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Atomless rmaximal sets
 Israel J. Math
"... Abstract. We focus on L(A), the filter of supersets of A in the structure of the computably enumerable sets under the inclusion relation, where A is an atomless rmaximal set. We answer a long standing question by showing that there are infinitely many pairwise nonisomorphic filters of this type. 1 ..."
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Abstract. We focus on L(A), the filter of supersets of A in the structure of the computably enumerable sets under the inclusion relation, where A is an atomless rmaximal set. We answer a long standing question by showing that there are infinitely many pairwise nonisomorphic filters of this type. 1.
ASYMPTOTIC DENSITY AND THE COARSE COMPUTABILITY BOUND
"... Abstract. For r ∈ [0, 1] we say that a set A ⊆ ω is coarsely computable at density r if there is a computable set C such that {n: C(n) = A(n)} has lower density at least r. Let γ(A) = sup{r: A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibilit ..."
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Abstract. For r ∈ [0, 1] we say that a set A ⊆ ω is coarsely computable at density r if there is a computable set C such that {n: C(n) = A(n)} has lower density at least r. Let γ(A) = sup{r: A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibility. For example, we show that if r ∈ (0, 1] there are sets A0, A1 such that γ(A0) = γ(A1) = r where A0 is coarsely computable at density r while A1 is not coarsely computable at density r. We show that a real r ∈ [0, 1] is equal to γ(A) for some c.e. set A if and only if r is leftΣ03. A surprising result is that if G is a ∆02 1generic set, and A 6T G with γ(A) = 1, then A is coarsely computable at density 1. 1.
LIMITS ON JUMP INVERSION FOR STRONG REDUCIBILITIES
"... Abstract. We show that Sacks ’ and Shoenfield’s analogs of jump inversion fail for both tt and wttreducibilities in a strong way. In particular we show that there is a ∆0 2 set B>tt ∅ ′ such that there is no c.e. set A with A ′ ≡wtt B. We also show that there is a Σ0 2 set C>tt ∅ ′ such that ..."
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Abstract. We show that Sacks ’ and Shoenfield’s analogs of jump inversion fail for both tt and wttreducibilities in a strong way. In particular we show that there is a ∆0 2 set B>tt ∅ ′ such that there is no c.e. set A with A ′ ≡wtt B. We also show that there is a Σ0 2 set C>tt ∅ ′ such that there is no ∆0 2 set D with D ′ ≡wtt C. 1.
Filters on computable posets
 In preparation
, 2006
"... We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a ∆02 maximal filter, and there is ..."
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We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a ∆02 maximal filter, and there is a computable poset with no Π01 or Σ 0 1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result that the principle “every countable poset has a maximal filter ” is equivalent to ACA0 over RCA0. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a Σ01 unbounded filter, and there is a computable poset with no Π 0 1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle “every countable poset has an unbounded filter ” is equivalent to ACA0 over RCA0. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets. 1
A ... Set With No Infinite Low Subset in . . .
, 2000
"... We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the prooftheoretic strength and eective content of versions of Ramsey's Theorem. ..."
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We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the prooftheoretic strength and eective content of versions of Ramsey's Theorem.
Interpreting N in the computably enumerable weak truth table degrees
"... We give a firstorder coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an im ..."
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We give a firstorder coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an important further problem is the question of the actual complexity of the theory. If the structure is arithmetical, then its theory can be interpreted in true arithmetic, i.e. Th(N; +; \Theta). Thus an upper bound is ; (!) , the complexity of Th(N; +; \Theta). Here an interpretation of theories is a manyone reduction based on a computable map defined on sentences in some natural way. An example of an arithmetical structure is D T ( ; 0 ), the Turing degrees of \Delta 0 2 sets. Shore [16] proved that true arithmetic can be interpreted in Th(D T ( ; 0 )). A stronger result is interpretability without parameters of a copy of (N; +; \Theta) in the structure (interpretability of struc...
MARTIN’S AXIOM AND EMBEDDINGS OF UPPER SEMILATTICES INTO THE TURING DEGREES
"... Abstract. It is shown that every locally countable upper semilattice of cardinality continuum can be embedded into the Turing degrees, assuming Martin’s Axiom. 1. ..."
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Abstract. It is shown that every locally countable upper semilattice of cardinality continuum can be embedded into the Turing degrees, assuming Martin’s Axiom. 1.
ON DOWNEY’S CONJECTURE MARAT M. ARSLANOV, ISKANDER SH. KALIMULLIN,
"... Abstract. We prove that the degree structures of the d.c.e. and the 3c.e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey. More specifically, we show that the following statement fails in the former but holds in the latter structure: There are degrees f> e&g ..."
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Abstract. We prove that the degree structures of the d.c.e. and the 3c.e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey. More specifically, we show that the following statement fails in the former but holds in the latter structure: There are degrees f> e> d> 0 such that any degree u ≤ f is either comparable with both e and d, or incomparable with both. 1. The Theorems In 1965, Putnam [Pu65] defined the nc.e. sets as a generalization of the c.e. (or computably enumerable) sets: Definition 1. Given an integer n> 0, we call a set A ⊆ ω nc.e. if there is a computable sequence of sets {As}s∈ω such that for all x ∈ ω, A0(x) = 0, A(x) = limsAs(x), and {s ∈ ω  As(x) 6 = As+1(x)}. (Note that a c.e. set is thus simply a 1c.e. set; and a 2c.e. set is a d.c.e. set, i.e., a difference of two c.e. sets.) These sets were first extensively studied (and extended to the αc.e. sets for computable ordinals α) by Ershov [Er68a, Er68b, Er70] and are nowadays often said to form the Ershov hierarchy, which stratifies the ∆02sets. The nc.e. degrees, i.e., the Turing degrees of the nc.e. sets, were first investigated by Lachlan (late 1960’s, unpublished), who showed that for any nc.e. degree d> 0, there is a c.e. degree a with d> a> 0, and
NONCUPPING AND RANDOMNESS
"... Abstract. Let Y ∈ ∆0 2 be MartinLöfrandom. Then there is a promptly simple set A such that, for each MartinLöfrandom set Z, Y ≤T A ⊕ Z ⇒ Y ≤T Z. When Y = Ω, one obtains a c.e. noncomputable set A which is not weakly MartinLöfcuppable. That is, for any MartinLöfrandom set Z, if ∅ ′ ≤T A ⊕ Z ..."
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Abstract. Let Y ∈ ∆0 2 be MartinLöfrandom. Then there is a promptly simple set A such that, for each MartinLöfrandom set Z, Y ≤T A ⊕ Z ⇒ Y ≤T Z. When Y = Ω, one obtains a c.e. noncomputable set A which is not weakly MartinLöfcuppable. That is, for any MartinLöfrandom set Z, if ∅ ′ ≤T A ⊕ Z then ∅ ′ ≤T Z. 1.