Helmut Schwichtenberg and Stanley S. Wainer

Year: 2012

ISBN-13: 9780521517690

480 pages. Hardcover.

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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 G�del’s theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to Π11�CA0. Ordinal analysis and the (Schwichtenberg�Wainer) subrecursive hierarchies play a central role and are used in proving the ‘modified finite Ramsey’ and ‘extended Kruskal’ independence results for PA and Π11�CA0. 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.

### Table of Contents

**Part 1. Basic proof theory and computability**

- Logic
- Recursion Theory
- Gödel’s Theorems

**Part 2. Provable recursion in classical systems**

- The Provably Recursive Functions of Arithmetic
- Accessible Recursive Functions, ID < ω and Π11–CA0

**Part 3. Constructive logic and complexity**

- Computability in Higher Types
- Extracting Computational Content From Proofs
- Linear Two-Sorted Arithmetic