Turing's famous 1936 paper introduced a formal definition of a computing
machine, a Turing machine. This model led to both the development of
actual computers and to computability theory, the study of what machines
can and cannot compute. This book presents classical computability
theory from Turing and Post to current results and methods, and their
use in studying the information content of algebraic structures, models,
and their relation to Peano arithmetic. The author presents the subject
as an art to be practiced, and an art in the aesthetic sense of inherent
beauty which all mathematicians recognize in their subject.
Part I gives a thorough development of the foundations of computability,
from the definition of Turing machines up to finite injury priority
arguments. Key topics include relative computability, and computably
enumerable sets, those which can be effectively listed but not
necessarily effectively decided, such as the theorems of Peano
arithmetic. Part II includes the study of computably open and closed
sets of reals and basis and nonbasis theorems for effectively closed
sets. Part III covers minimal Turing degrees. Part IV is an introduction
to games and their use in proving theorems. Finally, Part V offers a
short history of computability theory.
The author has honed the content over decades according to feedback from
students, lecturers, and researchers around the world. Most chapters
include exercises, and the material is carefully structured according to
importance and difficulty. The book is suitable for advanced
undergraduate and graduate students in computer science and mathematics
and researchers engaged with computability and mathematical logic.
