator). Proofs of these results often hinge on the ability to embed certain classes of structures (lattices, jumphierarchies, etc.) in certain ways, into the structure of Turing degrees. The first part of the dissertation presents two results which concern embeddings onto initial segments of D with known double jumps, in other words a double jump inversion of certain degree structures onto initial segments. These results may prove to be useful tools in uncovering decidability results for (fragments of) the theory of the Turing degrees in languages containing the double jump operator. The second part of the dissertation relates to the problem of characterizing the Turing degrees which have a strong minimal cover, an issue first raised by Spector in 1956. Ishmukhametov solved the problem for the recursively enumerable degrees, by showing that those which have a strong minimal cover are exactly the r.e. weakly recursive degrees. Here we show that this characterization fails outside the r.e. degrees, and also construct a minimal degree below 0 ′ which is not weakly recursive, thereby answering a question from Ishmukhametov’s paper.
1 Introduction Degrees of Unsolvability
, 2006
"... Modern computability theory began with Turing [Turing, 1936], where he introduced ..."
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Modern computability theory began with Turing [Turing, 1936], where he introduced
EMBEDDINGS INTO THE TURING DEGREES.
, 2007
"... The structure of the Turing degrees was introduced by Kleene and Post in 1954 [KP54]. Since then, its study has been central in the area of Computability Theory. One approach for analyzing the shape of this structure has been looking at the structures that can be ..."
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The structure of the Turing degrees was introduced by Kleene and Post in 1954 [KP54]. Since then, its study has been central in the area of Computability Theory. One approach for analyzing the shape of this structure has been looking at the structures that can be