Driven by the question 'What is the computational content of a (formal) proof?' this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory computability and GÃdel's theorems. Part II studies and classifies provable recursion in classical systems from fragments of Peano arithmetic up to Î11âCA0. Ordinal analysis and the (SchwichtenbergâWainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and Î11âCA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems a constructive theory TCF of computable functionals realizability Dialectica interpretation computationally significant quantifiers and connectives and polytime complexity in a two-sorted higher-type arithmetic with linear logic. Language: English