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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...
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
"... ..."
The atomic model theorem and type omitting
 Trans. Amer. Math. Soc
"... We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than ..."
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Cited by 5 (1 self)
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We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Π1 1 conservative over RCA0. A priority argument with Shore blocking shows that it is also Π1 1conservative over BΣ2. We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ1 but implies IΣ2 over BΣ2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ωmodel consisting of the recursive sets.
ON STRONGLY JUMP TRACEABLE REALS
"... Abstract. In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing ..."
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Cited by 4 (0 self)
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Abstract. In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e. strongly jump traceables is Π 0 4complete. §1. Introduction. One of the fundamental concerns of computability theory is in understanding the relative difficulty of computational problems as measured by Turing reducubility (≤T). The equivalence classes of the preordering ≤T are called Turing degrees, and it is long recognized that the fundamental operator on the structure of the Turing degrees is the jump operator. For a set A, the
On the Structures Inside TruthTable Degrees
 Forschungsberichte Mathematische Logik 29 / 1997, Mathematisches Institut, Universitat
, 1997
"... . The following theorems on the structure inside nonrecursive truthtable degrees are established: Degtev's result that the number of bounded truthtable degrees inside a truthtable degree is at least two is improved by showing that this number is infinite. There are even infinite chains and anti ..."
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Cited by 4 (2 self)
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. The following theorems on the structure inside nonrecursive truthtable degrees are established: Degtev's result that the number of bounded truthtable degrees inside a truthtable degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truthtable degrees inside the truthtable degrees which implies an affirmative answer to a question of Jockusch whether every truthtable degree contains an infinite antichain of manyone degrees. Some but not all truthtable degrees have a least bounded truthtable degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmunefree truthtable degrees) which consist only of 2subjective sets and do therefore not contain any objective set. Furthermore a truthtable degree consisting of three positive degrees is constructed where one positive degree consists of enum...
Highness and Bounding Minimal Pairs
, 1993
"... We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high 2 nonbounding r.e. degree. 0 Introduction An important topic in the study of recursively enumerable sets and degrees has been the interaction between the jump operator and the order theoretic properties o ..."
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Cited by 3 (2 self)
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We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high 2 nonbounding r.e. degree. 0 Introduction An important topic in the study of recursively enumerable sets and degrees has been the interaction between the jump operator and the order theoretic properties of an r. e. set A (in the lattice E of all r. e. sets) and of its degree a in R, the upper semilattice of the r. e. degrees. An early theme in this area was the idea that sets with "low" jumps should behave like the recursive sets while those with "high" jumps should exhibit properties like the complete sets. For example, in the lattice E of r. e. sets modulo finite sets, we know from Soare[23] that if A is low, i. e. A 0 j T ; 0 , then L (A), the lattice of r. e. supersets of A, is isomorphic to E . In R there are many instances of the low Partially supported by IGC of Vicoria University, Wellington and ARO through MSI, Cornell University, DAAL03C0027. y Partailly suppo...
The atomic model theorem
"... We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than ..."
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Cited by 3 (1 self)
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We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Π1 1 conservative over RCA0. A priority argument with Shore blocking shows that it is also Π 1 1conservative over BΣ2. We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ1 but implies IΣ2 over BΣ2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ωmodel consisting of the recursive sets.
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
Jumps of minimal degrees below 0
 J. London Math. Soc
, 1996
"... Abstract. We show that there is a degree a REA in and low over 0 ′ such that no minimal degree below 0 ′ jumps to a degree above a. We also show that every nonlow r.e. degree bounds a nonlow minimal degree. Introduction. An important and longstanding area of investigation in recursion theory has be ..."
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Cited by 2 (0 self)
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Abstract. We show that there is a degree a REA in and low over 0 ′ such that no minimal degree below 0 ′ jumps to a degree above a. We also show that every nonlow r.e. degree bounds a nonlow minimal degree. Introduction. An important and longstanding area of investigation in recursion theory has been the relationship between quantifier complexity of the definitions of sets in arithmetic as expressed by the jump operator and the basic notion of relative computability as expressed by the ordering of the (Turing) degrees. In this paper we