Relative Definability of $n$-Generics

Abstract

A set $G \subseteq \omega$ is $n$-generic for a positive integer $n$ if and only if every $\Sigma^0_n$ formula of $G$ is decided by a finite initial segment of $G$ in the sense of Cohen forcing. It is shown here that every $n$-generic set $G$ is properly $\Sigma^0_n$ in some $G$-recursive $X$. As a corollary, we also prove that for every $n > 1$ and every $n$-generic set $G$ there exists a $G$-recursive $X$ which is generalized ${\rm low}_n$ but not generalized ${\rm low}_{n-1}$. Thus we confirm two conjectures of Jockusch.