Diagonalization and Self-Reference

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ISBN: 9780198534501 | 0198534507
Cover: Hardcover
Copyright: 10/13/1994

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This book presents a systematic, unified treatment of fixed points as they occur in Godel's incompleteness proofs, recursion theory, combinatory logic, semantics, and metamathematics. Packed with instructive problems and solutions, the book offers an excellent introduction to the subject and highlights recent research.

Introduction to Self-Reference and Recursion
Introduction to Self-Reference
Quotation and Self-Reference
Self-Reference in a More General Setting
Some Classical Fixed Point Arguments Compared
Five fixed Point Arguments
A Unification
Quasi-Diagonalization
How to Silence a Universal Machine
Silencing a Universal Machine
Some Related Problems
Solutions to Problems
Some General Incompleteness Theorems
Incompleteness
Provability by Stages
Self-Reference in Arithmetic
Normalization and Tarskification
Some Special Devices
Generalizations
Introduction to Formal Systems and Recursion
Formal Representability
Recursive Enumerability
A Universal System and Its Applications
A Universal System
Enumeration and Iteration Theorems
Systems with Effective Properties
Arithmetization of Formal Systems
Arithmetization of Elementary Formal Systems
Double Enumeration
More on Arithmetization
Elementary Formal Systems and Incompleteness Proofs
Truth in Lo Cannot be Formalized
A Concrete Incompleteness Proof
Double Indexed Relational Systems
Indexed Relational Systems
Double Indexing
Effective Representation Systems
Effective Representation
Exact and Effective Rosser Systems
Fixed Point Theorems in General Setting
Sequential Systems
Definitions and Purpose
Preliminary Definitions
Some Fixed Point Properties
Summary and Applications
Strong Fixed Point Properties
The Recursion and Extended Recursion Properties
The Myhill Property and Related Properties
Summary and Applications
Multiple and Fixed Point Properties
Double Fixed Points
Double Recursion Properties
Double Myhill Properties and Related Results
Symmetric Functions and Nice Functions
Synchronization and Pairing Functions
Synchronized Fixed Point Properties
Pairing Functions
Synchronized Recursion Resumed
Some Further Relations Between Fixed Point Properties
Single and Double Fixed Point Properties Compared
More on Systems of Type 1*
Combinators and Sequential Systems
Fixed Point Properties of Combinatory Logic
Interdefinability of Some Combinators
Fixed Points
Multiple Fixed Points
Formal Combinatory Logic
The Axiom Systems CL and CLo.
A Second Variety of Fixed Point Theorems
Self-Reference in CLo
Fixed Point Theorems of the Second Type
Extended Sequential Systems
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Diagonalization and Self-Reference

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