Definable incompleteness and Friedberg splittings

Journal of Symbolic Logic 67 (2):679-696 (2002)
Abstract
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 E, yielding a strong answer to a question previously explored by Downey, Stob, and Soare about whether halves of Friedberg splittings must lie in the same orbit
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