Preliminaries 1. Π11-uniformization and Applications to Turing Degrees 2. Rigidity of Hyperdegrees 3. Basis Theorems and Π11-Hyperarithmetic 4. The Jump Operator 5. Independence Results in the Turing Degrees 6. Higher Randomness References Index

"This book serves two purposes, and does so very well. In Part I, it provides an exposition of the now-classical theory of definability in first-order arithmetic, in the form of the arithmetic hierarchy, and in second-order arithmetic, in the form of effective descriptive set theory. This part would be a good source on which to base a graduate course on this material. In Parts II–IV, by giving a coherent and systematic treatment spanning many modern examples, it illustrates how these classical ideas have evolved into powerful mathematical tools, which is valuable both to newcomers and to experts." Mathematical Reviews "This is a very well written book by researchers who contributed with significant results to the field, the treatment is mathematical rigourous, with important open problems, and an up-dated list of references. The book is suited for advanced courses and research." Zentralblatt für Mathematik