Introduction

We consider genericity in the context of arithmetic. A set A ⊆ ε ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤ a) denote the set of degrees which are recursive in a. Since the set of n-generic sets is comeager, if some property is satisfied in D(≤ a) with a any generic degree, then in the sense of Baire category, we can say that it is satisfied in D(≤ a) for almost every degree a. So the structure of generic degrees plays an important role when we study the structure of D, the set of all degrees. For example, Slaman and Woodin [38] showed that there is a generic degree a such that if f is an automorphism of D and f(a) = a then f is identity. In this paper we mainly survey D(≤ a) when a is n-generic, as well as the properties of generic degrees in D. We assume the reader is familiar with the basic results of degree theory and arithmetical forcing. Feferman [4], Hinman [8], Hinman [9], and Lerman [25] are good references in this area. Odifreddi [29] is a good survey for basic notions and results for forcing and reducibilities. Jockusch [11] is a pioneering work in this area.