A one-generic that does not compute a modulus for one-genericity
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.)
Marcia Groszek is a professor on the faculty of Dartmouth College.