This book is devoted to a deep and interesting study of sets and functions on natural numbers in terms of three properties intimately connected with the notion of computability: (1) closure (especially recursion and minimization), (2) reflexivity (universal and indexing functions), and (3) enumeration (Kleene’s normal form theorem). The presentation is organized around Church’s thesis, which can be stated in terms of minimal closure under suitable operations, and the hierarchy of structures matches the list of properties given above.

The author pays considerable attention to traditional aspects of recursive function theory, such as universal and indexing functions, the recursion theorem, and the classification of sets. The book’s great merit, however, is its presentation of facts about reflexive structures and the inductive definitions, with both finitary and nonfinitary inductions, at an elementary level. Previous textbooks covered this material only in part.

The book presents many of the author’s original results, especially in the area of hyperenumeration. In addition, some proofs of old results contain many new and enjoyable ideas. The exercises and historical notes at the end of every section should be very helpful to the reader.

The book does have two minor shortcomings. The new notation and terminology that are used extensively in this book deserve a special index of notation and a more elaborate glossary. Similarly, the bibliography should be enlarged considerably to offer an appropriate perspective on the domain. References such as Strong [1] and Hennie [2] should not have been omitted.

I found this text to be a well-written, algebraically oriented introduction to the general theory of computability. It can be used as a text for undergraduate students in mathematics and in computer science, and I will be using parts of it in my lectures.