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Incomputability in Nature
"... To what extent is incomputability relevant to the material Universe? We look at ways in which this question might be answered, and the extent to which the theory of computability, which grew out of the work of Godel, Church, Kleene and Turing, can contribute to a clear resolution of the current conf ..."
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Cited by 19 (11 self)
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To what extent is incomputability relevant to the material Universe? We look at ways in which this question might be answered, and the extent to which the theory of computability, which grew out of the work of Godel, Church, Kleene and Turing, can contribute to a clear resolution of the current confusion. It is hoped that the presentation will be accessible to the nonspecialist reader.
Effective presentability of Boolean algebras of CantorBendixson rank 1
 Journal of Symbolic Logic
, 1999
"... We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of CantorBendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even tho ..."
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Cited by 6 (6 self)
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We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of CantorBendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite CantorBendixson rank.
Generalized cohesiveness
 J. SYMBOLIC LOGIC
, 1997
"... We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey’s Theorem. An infinite set A of natural numbers is n–cohesive (respectively, n–r–cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2–c ..."
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Cited by 6 (3 self)
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We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey’s Theorem. An infinite set A of natural numbers is n–cohesive (respectively, n–r–cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2–coloring of the n–element sets of natural numbers. (Thus the 1–cohesive and 1–r–cohesive sets coincide with the cohesive and r–cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n–cohesive and n–r–cohesive sets. For example, we show that for all n ≥ 2, there exists a ∆ 0 n+1 n–cohesive set. We improve this result for n = 2 by showing that there is a Π0 2 2–cohesive set. We show that the n–cohesive and n–r–cohesive degrees together form a linear, non–collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n–cohesive degrees as exactly the degrees ≥ 0 (n+1) and show that each n–r–cohesive degree has jump> 0 (n).