Proof Theory The First Step into Impredicativity

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).

1 Historical Background.- 2 Primitive Recursive Functions and Relations.- 2.1 Primitive Recursive Functions.- 2.2 Primitive Recursive Relations.- 3 Ordinals.- 3.1 Heuristic.- 3.2 Some Basic Facts on Ordinals.- 3.3 Fundamentals of Ordinal Arithmetic.- 3.3.1 A Notation System for the Ordinals below epsilon nought.- 3.4 The Veblen Hierarchy.- 3.4.1 Preliminaries.- 3.4.2 The Veblen Hierarchy.- 3.4.3 A Notation System for the Ordinals below Gamma nought.- 4 Pure Logic.- 4.1 Heiristics.- 4.2 First and Second Order Logic.- 4.3 The Tait calculus.- 4.4 Trees and the Completeness Theorem.- 4.5 Gentzens Hauptsatz for Pure First Order Logic.- 4.6 Second Order Logic.- 5 Truth Complexities for Pi 1-1-Sentences.- 5.1 The language of Arithmetic.- 5.2 The Tait language for Second Order Arithmetic.- 5.3 Truth Complexities for Arithmetical Sentences.- 5.4 Truth Complexities for Pi 1-1-Sentences.- 6 Inductive Definitions.- 6.1 Motivation.- 6.2 Inductive Definitions as Monotone Operators.- 6.3 The Stages of an Inductive Definition.- 6.4 Arithmetically Definable Inductive Definitions.- 6.5 Inductive Definitions, Well-Orderings and Well-Founded Trees.- 6.6 Inductive Definitions and Truth Complexities.- 6.7 The Pi-1-1- Ordinal of a Theory.- 7 The Ordinal Analysis for Pean Arithmetic.- 7.1 The Theory PA.- 7.2 The Theory NT.- 7.3 The Upper Bound.- 7.4 The Lower Bound.- 7.5 The Use of Gentzen's Consistency Proof for Hilbert's Programme.- 7.5.1 On the Consistency of Formal and Semi-Formal Systems.- 7.5.2 The Consistency of NT.- 7.5.3 Kreisel's Counterexample.- 7.5.4 Gentzen's Consistency Proof in the Light of Hilbert's Programme.- 8 Autonomous Ordinals and the Limits of Predicativity.- 8.1 The Language L-kappa.- 8.2 Semantics for L-kappa.- 8.3 Autonomous Ordinals.- 8.4 The Upper Bound for Autonomous Ordinals.- 8.5 The Lower Bound for Autonomous Ordinals.- 9 Ordinal Analysis of the Theory for Inductive Definitions.- 9.1 The Theory ID1.- 9.2 The Language L infinity (NT).- 9.3 The Semi-FormalSystem for L infinity (NT).- 9.3.1 Semantical Cut-Elimination.- 9.3.2 Operator Controlled Derivations.- 9.4 The Collapsing Theorem for ID1.- 9.5 The Upper Bound.- 9.6 The Lower Bound.- 9.6.1 Coding Ordinals in L(NT).- 9.6.2 The Well-Ordering Proof.- 9.7 Alternative Interpretations for Omega.- 10 Provably Recursive Functions of NT.- 10.1 Provably Recursive Functions of a Theory.- 10.2 Operator Controlled Derivations.- 10.3 Iterating Operators.- 10.4 Cut Elimination for Operator Controlled Derivations.- 10.5 The Embedding of NT.- 10.6 Discussion.- 11 Ordinal Analysis for Kripke Platek Set Theory with infinity.- 11.1 Naive Set Theory.- 11.2 The Language of Set Theory.- 11.3 Constructible Sets.- 11.4 Kripke Platek Set Theory.- 11.5 ID1 as a Subtheory of Kp-omega.- 11.6 Variations of KP-omega and Axiom beta.- 11.7 The Sigma Ordinal of KP-omega.- 11.8 The Theory of Pi-2 Reflection.- 11.9 An Infinite Verification Calculus for the Constructible Hierarchy.- 11.10 A Semi-Formal System for Ramified Set Theory.- 11.11 The Collapsing Theorem for Ramified Set Theory.- 11.12 Ordinal Analysis for Kripke Platek Set Theory.- 12 Predicativity Revisited.- 12.1 Admissible Extensions.- 12.2 M-Logic.- 12.3 Extending Semi-Formal Systems.- 12.4 Asymmetric Interpretations.- 12.5 Reduction of T+ to T.- 12.6 The Theories KP n and KP 0-n.- 12.7 The Theories KPl 0 and KP i 0.- 13 Non-Monotone Inductive Definitions.- 13.1 Non-Monotone Inductive Definitions.- 13.2 Prewellorderings.- 13.3 The Theory for Pi 0-1 definable Fixed-Points.- 13.4 ID1 as a Sub-Theory of the Theory for Pi 0-1 definable Fixed-Points.- 13.5 The Upper Bound for the Proof theoretical Ordinal of Pi 0-1-FXP.- 14 Epilogue.