# Proof and computations [electronic resource] / Helmut Schwichtenberg, Stanley S. Wainer.

Author/Creator:
Schwichtenberg, Helmut, 1942-
Publication:
Cambridge : Cambridge University Press, 2012.
Series:
Perspectives in logic.
Perspectives in logic
Format/Description:
Book
1 online resource (482 p.)
Subjects:
Computable functions.
Proof theory.
Form/Genre:
Electronic books.
Language:
English
Summary:
This major graduate-level text provides a detailed, self-contained coverage of proof theory.
Contents:
1.4. Soundness and completeness of the classical fragment1.4.1. Models.; 1.4.2. Soundness of classical logic.; 1.4.3. Completeness of classical logic.; 1.4.4. Compactness and Löwenheim-Skolem theorems.; 1.5. Tait calculus; 1.6. Notes; Chapter 2: RECURSION THEORY; 2.1. Register machines; 2.1.1. Programs.; 2.1.2. Program constructs.; 2.1.3. Register machine computable functions.; 2.2. Elementary functions; 2.2.1. Definition and simple properties.; 2.2.2. Elementary relations.; 2.2.3. The class ε.; 2.2.4. Closure properties of ε.; 2.2.5. Coding finite lists.; 2.3. Kleene's normal form theorem
2.3.1. Program numbers.2.3.2. Normal form.; 2.3.3. Σo1-definable relations and μ-recursive functions.; 2.3.4. Computable functions.; 2.3.5. Undecidability of the halting problem.; 2.4. Recursive definitions; 2.4.1. Least fixed points of recursive definitions.; 2.4.2. The principles of finite support and monotonicity, and the effective index property.; 2.4.3. Recursion theorem.; 2.4.4. Recursive programs and partial recursive functions.; 2.4.5. Relativized recursion.; 2.5. Primitive recursion and for-loops; 2.5.1. Primitive recursive functions.; 2.5.2. Loop-programs.
2.5.3. Reduction to primitive recursion.2.5.4. A complexity hierarchy for Prim.; 2.6. The arithmetical hierarchy; 2.6.1. Kleene's second recursion theorem.; 2.6.2. Characterization of Σ01-definable and recursive relations.; 2.6.3. Arithmetical relations.; 2.6.4. Closure properties.; 2.6.6. Σ0r-complete relations.; 2.7. The analytical hierarchy; 2.7.1. Analytical relations.; 2.7.2. Closure properties.; 2.7.3. Universal Σ1r+1-definable relations.; 2.7.4. Σ1r-complete relations.; 2.8. Recursive type-2 functionals and well-foundedness; 2.8.1. Computation trees.; 2.8.2. Ordinal assignments
recursive ordinals.
Notes:
Description based upon print version of record.
Includes bibliographical references and index.
Contributor:
Association for Symbolic Logic.
Wainer, S. S.
ISBN:
1-107-22428-4
1-280-48485-3
1-139-22192-2
9786613579836
1-139-21710-0
1-139-21403-9
1-139-22363-1
1-139-22020-9
1-139-03190-2
OCLC:
775869745