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STUDIES IN PROOF THEORY 1

This book is intended to be an introduction to proof-theory. Starting with Hilbert, whose second problem (in the famous list of problems of 1900) was the consistency of number theory, we describe Hilbert’s program and its demise at the hands of Gödel. We then proceed with Gentzen's results (the Hauptsatz is the main result of part I, i.e. chapters 2, 3, 4) and their improvements and generalizations to ω -logic (chapters 5, 6, 7) and to more «advanced» logics (chapters 8-12). I have tried to cover the main ideas and tecniques of proof-theory. We start with Gödel’s results as our point of departure (which are proven in most good texts, and presumably not better presented here), and we arrive close to the analysis of the –comprehension schema which is, at the time we are writing this introduction (1982) the most advanced technique within the subject. I have tried to include here alI important results concerning ω-logic (and with these results as a basis, the enormous literature in the field is accessible), except what is concerned with systems of notations: I think that these constructions are superseded by what we present in III and IV; furthermore, the - -recent pubblication of a collective book covering these topics by the best specialists in the field convinced me not to try to compete with them in this book which is aIready too long.
The historical part is the weak point of this book; I know the main results and ideas mainly for having practiced them, and rarely from the standard sources(…)
 (From the Foreward)