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Generalized high degrees have the complementation property
 Journal of Symbolic Logic
"... Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order ..."
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Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order theoretic properties of a degree and its complexity of definition in arithmetic as expressed by the Turing jump operator which embodies a single step in the hierarchy of quantification. For example, there is a long history of results showing that 0 ′
Noncappable Enumeration Degrees Below 0
"... We prove that there exists a noncappable enumeration degree strictly below 0 0 e . Two notions of relative computability, Turing and enumeration reducibility, are basic to any natural finestructure theory for the classes of computable and incomputable objects. Of the theories for the correspondin ..."
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We prove that there exists a noncappable enumeration degree strictly below 0 0 e . Two notions of relative computability, Turing and enumeration reducibility, are basic to any natural finestructure theory for the classes of computable and incomputable objects. Of the theories for the corresponding degree structures (D D D and D D D e ), that for the Turing degrees is the better developed, mainly due to the depth of knowledge of specific local structure (see for example Lerman [Le83], Odifreddi [Od89] and Soare [So87]). Despite its importance (see [Co90]) in applications to nondeterministic computations, to relative computability involving partial information, in providing models of calculus, and in setting 1991 Mathematics Subject Classification. 03D30. Key words and phrases. Enumeration operator, enumeration degree, \Sigma 2 set. Research partially supported by British CouncilMURST grant no. ROM/889 /92/81, S.E.R.C. Research Grant no. GR/H 02165 and EC Human Capital and Mobili...
DISCONTINUOUS PHENOMENA
"... ABSTRACT. We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined. 1. ..."
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ABSTRACT. We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined. 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 .