Computability Theory

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, this is a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results that also serves as a guide to the direction of current research in the field. Computability Theory includes both the standard material for a first course in computability and more advanced looks at Turing degrees, forcing, Turing definability, and determinacy. The final chapter explores a variety of computability applications in mathematics and science.

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 THEORYAlgorithms and Algorithmic ContentHilbert's Programme Gödel, and the Discovery of IncomputabilityComputability and Unsolvability in the Real WorldMODELS OF COMPUTABILITY AND THE CHURCH-TURING THESISThe Recursive FunctionsChurch's Thesis, and the Computability of Sets and RelationsUnlimited Register MachinesTuring's MachinesChurch, Turing, and the Equivalence of ModelsLANGUAGE, PROOF AND COMPUTABLE FUNCTIONSPeano Arithmetic and its ModelsWhat Functions Can We Describe in a Theory?CODING, SELF-REFERENCE AND THE UNIVERSAL TURING MACHINERussell's ParadoxGödel NumberingsA Universal Turing MachineThe Fixed Point TheoremComputable ApproximationsENUMERABILITY AND COMPUTABILITYBasic NotionsThe Normal Form TheoremIncomputable Sets and the Unsolvability of the Halting Problem for Turing MachinesThe Busy Beaver functionTHE SEARCH FOR NATURAL EXAMPLES OF INCOMPUTABLE SETSThe Ubiquitous Creative Sets Some Less Natural Examples of Incomputable SetsHilbert's Tenth Problem and the Search for Really Natural ExamplesCOMPARING COMPUTABILITYMany-One ReducibilityThe Non-Computable Universe and Many-One Degrees Creative Sets RevisitedGÖDEL'S INCOMPLETENESS THEOREMSemi-Representability and C.E. SetsIncomputability and Gödel's TheoremDECIDABLE AND UNDECIDABLE THEORIESPA is Undecidable Other Undecidable Theories, and their Many-One EquivalenceSome Decidable TheoriesSECTION II: INCOMPUTABILITY AND INFORMATION CONTENTCOMPUTING WITH ORACLESOracle Turing MachinesRelativising, and Listing the Partial Computable FunctionalsIntroducing the Turing UniverseEnumerating with Oracles, and the Jump Operator The Arithmetical Hierarchy and Post's TheoremThe Structure of the Turing UniverseNONDETERMINISM, ENUMERATIONS AND POLYNOMIAL BOUNDSOracles versus Enumerations of Data Enumeration Reducibility and the Scott Model for Lambda Calculus The Enumeration Degrees,and the Natural Embedding of theTuring DegreesThe Structure of De and the Arithmetical HierarchyThe Medvedev LatticePolynomial Bounds and P =?NPSECTION III: MORE ADVANCED TOPICSPOST'S PROBLEM: IMMUNITY AND PRIORITYInformation Content and StructureImmunity PropertiesApproximation and PrioritySacks Splitting Theorem and Cone AvoidanceMinimal Pairs and Extensions of EmbeddingsThe |3 Theory - Information Content RegainedHigher Priority and Maximal SetsThe Computability of TheoriesFORCING AND CATEGORYForcing in Computability TheoryBaire Space, Category and Measuren-Genericity and Applications Forcing with Trees, and Minimal DegreesAPPLICATIONS OF DETERMINACYGale-Stewart Games An Upper Cone of Minimal Covers Borel and Projective Determinacy, and the Global Theory of DTHE COMPUTABILITY OF THEORIESFeferman's TheoremTruth versus ProvabilityComplete extensions of Peano Arithmetic and Classes The Low Basis TheoremArslanov's Completeness CriterionA Priority-Free Solution to Post's ProblemRandomnessCOMPUTABILITY AND STRUCTUREComputable ModelsComputability and Mathematical StructuresEffective Ramsey TheoryComputability in Analysis Computability and Incomputability in ScienceFURTHER READING INDEX