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Automorphisms of the lattice of recursively enumerable sets: Orbits, Adv
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to
Countable thin Π0 1 classes
 Annals of Pure and Applied Logic
, 1993
"... Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive CantorBendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is ..."
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Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive CantorBendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is shown that, for all ordinals α>1, no set of degree ≥ 0 ′ ′ can be a member of any thin Π0 1 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π0 1 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree (not necessarily r.e.) that contains a set which is of rank one in some thin Π0 1 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π0 1 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π0 1 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree 0 ′ ′. Introduction.
Degree theoretic definitions of the low_2 recursively enumerable sets
 J. SYMBOLIC LOGIC
, 1995
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The complexity of orbits of computably enumerable sets
 BULLETIN OF SYMBOLIC LOGIC
, 2008
"... The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; ..."
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The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E; for all finite α ≥ 9, there is a properly ∆0 α orbit (from the proof).
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.
Extending and Interpreting Post’s Programme
, 2008
"... Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic in ..."
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Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals which produced richness of local computabilitytheoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Posts programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Posts programme, and look at two interesting developments of Posts approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computabilitytheoretic infrastructure. We will conclude by viewing Posts programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world.
Sorbi A., Algebraic properties of Rogers semilattices of arithmetical numberings
 In Computability and Models, S.B. Cooper and S.S. Goncharov eds.—Kluwer
, 2003
"... Abstract We investigate initial segments and intervals of Rogers semilattices of arithmetical families. We prove that there exist intervals with different algebraic properties; the elementary theory of any Rogers semilattice at arithmetical level n ≥ 2 is hereditarily undecidable; the class of all R ..."
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Abstract We investigate initial segments and intervals of Rogers semilattices of arithmetical families. We prove that there exist intervals with different algebraic properties; the elementary theory of any Rogers semilattice at arithmetical level n ≥ 2 is hereditarily undecidable; the class of all Rogers semilattices of a fixed level n ≥ 2 has an incomplete theory. ∗All authors were partially supported by grant INTASRFBR Computability and Models no.
Immunity Properties and the nC.E. Hierarchy
 in Theory and Applications of Models of Computation, Third International Conference on Computation and Logic, TAMC 2006, Beijing, May 2006, Proceedings, (JinYi
"... Abstract. We extend Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets, and characterise, in the spirit of Post [9], the degrees of the immune and hyperimmune d.c.e. sets. We also show that no properly d.c.e. set can be hhimmune, and indicate how to generalise these results to nc ..."
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Cited by 2 (1 self)
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Abstract. We extend Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets, and characterise, in the spirit of Post [9], the degrees of the immune and hyperimmune d.c.e. sets. We also show that no properly d.c.e. set can be hhimmune, and indicate how to generalise these results to nc.e. sets, n> 2. 1
On Lachlan's major subdegree problem
 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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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