Degrees of Unsolvability Local and Global Theory [Hardcover]

This volume presents a systematic study of the interaction between local and global degree theory.This volume presents a systematic study of the interaction between local and global degree theory. It introduces the reader to the fascinating combinatorial methods of recursion theory while simultaneously showing how to use these methods to prove global theorems about degrees.This volume presents a systematic study of the interaction between local and global degree theory. It introduces the reader to the fascinating combinatorial methods of recursion theory while simultaneously showing how to use these methods to prove global theorems about degrees.Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the eleventh publication in the Perspectives in Logic series, Manuel Lerman presents a systematic study of the interaction between local and global degree theory. He introduces the reader to the fascinating combinatorial methods of recursion theory while simultaneously showing how to use these methods to prove global theorems about degrees. The intended reader will have already taken a graduate-level course in recursion theory, but this book will also be accessible to those with some background in mathematical logic and a feeling for computability. It will prove a key reference to enable readers to easily locate facts about degrees and it will direct them to further results.Introduction; Part I. The Structure of the Degrees: 1. Recursive functions; 2. Embeddings and extensions of embeddings in the degrees; 3. The jump operator; 4. High/low hierarchies; Part II. Countable Ideals of Degrees: 5. Minimal degrees; 6. Finite distributive lattices; 7. Finite lattices; 8. Countable usls; Part III. Initial Segments ofD and the Jump Operator: 9. Minimal degrees and high/low hielã"