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Definability in the Turing Degrees
 J. Symbolic Logic
, 1986
"... . Suppose that R is a countable relation on the Turing degrees. Then R can be defined in D, the Turing degrees with # T , by a first order formula with finitely many parameters. The parameters are built by means of a notion of forcing in which the conditions are essentially finite. The co ..."
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. Suppose that R is a countable relation on the Turing degrees. Then R can be defined in D, the Turing degrees with # T , by a first order formula with finitely many parameters. The parameters are built by means of a notion of forcing in which the conditions are essentially finite. The conditions in the forcing partial specify finite initial segments of the generic reals and impose a infinite constraint on further extensions. In section 3, this result is applied to show that any elementary function from D to D is an automorphism. Other applications are given toward the rigidity question for D. By observing that a single jump is all that is needed to meet the relevant dense sets, it is also shown that the recursively enumerable degrees can be defined from finitely many parameters in the structure consisting of the degrees below 0 # with # T . 1. Introduction Definability has provided the most fruitful approach to understanding the modeltheoretic structure ...
The theory of the degrees below 0
 J. London Math. Soc
, 1981
"... Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppers ..."
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Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppersemilattice. The local theory concerns
Degree theoretic definitions of the low_2 recursively enumerable sets
 J. SYMBOLIC LOGIC
, 1995
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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.
Low upper bounds of ideals
"... Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1. ..."
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Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1.
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 COMPUTABLY ENUMERABLE DEGREES
"... Abstract. We discuss the status of the problem of characterizing the finite (weak) lattices which can be embedded into the computably enumerable degrees. In particular, we summarize the current status of knowledge about the problem, provide an overview of how to prove these results, discuss directio ..."
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Abstract. We discuss the status of the problem of characterizing the finite (weak) lattices which can be embedded into the computably enumerable degrees. In particular, we summarize the current status of knowledge about the problem, provide an overview of how to prove these results, discuss directions which have been pursued to try to solve the problem, and present some related open questions. 1.
Notre Dame Journal of Formal Logic Embedding and Coding Below a 1Generic Degree
"... Abstract We show that the theory of D( � g), where g is a 2generic or a 1generic degree below 0 ′ , interprets true first order arithmetic. To this end we show that 1genericity is sufficient to find the parameters needed to code a set of degrees using Slaman and Woodin’s method of coding in Turin ..."
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Abstract We show that the theory of D( � g), where g is a 2generic or a 1generic degree below 0 ′ , interprets true first order arithmetic. To this end we show that 1genericity is sufficient to find the parameters needed to code a set of degrees using Slaman and Woodin’s method of coding in Turing degrees. We also prove that any recursive lattice can be embedded below a 1generic degree preserving top and bottom. 1