Proofs and Computation

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

  1. Logic
  2. Recursion Theory
  3. Gödel’s Theorems

Part 2. Provable recursion in classical systems

  1. The Provably Recursive Functions of Arithmetic
  2. Accessible Recursive Functions, ID < ω and Π11–CA0

Part 3. Constructive logic and complexity

  1. Computability in Higher Types
  2. Extracting Computational Content From Proofs
  3. Linear Two-Sorted Arithmetic