Preface

This is the second volume of a graduate course in set theory. Volume I covered

the basics of modern set theory and was primarily aimed at beginning graduate

students. Volume II is aimed at more advanced graduate students and research

mathematicians specializing in fields other than set theory. It contains short but

rigorous introductions to various set-theoretic techniques that have found applica-

tions outside of set theory. Although we think of Volume II as a natural continuation

of Volume

I,1

each volume is sufficiently self-contained to be studied separately.

The main prerequisite for Volume II is a knowledge of basic naive and axiomatic

set

theory.2

Moreover, some knowledge of mathematical logic and general topology

is indispensible for reading this volume. A minicourse in mathematical logic was

given in Chapters 5 and 6 of Volume I, and we include an appendix on general

topology at the end of this volume. Our terminology is fairly standard. For the

benefit of those readers who learned their basic set theory from a different source

than our Volume I we include a short section on somewhat idiosyncratic notations

introduced in Volume I. In particular, some of the material on mathematical logic

covered in Chapter 5 is briefly reviewed. The book can be used as a text in the

classroom as well as for self-study.

We tried to keep the length of the text moderate. This may explain the ab-

sence of many a worthy theorem from this book. Our most important criterion for

inclusion of an item was frequency of use outside of pure set theory. We want to em-

phasize that "item" may mean either an important concept (like "equiconsistency

with the existence of a measurable cardinal"), a theorem (like Ramsey's Theorem),

or a proof technique (like the craft of using Martin's Axiom). Therefore, we occa-

sionally illustrate a technique by proving a somewhat marginal theorem. Of course,

the "frequency of use outside set theory" is based on our subjective perceptions.

At the end of most chapters there are "Mathographical Remarks." Their pur-

pose is to show where the material fits in the history and literature of the subject.

We hope they will provide some guidance for further reading in set theory. They

should not be mistaken for "scholarly remarks" though. We did not make any effort

whatsoever to trace the theorems of this book to their origins. However, each of the

theorems presented here can also be found in at least one of the more specialized

texts reviewed in the "Mathographical Remarks." Therefore, we do not feel guilty

of severing chains of historical evidence.

1 This is the reason why the present volume starts with Chapter 13.

2Possible

alternatives to our Volume I are such texts as: K. Devlin, The Joy of Sets. Funda-

mentals of Contemporary Set Theory, Springer Verlag, 1993; A. Levy; Basic Set Theory, Springer

Verlag, 1979; or J. Roitman, Introduction to Modern Set Theory, John Wiley, 1990.

ix