Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences.

Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level.

The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science.

Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way.

SECTION I: COMPUTABILITY, AND UNSOLVABLE PROBLEMS

HILBERT AND THE ORIGINS OF COMPUTABILITY THEORY

Algorithms and Algorithmic Content

Hilbert's Programme

Gödel, and the Discovery of Incomputability

Computability and Unsolvability in the Real World

MODELS OF COMPUTABILITY AND THE CHURCH-TURING THESIS

The Recursive Functions

Church's Thesis, and the Computability of Sets and Relations

Unlimited Register Machines

Turing's Machines

Church, Turing, and the Equivalence of Models

LANGUAGE, PROOF AND COMPUTABLE FUNCTIONS

Peano Arithmetic and its Models

What Functions Can We Describe in a Theory?

CODING, SELF-REFERENCE AND THE UNIVERSAL TURING MACHINE

Russell's Paradox

Gödel Numberings

A Universal Turing Machine

The Fixed Point Theorem

Computable Approximations

ENUMERABILITY AND COMPUTABILITY

Basic Notions

The Normal Form Theorem

Incomputable Sets and the Unsolvability of the Halting Problem for Turing Machines

The Busy Beaver function

THE SEARCH FOR NATURAL EXAMPLES OF INCOMPUTABLE SETS

The Ubiquitous Creative Sets

Some Less Natural Examples of Incomputable Sets

Hilbert's Tenth Problem and the Search for Really Natural Examples

COMPARING COMPUTABILITY

Many-One Reducibility

The Non-Computable Universe and Many-One Degrees

Creative Sets Revisited

GÖDEL'S INCOMPLETENESS THEOREM

Semi-Representability and C.E. Sets

Incomputability and Gödel's Theorem

DECIDABLE AND UNDECIDABLE THEORIES

PA is Undecidable

Other Undecidable Theories, and their Many-One Equivalence

Some Decidable Theories

SECTION II: INCOMPUTABILITY AND INFORMATION CONTENT

COMPUTING WITH ORACLES

Oracle Turing Machines

Relativising, and Listing the Partial Computable Functionals

Introducing the Turing Universe

Enumerating with Oracles, and the Jump Operator

The Arithmetical Hierarchy and Post's Theorem

The Structure of the Turing Universe

NONDETERMINISM, ENUMERATIONS AND POLYNOMIAL BOUNDS

Oracles versus Enumerations of Data

Enumeration Reducibility and the Scott Model for Lambda Calculus

The Enumeration Degrees,and the Natural Embedding of the

Turing Degrees

The Structure of De and the Arithmetical Hierarchy

The Medvedev Lattice

Polynomial Bounds and P =?NP

SECTION III: MORE ADVANCED TOPICS

POST'S PROBLEM: IMMUNITY AND PRIORITY

Information Content and Structure

Immunity Properties

Approximation and Priority

Sacks Splitting Theorem and Cone Avoidance

Minimal Pairs and Extensions of Embeddings

The |3 Theory - Information Content Regained

Higher Priority and Maximal Sets

The Computability of Theories

FORCING AND CATEGORY

Forcing in Computability Theory

Baire Space, Category and Measure

n-Genericity and Applications

Forcing with Trees, and Minimal Degrees

APPLICATIONS OF DETERMINACY

Gale-Stewart Games

An Upper Cone of Minimal Covers

Borel and Projective Determinacy, and the Global Theory of D

THE COMPUTABILITY OF THEORIES

Feferman's Theorem

Truth versus Provability

Complete extensions of Peano Arithmetic and Classes

The Low Basis Theorem

Arslanov's Completeness Criterion

A Priority-Free Solution to Post's Problem

Randomness

COMPUTABILITY AND STRUCTURE

Computable Models

Computability and Mathematical Structures

Effective Ramsey Theory

Computability in Analysis

Computability and Incomputability in Science

FURTHER READING

INDEX

"A very nice volume indeed. Although primarily a textbook, it lives up to the author's aim to have 'plenty here to interest and inform everyone, from the beginner to the expert.' … Cooper writes in an informal style, emphasizing the ideas underlying the techniques. All the standard topics and classic results are here. … Students will find useful pointers to the literature and an abundance of exercises woven into the text."

- Zentralblatt MATH, 1041

"[It] provides not only a reference repository of well-crafted proofs or proof-outlines for a large number of basic and beyond-basic facts in several areas of computability theory, but can also serve well as the textual basis for a course on the subject…"

- Mathematical Reviews, 2005h