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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
1995], Degree theoretic definitions of the low 2 recursively enumerable sets
 J. Symbolic Logic
, 1995
"... 1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) mac ..."
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Cited by 7 (5 self)
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1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) machines, ϕe, can be used; access to
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.
Beyond Gödel's Theorem: Turing Nonrigidity Revisited
 In Logic Colloquium ’95
, 1998
"... 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 ..."
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Cited by 3 (3 self)
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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
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.