Computability Theory

Zusatztext "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 Klappentext 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 IND Zusammenfassung 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. Inhaltsverzeichnis 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 TheoremIncomputa...

"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

Autorentext
Cooper, S. Barry; Cooper, S. Barry

Klappentext
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.

Zusammenfassung
Offers an introduction to contemporary computability theory, techniques, and results. This book places the basic concepts and techniques of computability theory in their historical, philosophical and logical context. It includes a chapter that explores a variety of computability applications to mathematics and science.

Inhalt
COMPUTABILITY, AND UNSOLVABLE PROBLEMS: Hilbert and the Origins of Computability Theory. Models of Computability and the Church-Turing Thesis. Language, Proof and Computable Functions. Coding, Self-Reference and Diagonalisation. Enumerability and Computability. The Search for Natural Examples of Incomputable Sets. Comparing Computability. Gödel's Incompleteness Theorem. Decidable and Undecidable Theories. INCOMPUTABILITY AND INFORMATION CONTENT: Computing with Oracles. Nondeterminism, Enumerations and Polynomial Bounds. MORE ADVANCED TOPICS: Post's Problem: Immunity and Priority. The Computability of Theories. Forcing and Category. Applications of Determinacy. Computability and Structure.

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