Kahle R., Strahm T., Studer T. (eds.) Advances in Proof Theory

Basel: Birkhäuser, 2016. — 430 p.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.A Survey on Ordinal Notations Around the Bachmann-Howard Ordinal
About Truth and Types
Lindenbaum’s Lemma via Open Induction
Ordinal Analysis of Intuitionistic Power and Exponentiation Kripke Platek Set Theory
Machine-Checked Proof-Theory for Propositional Modal Logics
Intuitionistic Decision Procedures Since Gentzen
The Operational Perspective: Three Routes
Some Remarks on the Proof-Theory and the Semantics of Infinitary Modal Logic
From Subsystems of Analysis to Subsystems of Set Theory
Restricting Initial Sequents: The Trade-Offs Between Identity, Contraction and Cut
Higman’s Lemma and Its Computational Content
How to Reason Coinductively Informally
Pointwise Transfinite Induction and a Miniaturized Predicativity