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Maximal theories

Annals of Pure and Applied Logic 33 (C):245-282 (1987)

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  1. Effectively closed sets and enumerations.Paul Brodhead & Douglas Cenzer - 2008 - Archive for Mathematical Logic 46 (7-8):565-582.
    An effectively closed set, or ${\Pi^{0}_{1}}$ class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of ${\Pi^{0}_{1}}$ classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of ${\Pi^{0}_{1}}$ classes and (...)
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  • Sound, totally sound, and unsound recursive equivalence types.R. G. Downey - 1986 - Annals of Pure and Applied Logic 31:1-20.
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  • Undecidability of L(F∞) and other lattices of r.e. substructures.R. G. Downey - 1986 - Annals of Pure and Applied Logic 32:17-26.
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  • Countable thin Π01 classes.Douglas Cenzer, Rodney Downey, Carl Jockusch & Richard A. Shore - 1993 - Annals of Pure and Applied Logic 59 (2):79-139.
    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 (...)
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  • The upward closure of a perfect thin class.Rod Downey, Noam Greenberg & Joseph S. Miller - 2008 - Annals of Pure and Applied Logic 156 (1):51-58.
    There is a perfect thin class whose upward closure in the Turing degrees has full measure . Thus, in the Muchnik lattice of classes, the degree of 2-random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [S. Simpson, Mass problems and randomness, Bulletin of Symbolic Logic 11 1–27].
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  • Degrees containing members of thin Π10 classes are dense and co-dense.Rodney G. Downey, Guohua Wu & Yue Yang - 2018 - Journal of Mathematical Logic 18 (1):1850001.
    In [Countable thin [Formula: see text] classes, Ann. Pure Appl. Logic 59 79–139], Cenzer, Downey, Jockusch and Shore proved the density of degrees containing members of countable thin [Formula: see text] classes. In the same paper, Cenzer et al. also proved the existence of degrees containing no members of thin [Formula: see text] classes. We will prove in this paper that the c.e. degrees containing no members of thin [Formula: see text] classes are dense in the c.e. degrees. We will (...)
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  • (1 other version)Initial segments of the lattice of Π10 classes.Douglas Cenzer & Andre Nies - 2001 - Journal of Symbolic Logic 66 (4):1749-1765.
    We show that in the lattice E Π of Π 0 1 classes there are initial segments [ $\emptyset$ , P] = L(P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a Π 0 1 class P such that L is isomorphic to the lattice L(P)*, which is L(P), modulo finite differences. For the 2-element lattice, we obtain (...)
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