Blog Archives

Topic Archive: effective genericity

NERDS: New England Recursion & Definability SeminarSunday, November 6, 20162:50 pmScience Center, Room E104‚Äč, Wellesley College, Wellesley, MA

Rose Weisshaar

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.