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  1. Definability, Automorphisms, and Dynamic Properties of Computably Enumerable Sets.Leo Harrington & Robert I. Soare - 1996 - Bulletin of Symbolic Logic 2 (2):199-213.
    We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility).
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  • Definable Incompleteness and Friedberg Splittings.Russell Miller - 2002 - Journal of Symbolic Logic 67 (2):679-696.
    We define a property R(A 0 , A 1 ) in the partial order E of computably enumerable sets under inclusion, and prove that R implies that A 0 is noncomputable and incomplete. Moreover, the property is nonvacuous, and the A 0 and A 1 which we build satisfying R form a Friedberg splitting of their union A, with A 1 prompt and A promptly simple. We conclude that A 0 and A 1 lie in distinct orbits under automorphisms of (...)
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  • Orbits of Computably Enumerable Sets: Low Sets Can Avoid an Upper Cone.Russell Miller - 2002 - Annals of Pure and Applied Logic 118 (1-2):61-85.
    We investigate the orbit of a low computably enumerable set under automorphisms of the partial order of c.e. sets under inclusion. Given an arbitrary low c.e. set A and an arbitrary noncomputable c.e. set C, we use the New Extension Theorem of Soare to construct an automorphism of mapping A to a set B such that CTB. Thus, the orbit in of the low set A cannot be contained in the upper cone above C. This complements a result of Harrington, (...)
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  • Duality, Non-Standard Elements, and Dynamic Properties of R.E. Sets.V. Yu Shavrukov - 2016 - Annals of Pure and Applied Logic 167 (10):939-981.
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