Genericity in the enumeration degrees

In this thesis we study the notion of enumeration 1-genericity, various basic properties of it and its relationship with 1-genericity. We also study the problem of avoiding uniformity in the �02 enumeration degrees. In Chapter 2 we give a brief background survey of the notion of genericity in the context of the Turing degrees as well as in the enumeration degrees. Chapter 3 presents a brief overview of the relationship between noncupping and genericity in the enumeration degrees. We give a result that will be useful in proving the existence of prime ideals of �02 enumeration degrees in Chapter 5, namely, we show the existence of a 1-generic enumeration degree 0e < a < 00 e which is noncuppable and low2. In Chapter 4 we investigate the property of incomparability relative to a class of degrees of a speci�c level of the Arithmetical Hierarchy. We show that for every uniform �02 class of enumeration degrees C, there exists a high �02 enumeration degree c which is incomparable with any degree b 2 C such that b =2 f0e; 00 e g. Chapter 5 is devoted to the introduction of the notions of \enumeration 1- genericity" and \symmetric enumeration 1-genericity". We study the distribution of the enumeration 1-generic degrees and show that it resembles to some extent the distribution of the class of 1-generic degrees. We also present an application of enumeration 1-genericity to show the existence of prime ideals of �02 enumeration degrees. We then look at the relationship between enumeration 1-genericity and highness. Finally, in Chapter 6 we present two di�erent approaches to the problem of separating the class of the enumeration 1-generic degrees from the class of 1-generic degrees. One of them is by showing the existence of a non trivial enumeration 1- generic set which is not 1-generic and the other is by proving that there exists a property that both classes do not share, namely, nonsplitting.