Set Theory for the Working Mathematician


Krzysztof Ciesielski

London Math Society Student Texts 39

Cambridge University Press, 1997.

Hardback ISBN 0-521-59441-3; paperback ISBN 0-521-59465-0.


The course presented in this text concentrates on the typical methods of modern set theory: transfinite induction, Zorn's lemma, the continuum hypothesis, Martin's axiom, the diamond principle $\diamondsuit$, and elements of forcing. The choice of the topics and the way in which they are presented is subordinate to one purpose -- to get the tools that are most useful in applications, especially in abstract geometry, analysis, topology, and algebra. In particular, most of the methods presented in this course are accompanied by many applications in abstract geometry, real analysis, and, in a few cases, topology and algebra. Thus the text is dedicated to all readers that would like to apply set-theoretic methods outside set theory.

The course is presented as a textbook that is appropriate for either a lower-level graduate course or an advanced undergraduate course. However, the potential readership should also include mathematicians whose expertise lies outside set theory but who would like to learn more about modern set-theoretic techniques that might be applicable in their field.

The reader of this text is assumed to have a good understanding of abstract proving techniques, and of the basic geometric and topological structure of the n-dimensional Euclidean space Rn. In particular, a comfort in dealing with the continuous functions from Rn into Rn is assumed. A basic set-theoretic knowledge is also required. This includes a good understanding of the basic set operations (union, intersection, Cartesian product of arbitrary families of sets, and difference of two sets), abstract functions (the operations of taking images and preimages of sets with respect to functions), and elements of the theory of cardinal numbers (finite, countable, and uncountable sets.) Most of this knowledge is included in any course in analysis, topology, or algebra. These prerequisites are also discussed briefly in Part I of the text.

The book is organized as follows. Part I introduces the reader to axiomatic set theory and uses it to develop basic set theoretical concepts. In particular, Chapter 1 contains a necessary logic background, discusses the most fundamental axioms of ZFC, and uses them to define basic set theoretic operations. In Chapter 2 the notions of relation, function, and Cartesian product are defined within the framework of ZFC theory. The related notions are also introduced and their fundamental properties are discussed. Chapter 3 describes the set theoretic interpretation of the sets of natural, integer, and real numbers. Most of the facts presented in Part I is left without the proofs.

Part II deals with the fundamental concepts of "classical set theory." The ordinal and cardinal numbers are introduced and their arithmetic is developed. The theorem on definition by recursion is proved and used to prove Zorn's Lemma. Section 4.4 contains some standard applications of the Zorn's Lemma in analysis, topology and algebra.

Part III is designed to familiarize the reader with proofs by transfinite induction. In particular, Section 6.1 illustrates a typical transfinite induction construction and the diagonalization argument by describing several constructions of the subsets of Rn with different strange geometric properties. The two remaining sections of Chapter 6 introduce the basic elements of descriptive set theory and discuss Borel and Lebesgue measurable sets, and the sets with Baire property. Chapter 7 is designed to help the reader to master the recursive definitions technique. Most of the examples presented there concern real functions, and in many cases consist of the newest research results in this area.

Part IV is designed to introduce the tools of "modern set theory:" Martin's Axiom, Diamond Principle and forcing method. The overall idea behind their presentation is to introduce them as the natural refinements of the method of transfinite induction. Thus, based on the solid foundation built in Part III, the forcing notions and forcing arguments presented there are obtained as "transformed" transfinite induction arguments. In particular, the more standard axiomatic approach to these methods is described in Chapter 8, where Martin's Axiom and Diamond Principle are introduced and discussed. Chapter 9 is the most advanced part of this text and describes the forcing method. Section 9.1 consists of some additional prerequisites, mainly logical, necessary to follow the other sections. In Section 9.2 the main theoretical basis for the forcing theory is introduced while proving the consistency of ZFC and the negation of CH. In Section 9.3 it is constructed a generic model for ZFC+"Diamond Principle." (Thus, also for ZFC+CH.) Section 9.4 discusses the Product Lemma and uses it to deduce few more properties of the Cohen model, i.e., the model from Section 9.2. The book is finished with Section 9.5 in which it is proved the simultaneous consistency of Martin's Axiom and the negation of Continuum Hypothesis. This proof, done by the iterated forcing, shows that even in the world of "sophisticated recursion method" of forcing the "classical" recursion technique is still a fundamental method of set theory -- the desired model is obtained by constructing forcing extensions by transfinite induction.

It is also worth to point out here that the readers with different background will certainly be interested in different parts of this text. Most of advanced graduate students as well as mathematical researchers using this book will almost certainly just skim Part I. The same may be also true for some of these readers for at least some portion of Part II. Part III and the first chapter of Part IV should be considered as the core of this text and are written to the widest group of readers. Finally, the last chapter (concerning forcing) is the most difficult and logic oriented, and will be probably interested only to the most dedicated readers. It certainly can be excluded from any undergraduate course taught based on this text.