# Blog Archives

# Topic Archive: genericity

# Comparing notions of effective genericity

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.

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

A function *g ∈ 2 ^{ω}* is 1-generic if, for every recursively enumerable

*W⊂ 2*, there is some

^{< ω}*σ⊂ g*such that either

*σ∈W*or σ has no extensions in

*W*. That is, any possible Σ

^{0}

_{1}property of

*g*is either forced to be true or forced to be false by some finite initial segment of

*g*. A function

*f∈ω*is a modulus for 1-genericity if, whenever

^{ω}*h*pointwise dominates

*f*, then

*h*computes some 1-generic. If

*g*is 1-generic and either Δ

^{0}

_{2}-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.)