Countable thin Π01 classes

Annals of Pure and Applied Logic 59 (2):79-139 (1993)

Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that no set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree that contains a set which is of rank one in some thin Π01 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 Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 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”
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DOI 10.1016/0168-0072(93)90001-t
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References found in this work BETA

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Citations of this work BETA

Initial Segments of the Lattice of Π10 Classes.Douglas Cenzer & Andre Nies - 2001 - Journal of Symbolic Logic 66 (4):1749-1765.
Effectively Closed Sets and Enumerations.Paul Brodhead & Douglas Cenzer - 2008 - Archive for Mathematical Logic 46 (7-8):565-582.
On the Cantor-Bendixon Rank of Recursively Enumerable Sets.Peter Cholak & Rod Downey - 1993 - Journal of Symbolic Logic 58 (2):629-640.

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