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Parameter Definability in the Recursively Enumerable Degrees
"... 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 definabl ..."
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Cited by 34 (13 self)
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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...
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 8 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Degree structures: Local and global investigations
 Bulletin of Symbolic Logic
"... $1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead. ..."
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Cited by 6 (2 self)
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$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.
Natural Definability in Degree Structures
"... . 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 ..."
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Cited by 5 (1 self)
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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 ordertheoretically 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...
A Splitting Theorem for nREA Degrees
"... We prove that, for any D, A and U with D > T A # U and r.e. in A# U , there are pairs X 0 , X 1 and Y 0 , Y 1 such that D # T X 0 #X 1 ; D # T Y 0 # Y 1 ; and, for any i and j from {0, 1} and any set B, if X i #A # T B and Y j # A # T B then A # T B. We then deduce that for any degrees d, a, and b s ..."
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Cited by 5 (5 self)
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We prove that, for any D, A and U with D > T A # U and r.e. in A# U , there are pairs X 0 , X 1 and Y 0 , Y 1 such that D # T X 0 #X 1 ; D # T Y 0 # Y 1 ; and, for any i and j from {0, 1} and any set B, if X i #A # T B and Y j # A # T B then A # T B. We then deduce that for any degrees d, a, and b such that a and b are recursive in d, a ## T b, and d is nREA in to a, d can be split over a avoiding b. This shows that the Main Theorem of Cooper [1990] and [1993] is false.
Direct and local definitions of the Turing jump
, 2008
"... We show that there are 5 formulas in the language of the Turing degrees, D, ..."
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Cited by 1 (1 self)
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We show that there are 5 formulas in the language of the Turing degrees, D,
Differences of Computably Enumerable Sets
, 1999
"... We consider the lower semilattice D of differences of c.e. sets under inclusion. It is shown that D is not distributive as a semilattice, and that the c.e. sets form a definable subclass. 1 Introduction A persistent open problem about the lattice E of computably enumerable (c.e.) sets under inclusi ..."
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We consider the lower semilattice D of differences of c.e. sets under inclusion. It is shown that D is not distributive as a semilattice, and that the c.e. sets form a definable subclass. 1 Introduction A persistent open problem about the lattice E of computably enumerable (c.e.) sets under inclusion is to determine the least number k such that the \Sigma k theory is undecidable. Lachlan [6] proved that the \Sigma 2 theory in the language of lattices is decidable, while one of the various known proofs of undecidability for Th(E), in that case due to Harrington, shows that in fact the \Sigma 8 theory in the language of lattices is undecidable (see [10], p. 381 for a sketch of that proof). Thus a very unsatisfying gap of 6 quantifier alternations remains. The reason why the undecidability proofs are so "bad" is that the coding used is very indirect. For instance, first one codes the class of finite symmetric graphs (which has an hereditarily undecidable \Sigma 2 theory) in the cl...