The Aim Of This Volume Is To Collect Original Contributions By The Best Specialists From The Area Of Proof Theory, Constructivity, And Computation And Discuss Recent Trends And Results In These Areas. Some Emphasis Will Be Put On Ordinal Analysis, Reductive Proof Theory, Explicit Mathematics And Type-theoretic Formalisms, And Abstract Computations. The Volume Is Dedicated To The 60th Birthday Of Professor Gerhard Jäger, Who Has Been Instrumental In Shaping And Promoting Logic In Switzerland For The Last 25 Years. It Comprises Contributions From The Symposium “advances In Proof Theory”, Which Was Held In Bern In December 2013. Proof Theory Came Into Being In The Twenties Of The Last Century, When It Was Inaugurated By David Hilbert In Order To Secure The Foundations Of Mathematics. It Was Substantially Influenced By Gödel's Famous Incompleteness Theorems Of 1930 And Gentzen's New Consistency Proof For The Axiom System Of First Order Number Theory In 1936. Today, Proof Theory Is A Well-established Branch Of Mathematical And Philosophical Logic And One Of The Pillars Of The Foundations Of Mathematics. Proof Theory Explores Constructive And Computational Aspects Of Mathematical Reasoning; It Is Particularly Suitable For Dealing With Various Questions In Computer Science. . W. Buchholz: A Survey On Ordinal Notations Around The Bachmann-howard Ordinal -- A. Cantini: About Truth And Types -- R. Dyckhoff: Intuitionistic Decision Procedures Since Gentzen -- S. Feferman: The Operational Perspective -- R. Gore: Formally Verified Proof-theory Using Isabelle/hol -- P. Minari: Analytic Equational Proof Systems For Combinatory Logic And Lambda Calculus -- W. Pohlers: From Subsystems Of Classical Analysis To Subsystems Of Set Theory - A Personal Account -- M. Rathjen: Ordinal Analysis And Witness Extraction -- P. Schuster: Logic Completeness Via Open Induction -- H. Schwichtenberg: On The Computational Content Of Higman's Lemma -- P. Schroeder-heister: Tba -- A. Setzer: Tba -- S. Wainer: On Weak Pointwise Induction, And A Miniaturized Predicativity. Edited By Reinhard Kahle, Thomas Strahm, Thomas Studer. Front Matter....Pages i-xii A Survey on Ordinal Notations Around the Bachmann-Howard Ordinal....Pages 1-29 About Truth and Types....Pages 31-64 Lindenbaum’s Lemma via Open Induction....Pages 65-77 Ordinal Analysis of Intuitionistic Power and Exponentiation Kripke Platek Set Theory....Pages 79-172 Machine-Checked Proof-Theory for Propositional Modal Logics....Pages 173-243 Intuitionistic Decision Procedures Since Gentzen....Pages 245-267 The Operational Perspective: Three Routes....Pages 269-289 Some Remarks on the Proof-Theory and the Semantics of Infinitary Modal Logic....Pages 291-318 From Subsystems of Analysis to Subsystems of Set Theory....Pages 319-338 Restricting Initial Sequents: The Trade-Offs Between Identity, Contraction and Cut....Pages 339-351 Higman’s Lemma and Its Computational Content....Pages 353-375 How to Reason Coinductively Informally....Pages 377-408 Pointwise Transfinite Induction and a Miniaturized Predicativity....Pages 409-425 The volume is dedicated to the 60th birthday of Professor Gerhard Jager, who has been instrumental in shaping and promoting logic in Switzerland for the last 25 years. It comprises contributions from the symposium Advances in Proof Theory, which was held in Bern in December 2013
