This is a major and comprehensive treatise on classical recursion theory. It is a self-contained, highly readable, and excellently organized book whose coverage extends from the foundations of the theory of computable functions to the frontiers of recursively enumerable (r.e.) degree theory. Soare shows how vastly the subject has grown since the publication, in 1967, of the principal forerunner, a work by Rogers [1] (which has, happily, just returned to print itself). Like Rogers, Soare blends full rigor with suggestive prose, which is well designed to enhance the reader’s intuitions. This preserves the flavor that Post imparted to the theory in the forties.

Soare’s book, like Rogers’s, has 16 chapters. The chapters fall into four parts, determined as much by the escalating methods of argument demanded as by the topics treated: fundamentals of recursion theory, finite injury arguments, methodology, and advanced topics. Throughout, priority arguments are unpacked and analyzed in terms of the various requirements that they need to fulfill.

The first two parts correspond roughly to what was known 20 years ago; they comprise less than one-third of the text. In Soare’s own felicitous words, “An initial segment of this book is intended as an introduction to the theory of recursive functions and r.e. sets.” Part 1 provides this routinely but well, covering the recursion theorem, Turing reducibility, and the arithmetical hierarchy. Part 2 presents Post’s problem and the Friedberg-Muchnik solution to it and then a chapter on oracle constructions of non-r.e. degrees.

Part 3 introduces the infinite injury priority method and applies it in the study of the lattice E of r.e. sets under inclusion, the investigation of the degrees of maximal, high, and low sets, and the classification of index sets. Part 4 uses what are called O‴ priority arguments and discusses areas of current research, leaving us with several open problems. Among them are What is the degree of the elementary theory of E? and, Which classes of r.e. degrees are definable in the semilattice of r.e. degrees?

The book contains a great variety of exercises, most of which record research results. Hints are given liberally. Authorship is attributed wherever possible, and a valuable bibliography of references extends to 30 pages. The book has an especially helpful notation index spanning ten pages. It will greatly ease the browser’s burden.

The book does not deal with complexity of computation, nor does it move into generalized recursion theory. Except for the one chapter on non-r.e. degrees, it remains within the range staked out by its title. The unrestricted theory of degrees is treated in Lerman’s work [2], which is a book in the same series as Soare’s and a plausible companion to it.