Book Review: Set theory, An introduction to independence proofs
C. Ward Henson · Bulletin of the American Mathematical Society · 1984
This book is intended as a text for beginning graduate courses in axiomatic set theory.It is thoughtfully constructed and very well written and this reviewer has used it successfully for the intended purpose.It could equally well serve as the basis for self-study by mathematicians whose work requires set-theoretic tools or is sensitive to the axioms of set theory.In such a course of study one first covers the basics, including cardinal and ordinal numbers and the methods of proof and definition by induction.This needs to be done in a formalized setting, based on a system of axioms, if one intends to be precise about foundational matters or to discuss independence results.Thus one also needs to discuss philosophical issues and to give some motivation for the choice of axioms.( Herethe system of axioms studied is ZFC-the axioms of Zermelo and Fraenkel, with the Axiom of Choice.)All this is efficiently presented by Professor Kunen in Chapter 1.Although the preface states that he assumed his readers to be familiar (at an undergraduate level) with ordinals and cardinals, the author has done a good job of explaining these basic matters.The graduate students to whom I taught set theory using this book had little trouble there, even though most of them were studying set theory for the first time.The second general topic in any set theory course of mine is Godel's universe L of constructive sets [Gl].Not only is this a central aspect of axiomatic set theory and an important ingredient in many independence proofs, but its treatment is pedagogically very important, requiring as it does such important technical concepts as absoluteness.Informally, one defines the successive levels L a of the universe L by induction on the ordinal number a as follows: L 0 is the empty set; L a+l is the collection of subsets of L a which are first-order definable in the model (L a , G), allowing parameters; for limit ordinals X, L x is the union of L a for all a < X.Then L is the union of the levels L a for all ordinals a. Remarkably (L, G) satisfies all the ZFC axioms and the generalized continuum hypothesis as well, along with a number of other important mathematical principles.To prove this carefully one needs to bring the definition of L into formalized set theory by giving a treatment within ZFC of first-order definability.Or one can follow the lead of GodePs monograph [G2] and rework the definition of L a+l to remove the motivating idea of definability while smoothing the technical difficulties somewhat.My own choice when teaching a set theory course is to stay close to the informal definition and to beg the technical questions on the grounds that the motivation is an important aid to understanding and that the technicalities can always be returned to later.