Wolfram Pohlers

Proof Theory (eBook, PDF)

The First Step into Impredicativity

• Format: PDF

Produktbeschreibung

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).

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Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 374
• Erscheinungstermin: 1. Oktober 2008
• Englisch
• ISBN-13: 9783540693192
• Artikelnr.: 44129002

Autorenporträt

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.

Inhaltsangabe

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.

Rezensionen

From the reviews:

"Proof Theory takes various axiom systems ... that treat induction in different ways and analyzes them from the ordinal viewpoint to gauge their relative strengths. ... This new version includes several developments in the field that have occurred over the twenty years since the original. Although the current book, appearing in the Universitext series, claims to be 'pitched at undergraduate/graduate level,' an undergraduate course out of Proof theory would be ambitious indeed." (Leon Harkleroad, The Mathematical Association of America, March, 2009)

"The book is addressed primarily to students of mathematical logic interested in the basics of proof theory, and it can be used both for introductory and advanced courses in proof theory. ... this book may be recommended to a larger circle of readers interested in proof theory." (Branislav Boricic, Zentrablatt MATH, Vol. 1153, 2009)

"This is a textbook-an excellent one-on proof theory, starting from the very elementary (heuristic accounts of sets, ordinals, logic, etc.), and going into a sophisticated area (impredicativity). ... The author's main tool is enquiry into truth complexity and ordinal analysis." (M. Yasuhara, Mathematical Reviews, Issue 2010 a)