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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...
Uniform almost everywhere domination
 Journal of Symbolic Logic
, 2006
"... ABSTRACT. We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed t ..."
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Cited by 25 (1 self)
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ABSTRACT. We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the prooftheoretic strength of the regularity of Lebesgue measure for G δ sets. Our constructions essentially settle the reverse mathematical classification of this principle. 1.
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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Cited by 19 (3 self)
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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 D(≤, ∨, ′ ) is undecidable
 In Proceedings of Logic Colloquium
, 2003
"... We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable. ..."
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Cited by 2 (0 self)
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We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.
On Lachlan's major subdegree problem, to
 in: Set Theory and the Continuum, Proceedings of Workshop on Set Theory and the Continuum
, 1989
"... The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become a longstanding open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the ..."
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Cited by 2 (2 self)
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The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become a longstanding open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b> a if for any c.e. degree x, 0 ′ = b ∨ x if and only if 0 ′ = a ∨ x. In this paper, we show that every c.e. degree b ̸ = 0 or 0 ′ has a major subdegree, answering Lachlan’s question affirmatively. 1
PSEUDOJUMP INVERSION AND SJTHARD SETS
, 2011
"... Abstract. There are noncomputable c.e. sets, computable from every SJThard c.e. set. This yields a natural pseudojump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone. 1. ..."
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Cited by 1 (1 self)
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Abstract. There are noncomputable c.e. sets, computable from every SJThard c.e. set. This yields a natural pseudojump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone. 1.
Index sets for . . .
, 1997
"... ... class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of ..."
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... class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of