Comparing notions of effective genericity
Notre Dame University
In recent work, Cholak, Dzhafarov, Hirst and Slaman showed that for n ≥ 3, every Mathias n-generic computes a Cohen n-generic. It is natural to wonder how other types of generic objects compare to one another. We consider generics for an effective version of Hechler forcing. Adapting a method developed by Cholak, Dzhafarov, and Soskova, we show that for n ≥ 3, every Mathias n-generic computes a Hechler n-generic, and every Hechler n-generic computes a Mathias n-generic. Finally, we explore the (open) question of whether, for n ≥ 3, the Mathias n-generics and the Hechler n-generics occupy exactly the same Turing degrees.
Rose Weisshaar is a doctoral student at Notre Dame University, under the supervision of Peter Cholak.