The kernel of this book consists of a series of lectures on in?nitary proof theory which I gave during my time at the Westfalische ¨ Wilhelms-Universitat ¨ in Munster ¨ . It was planned as a successor of Springer
Lecture Notes in Mathematics 1407. H- ever, when preparing it, I decided to also include material which has not been treated in SLN 1407. Since the appearance of SLN 1407 many innovations in the area of - dinal analysis have
taken place. Just to mention those of them which are addressed in this book: Buchholz simpli?ed local predicativity by the invention of operator controlled derivations (cf. Chapter 9, Chapter 11); Weiermann detected
applications of methods of impredicative proof theory to the characterization of the provable recursive functions of predicative theories (cf. Chapter 10); Beckmann improved Gentzen's boundedness theorem (which appears as
Stage Theorem (Theorem 6. 6. 1) in this book) to Theorem 6. 6. 9, a theorem which is very satisfying in itself - though its real importance lies in the ordinal analysis of systems, weaker than those treated here. Besides these
innovations I also decided to include the analysis of the theory (? -REF) as an example of a subtheory of set theory whose ordinal analysis only 2 0 requires a ?rst step into impredicativity. The ordinal analysis of(? -FXP) of
non- 0 1 0 monotone? -de?nable inductive de?nitions in Chapter 13 is an application of the 1 analysis of(? -REF).