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The Turing Universe Is Not Rigid
"... A nontrivial automorphism of the Turing degrees is constructed. ..."
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A nontrivial automorphism of the Turing degrees is constructed.
Definability in the Enumeration Degrees
 Sacks Symposium
, 1997
"... We prove that every countable relation on the enumeration degrees, E, is uniformly definable from parameters in E. Consequently, the first order theory of E is recursively isomorphic to the second order theory of arithmetic. By an e#ective version of coding lemma, we show that the first order theory ..."
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We prove that every countable relation on the enumeration degrees, E, is uniformly definable from parameters in E. Consequently, the first order theory of E is recursively isomorphic to the second order theory of arithmetic. By an e#ective version of coding lemma, we show that the first order theory of the enumeration degrees of the # 0 2 sets is not decidable. 1 Introduction Definition 1.1 The following terms specify the variations on relative enumerability. . An recursive enumeration procedure U is a recursively enumerable collection of pairs #a, b# in which a and b are finite subsets of N. . U(A) is equal to B if B = {n : (#a)(#b)[a # A and #a, b# # U and n # b]} . . Say that B is enumeration reducible to A (A # e B) if there is a recursive enumeration procedure U such that U(A) = B. Similarly, A is enumeration equivalent to B (A # e B) if A # e B and B # e A. Definition 1.2 The enumeration degrees are defined as follows. . The enumeration degree of ...
Bounding and nonbounding minimal pairs in the enumeration degrees
 J. Symbolic Logic
"... Abstract. We show that every nonzero ∆ 0 2 edegree bounds a minimal pair. On the other hand, there exist Σ 0 2 edegrees which bound no minimal pair. 1. ..."
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Abstract. We show that every nonzero ∆ 0 2 edegree bounds a minimal pair. On the other hand, there exist Σ 0 2 edegrees which bound no minimal pair. 1.
GOODNESS IN THE ENUMERATION AND SINGLETON DEGREES
"... Abstract. We investigate and extend the notion of a good approximation with respect to the enumeration (De) and singleton (Ds) degrees. We refine two results by Griffith, on the inversion of the jump of sets with a good approximation, and we consider the relation between the double jump and index s ..."
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Abstract. We investigate and extend the notion of a good approximation with respect to the enumeration (De) and singleton (Ds) degrees. We refine two results by Griffith, on the inversion of the jump of sets with a good approximation, and we consider the relation between the double jump and index sets, in the context of enumeration reducibility. We study partial order embeddings ιs and ι̂s of, respectively, De and DT (the Turing degrees) into Ds, and we show that the image of DT under ι̂s is precisely the class of retraceable singleton degrees. We define the notion of a good enumeration, or singleton, degree to be the property of containing the set of good stages of some good approximation, and we show that ιs preserves the latter, as also other naturally arising properties such as that of totality or of being Γ0n, for Γ ∈ {Σ,Π,∆} and n> 0. We prove that the good enumeration and singleton degrees are immune and that the good Σ02 singleton degrees are hyperimmune. Finally we show that, for singleton degrees as < bs such that bs is good, any countable partial order can be embedded in the interval (as, bs). 1.
On the jump classes of noncuppable enumeration degrees
 the Journal of Symbolic Logic
"... Abstract. We prove that for every Σ02 enumeration degree b there exists a noncuppable Σ02 degree a> 0e such that b ′ ≤e a ′ and a′′≤e b′′. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding ..."
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Abstract. We prove that for every Σ02 enumeration degree b there exists a noncuppable Σ02 degree a> 0e such that b ′ ≤e a ′ and a′′≤e b′′. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding ι: DT → De, that there exist Σ02 noncuppable enumeration degrees at every possible—i.e. above low1—level of the high/low jump hierarchy in the context of De. 1.
Noncappable Enumeration Degrees Below ...
"... We prove that there exists a noncappable enumeration degree strictly below 0 0 e . ..."
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We prove that there exists a noncappable enumeration degree strictly below 0 0 e .
Relative Set Genericity
"... A set of natural numbers is generic relatively a set B if and only if it is the preimage of some set A using a Bgeneric Bregular enumeration such that both A and its complement are ereducible to B: ..."
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A set of natural numbers is generic relatively a set B if and only if it is the preimage of some set A using a Bgeneric Bregular enumeration such that both A and its complement are ereducible to B:
Decidability and Definability in the Σ 0 2Enumeration Degrees By
, 2005
"... Enumeration reducibility was introduced by Friedberg and Rogers in 1959 as a positive reducibility between sets. The enumeration degrees provide a wider context in which to view the Turing degrees by allowing us to use any set as an oracle instead of just total functions. However, in spite of the f ..."
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Enumeration reducibility was introduced by Friedberg and Rogers in 1959 as a positive reducibility between sets. The enumeration degrees provide a wider context in which to view the Turing degrees by allowing us to use any set as an oracle instead of just total functions. However, in spite of the fact that there are several applications of enumeration reducibility in computable mathematics, until recently relatively little research had been done in this area. In Chapter 2 of my thesis, I show that the ∀∃∀fragment of the first order theory of the Σ 0 2enumeration degrees is undecidable. I then show how this result actually demonstrates that the ∀∃∀theory of any substructure of the enumeration degrees which contains the ∆ 0 2degrees is undecidable. In Chapter 3, I present current research that Andrea Sorbi and I are engaged in, in regards to classifying properties of nonsplitting Σ 0 2degrees. In particular I give proofs that there is a properly Σ 0 2enumeration degree and that every ∆ 0 2enumeration degree bounds a nonsplitting ∆ 0 2degree.