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Review: Michael Machtey, Paul Young, An Introduction to the General Theory of Algorithms

As to contents, the title of the book is not at all misleading; most aspects of set theory are in fact covered. The book starts with a chapter on the Zermelo-Fraenkel axioms and a development of basic set theory, followed by a chapter mostly about constructibility. Then comes forcing, developed from scratch and taken as far as a discussion of iterated forcing and Martin's axiom. Chapters 5 and 6 deal with large cardinal theory, including all our old favourites from inaccessible to supercompact and beyond. The usual 0" theory is also covered in this section. Finally there is a chapter on descriptive set theory, including a discussion of determinacy. There are errors in the book, and numerous misprints (missing primes and the like), but since the book is aimed at a sophisticated audience, these should not prove too much of a problem. The book is supplied with brief historical references, which may well bring the author some complaints, but should be of use to many readers. Although no book can cover all set theory, Jech's coverage does manage to be everywhere dense, and as a reference text it is extremely well written. KEITH J. DEVLIN