Blog Archives

Topic Archive: 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.

NERDS: New England Recursion & Definability SeminarSaturday, April 2, 201610:45 amLocklin Hall, Room 232, Springfield College

A one-generic that does not compute a modulus for one-genericity

Dartmouth College

A function g ∈ 2ω is 1-generic if, for every recursively enumerable W⊂ 2< ω, there is some σ⊂ g such that either σ∈W or σ has no extensions in W. That is, any possible Σ01 property of g is either forced to be true or forced to be false by some finite initial segment of g. A function f∈ωω is a modulus for 1-genericity if, whenever h pointwise dominates f, then h computes some 1-generic. If g is 1-generic and either Δ02-definable or 2-generic, then g computes a modulus for 1-genericity. We give a priority argument to produce a 1-generic g that does not compute a modulus for 1-genericity.

(Joint work with Theodore A. Slaman.)