ABSTRACT
Gödel's Incompleteness Theorems are among the most significant results in the foundation of mathematics. These results have a positive consequence: any system of axioms for mathematics that we recognize as correct can be properly extended by adding as a new axiom a formal statement expressing that the original system is consistent. This suggests that our mathematical knowledge is inexhaustible, an essentially philosophical topic to which this book is devoted. Basic material in predicate logic, set theory and recursion theory is presented, leading to a proof of incompleteness theorems. The inexhaustibility of mathematical knowledge is treated based on the concept of transfinite progressions of theories as conceived by Turing and Feferman. All concepts and results necessary to understand the arguments are introduced as needed, making the presentation self-contained and thorough.
CHAPTER l. INTRODUCTION . CHAPTER 2. ARITHMETICAL PRELIMINARIES CHAPTER 3. PRIMES AND PROOFS . CHAPTER 4. THE LANGUAGE OF ARITHMETIC CHAPTER 5. THE LANGUAGE OF ANALYSIS CHAPTER 6. ORDINALS AND INDUCTIVE DEFINITIONS. CHAPTER 7. FORMAL LANGUAGES AND THE DEFINITION OF TRUTH CHAPTER 8. LOGICC AND THEORIES. CHAPTER 9. PEA.NO ARTHMETIC AND COMPUTABILl1Y CHAPTER 9. PEA.NO ARTHMETIC AND COMPUTABILl1Y CHAPTER 10. ELEMENTARY AND CLASSICAL ANALYSIS CHAPTER 11. THE RECURSION THEOREM AND ORDINAL NOTATIONS CHAPTER 12. THE INCOMPLETENESS THEOREMS CHAPTER 13. ITERATED CONSISTENCY CHAPTER 14. ITERATED REFLECTION CHAPTER 15. ITERATED ITERATION AND INEXHAUSTIBILITY
