Aspects of Incompleteness

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 tenth publication in the Lecture Notes in Logic series, Per Lindström presents some of the main topics and results in general metamathematics. In addition to standard results of Gödel et al. on incompleteness, (non-)finite axiomatizability, and interpretability, this book contains a thorough treatment of partial conservativity and degrees of interpretability. It comes complete with exercises, and will be useful as a textbook for graduate students with a background in logic, as well as a valuable resource for researchers. Contents CHAPTER 0. Introduction CHAPTER 1. Preliminaries Exercises Notes CHAPTER 2. Incompleteness §1. Incompleteness §2. Consistency statements §3. Independent formulas §4. The length of proofs Exercises Notes CHAPTER 3. Numerations of r.e. sets §1. Numerations of r.e. sets §2. Types of independence Exercises Notes CHAPTER 4. Axiomatizations §1. Finite and bounded axiomatizability; reflection principles §2. Irredundant axiomatizability Exercises Notes CHAPTER 5. Partial conservativity Exercises Notes CHAPTER 6. Interpretability §1. Interpretability §2. Faithful Interpretability Exercises Notes CHAPTER 7. Degrees of interpretabilty §1. Algebraic properties §2. A classification of degrees §3. Σ 1 and Π 1 degrees Exercises Notes CHAPTER 8. Generalizations §1. Incompleteness §2. Axiomatizations §3. Interpretability Notes REFERENCES INDEX NOTATION "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 tenth publication in the 'Lecture Notes in Logic' series, Per Lindstrom presents some of the main topics and results in general metamathematics. In addition to standard results of Godel et al. on incompleteness, (non-)finite axiomatizability, and interpretability, this book contains a thorough treatment of partial conservativity and degrees of interpretability. It comes complete with exercises, and will be useful as a textbook for graduate students with a background in logic, as well as a valuable resource for researchers."--Back cover 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 tenth publication in the 'Lecture Notes in Logic' series, Per Lindstrom presents some of the main topics and results in general metamathematics. In addition to standard results of Godel et al. on incompleteness, (non- )finite axiomatizability, and interpretability, this book contains a thorough treatment of partial conservativity and degrees of interpretability. It comes complete with exercises, and will be useful as a textbook for graduate students with a background in logic, as well as a valuable resource for researchers This book presents some of the main areas and results of general metamathematics. In addition to standard results of Gödel et al. on incompleteness, (non- )finite axiomatizability, interpretability, etc., it contains a thorough treatment of partial conservativity and degrees of interpretability. The method of arithmetization plays an all important role in the book. The reader is assumed to be acquainted with this method and with the elements of recursion theory. The book will be useful as a textbook for a graduate student course - there are many exercises - and as a reference text for researchers This volume presents some of the main areas and results of general metamathematics. In addition to standard results of Gödel et al. on incompleteness, (non-)finite axiomatizability, and interpretability, the book contains a thorough treatment of partial conservativity and degrees of interpretability. The method of arithmetization also plays an important role.