Computability TheoryComputability 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. |
Contents
Hilbert and the Origins of Computability Theory | 3 |
Models of Computability and the ChurchTuring Thesis | 11 |
Language Proof and Computable Functions | 45 |
Coding SelfReference and the Universal Turing Machine | 61 |
Enumerability and Computability | 69 |
The Search for Natural Examples of Incomputable Sets | 87 |
Comparing Computability and the Ubiquity of Creative Sets | 101 |
Gödels Incompleteness Theorem | 117 |
Nondeterminism Enumerations and Polynomial Bounds | 173 |
Immunity and Priority | 219 |
Forcing and Category | 273 |
Applications of Determinacy | 311 |
The Computability of Theories | 321 |
Computability and Structure | 343 |
Further Reading | 383 |
| 389 | |
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Common terms and phrases
1-generic 1-random A-term a₁ algorithm arithmetical hierarchy Assume axiomatisable axioms basic c.e. degrees c.e. sets Church-Turing Thesis comeager complete computability theory computable functions computable relation computably enumerable construction COROLLARY countable Deduce define DEFINITION defn deg(A degree structure dominates e-degree EXAMPLE EXERCISE exists finite function f given gives Gödel numbers Gödel's Gödel's Incompleteness Theorem Graph(f halting halting problem hence Hilbert's Hint hyperimmune incomputable sets induction infinite information content input isomorphic jump lambda calculus lattice Lemma linear ordering logic mathematics minimal cover minimal degree notation numbers oracle oracle Turing machine order theory ordinals outcomes p.c. function polynomial Post's Theorem primitive recursive primitive recursive functions problem PROOF Let prove putable real numbers Recursion Theory recursive functions relativised REMARK restraints result satisfied sequence Show SOLUTION strategy strings subset Theorem tree Turing computable Turing degrees Turing machine URM program write σο