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doi:10.1112/jlms/jdm016 THE JUMP OF A ΣnCUT
"... Let n � 1. We study the prooftheoretic strength of jump classes of the Turing degrees from the point of view of fragments of Peano arithmetic. By investigating the jump of a Σn definable cut in a model of Δninduction, we show that over the base theory PA − +Δninduction, the existence of a nontri ..."
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Let n � 1. We study the prooftheoretic strength of jump classes of the Turing degrees from the point of view of fragments of Peano arithmetic. By investigating the jump of a Σn definable cut in a model of Δninduction, we show that over the base theory PA − +Δninduction, the existence of a nontrivial lown Turing degree is equivalent to Σninduction. 1.
ON THE QUOTIENT STRUCTURE OF COMPUTABLY ENUMERABLE DEGREES MODULO THE NONCUPPABLE IDEAL
"... Abstract. We show that minimal pairs exist in the quotient structure of R modulo the ideal of noncuppable degrees. In the study of mathematical structures it is very common to form quotient structures by identifying elements in some equivalence classes. By varying the equivalence relations, the corr ..."
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Abstract. We show that minimal pairs exist in the quotient structure of R modulo the ideal of noncuppable degrees. In the study of mathematical structures it is very common to form quotient structures by identifying elements in some equivalence classes. By varying the equivalence relations, the corresponding quotient structures often reveal certain hidden features of the original structure. In this paper, we focus on of the upper semilattice of computably enumerable degrees and the equivalence relations are induced by definable ideals. We begin with introducing some notations and terminologies. Let R be the class of computably enumerable degrees or simply c.e. degrees. Definition 1. We say that a nonempty subset I of R is an ideal of R if I is downward closed and closed under join. In other words, the following conditions are satisfied. (a) If a is in I and b ≤ a then b is in I; (b) If a and b are in I, then their least upper bound, denoted by a ∨ b, is in I. We say that an ideal I is definable if there is a firstorder formula ϕ(x) over the partial order language L = {≤} such that a c.e. degree a ∈ I if and only if R  = ϕ(a). Each ideal I of R naturally induced an equivalence relation ≡I as follows. For any two c.e. degrees a and b, define a ≤I b if and only if ∃x ∈ I(a ≤T b ∨ x),