Proof Theory

The First Step into Impredicativity
1. Auflage 1989. 2. printing 2008
ISBN: 978-3-540-69318-5
Verlag: Springer

Buch, Englisch, 374 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 1200 g

Reihe: Universitext

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).

Zielgruppe

Research

Autoren/Hrsg.

Pohlers, Wolfram

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik

Inhaltsverzeichnis

Historical Background.- Primitive Recursive Functions and Relations.- Ordinals.- Pure Logic.- Truth Complexity for ?11-Sentences.- Inductive Definitions.- The Ordinal Analysis for PA.- Autonomous Ordinals and the Limits of Predicativity.- Ordinal Analysis of the Theory for Inductive Definitions.- Provably Recursive Functions of NT.- Ordinal Analysis for Kripke–Platek Set Theory with Infinity.- Predicativity Revisited.- Nonmonotone Inductive Definitions.- Epilogue.

Über Autor(innen)

Wolfram Pohlers (born 1943) is Full Professor and Director of the Institute for Mathematical Logic and Foundational Resarch at the Westfälische Wilhelms-Universität in Münster, Germany. He received his scientific training at the University of Munich where he worked as an Associate Professor from 1980 to 1985. From 1989 to 1990 he was a visiting scholar at the MSRI in Berkley and in 2005 he taught at the Ohio State University in Columbus.