Part I, The Roots of Computability Theory.- Introduction.- The Foundational Crisis of Mathematics.- Formalism.- Hilbert’s Attempt at Recovery.- Part II, Classicial Computability Theory.- The Quest for a Formalization.- The Turing Machine.- The First Basic Results.- Incomputable Problems.- Methods of Proving Incomputability.- Part III, Relative Computability.- Computation with External Help.- Degrees of Unsolvability.- The Turing Hierarchy of Unsolvability.- The Class D of Degrees of Unsolvability.- C.E. Degrees and the Priority Method.- The Arithmetical Hierarchy.- Part IV, Back to the Roots.- Computability (Church-Turing) Thesis Revisited.- Further Reading.- App. A, Mathematical Background.- App. B, Notation Index.- Glossary.- References.- Index.

This book offers an original and informative view of the development of fundamental concepts of computability theory. The treatment is put into historical context, emphasizing the motivation for ideas as well as their logical and formal development. In Part I the author introduces computability theory, with chapters on the foundational crisis of mathematics in the early twentieth century, and formalism. In Part II he explains classical computability theory, with chapters on the quest for formalization, the Turing Machine, and early successes such as defining incomputable problems, c.e. (computably enumerable) sets, and developing methods for proving incomputability. In Part III he explains relative computability, with chapters on computation with external help, degrees of unsolvability, the Turing hierarchy of unsolvability, the class of degrees of unsolvability, c.e. degrees and the priority method, and the arithmetical hierarchy. Finally, in the new Part IV the author revisits the computability (Church-Turing) thesis in greater detail. He offers a systematic and detailed account of its origins, evolution, and meaning, he describes more powerful, modern versions of the thesis, and he discusses recent speculative proposals for new computing paradigms such as hypercomputing. This is a gentle introduction from the origins of computability theory up to current research, and it will be of value as a textbook and guide for advanced undergraduate and graduate students and researchers in the domains of computability theory and theoretical computer science.This new edition is completely revised, with almost one hundred pages of new material. In particular the author applied more up-to-date, more consistent terminology, and he addressed some notational redundancies and minor errors. He developed a glossary relating to computability theory, expanded the bibliographic references with new entries, and added the new part described above and other new sections.

“For the prepared, this [1st ed.] is a wonderful blending of history and philosophy, foundations and computation, math and the mystery of the incomplete, undecidable, unknowable, incomputable. … a willing reader will be happy.” (Benjamin Wells, ACM Computing Reviews, April, 2016)“The book [1st ed.] incorporates many historical, philosophical, and practical concerns closely related to the definitions and core results of the subject. In addition, the book covers enough ground to be able to discuss the finite injury priority method. … The tone is friendly and accessible, and a reader can learn some of the fundamental aspects of computability theory and its history without getting bogged down in technical details.” (Joseph R. Mileti, Mathematical Reviews, September, 2016)“The main purpose of this book [1st ed.] is to provide the reader with a clear and deep understanding of the foundations of computability theory. This goal is fully achieved, because the book is well-written and its reading is very pleasant. … It is a good textbook for undergraduate or beginning graduate students in computer science or mathematics, and I suggest to use it.” (Patrizio Cintioli, zbMATH 1339.03001, 2016)  

Original, informative view of the development of the fundamental concepts of computability theory Avoids excessive formalism, offering instead intuitive understanding Characterized by appreciation of historical context and motivation of the logical development of the theory presented Completely revised new edition, with added glossary, extra bibliographic references, and a new chapter on the computability (Church-Turing) thesis