The Incompleteness Phenomenon: A New Course in Mathematical Logic

This book is a course in Mathematical Logic. It is divided into four chapters which can be taught in two semesters. The first two chapters provide a basic background in mathematical logic. All details are explained for students not so familiar with the abstract method used in mathematical logic. The last two chapters are more sophisticated, and here we assume that the reader will be able to fill in more details; in fact, this ability is an essential step for this sphere of mathematical thinking. Mathematical logic is the most abstract branch of mathematical thought, the most abstract human discipline. The main objective in this area is to understand the logic implicit in all mathematical thought. The difference between logic (considered as a branch of philosophy) and mathematical logic is that in mathematical logic we use and develop mathematical methods. That is, we use mathematical theorems to investigate and explain the logic implicit in mathematics. It should be clear that some of the results can also shed light on more general questions in epistemology and philosophy of science, but this is not the subject of this book. The main result in basic mathematical logic is that every "reasonable" mathematical system is intrinsically incomplete. This means that axiom systems cannot capture all semantical truths. This can also be expressed in the following way: If we assume that the human mind works in a way similar to an ideal computer, then there are mathematical problems which can never be solved by mathematicians. This is one aspect of Godel's famous incompleteness theorem, and the study of this phenomenon of incompleteness will be the main focus of this book. We think that the material of this book should be part of the basic background of every student in any discipline which employs deductive and formal reasoning as a part of its methodology. This definitely includes a large part of the social sciences.