Indifferent sets for genericity

Journal of Symbolic Logic 78 (1):113-138 (2013)
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Abstract

This paper investigates indifferent sets for comeager classes in Cantor space focusing of the class of all 1-generic sets and the class of all weakly 1-generic sets. Jockusch and Posner showed that there exist 1-generic sets that have indifferent sets [10]. Figueira, Miller and Nies have studied indifferent sets for randomness and other notions [7]. We show that any comeager class in Cantor space contains a comeager class with a universal indifferent set. A forcing construction is used to show that any 1-generic set, or weakly 1-generic set, has an indifferent set. Such an indifferent set can by computed by any set in GL2 which bounds the (weakly) 1-generic. We show by approximation arguments that some, but not all, ∆0 1-generic sets can compute an indifferent set 2 for themselves. We show that all ∆0 weakly 1-generic sets can compute 2 an indifferent set for themselves. Additional results on indifferent sets, including one of Miller, and two of Fitzgerald, are presented.

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A Hierarchy of Computably Enumerable Degrees.Rod Downey & Noam Greenberg - 2018 - Bulletin of Symbolic Logic 24 (1):53-89.

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