Supremum in the Lattice of Interpretability [MSc Thesis]

TABLE OF CONTENTS ii
CHAPTER 1. Introduction 1
1. Background 2
2. Overview 3
3. Genesis 4
CHAPTER 2. Preliminaries 5
1. Provability Logic 5
2. Interpretability Logic 11
3. Degrees of Interpretability 18
CHAPTER 3. Uniform Suprema in Arithmetic 22
1. Implementations of the Supremum in PA 23
2. Arithmetical Preliminaries 25
3. A True but Unprovable Principle for the Supremum 28
4. ˇSvejdar’s Implementation of the Supremum 29
5. Visser’s Implementation of the Supremum 37
6. Table of Properties of u and f 45
CHAPTER 4. Semantics for ILMS 46
1. The Logic ILMS 46
2. Axiom S and Structural Properties of Models 47
3. Modest Modal Semantics 49
4. The Impossibility of a Structural Characterization 54
5. Quest for an Extension Lemma 59
6. Arithmetical Completeness for a Simple Language 62
CHAPTER 5. A Relational Semantics for f 64
1. Introducing the Semantics 64
2. Coping with Non-Monotonicity 66
3. Other Properties of ILMSf–Frames 67
4. A Problem 71
CHAPTER 6. Conclusions and Future Research 72
1. Summary 72
2. Questions for Future Research 72
Appendix A. Modal Completeness of ILM by the Construction Method 75
1. The System ILM (Remainder) 75
2. Modal Completeness: Introduction 76
3. Preparing the Construction 76
4. Overview 78
5. Quasi–Frames 79
6. Elimination of Problems and Deficiencies 81
7. Rounding up 83
Bibliography 84