This text grew out of lecture notes written1 by Martin Weese of Humboldt
Universitat, Berlin, Germany in 1992/93. Winfried Just used part of these notes
during the same academic year in a lecture course on set theory for first and second
year graduate students at Ohio University, Athens, Ohio. While doing this, the
idea of writing up an English text of the lecture crossed his mind. The idea proved
irresistible enough to result in this book.
The genesis of this text accounts for some departures from the traditional text-
book format. Generally speaking, this is not so much a textbook that could form
the backbone of a lecture, but rather the text of the lecture itself. Accordingly,
our language is perhaps closer to spoken, colloquial. English than to the standard
style of mathematical writing. Frequently, the text takes on the form of a dialogue
between the authors and the reader. The exercises form an integral part of this
dialogue and are not relegated to the end of sections.
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.
We do feel some remorse for the total exclusion of descriptive set theory from
this text. This is a very important branch of set theory, and it overlaps with several
other areas of mathematics, most notably topology and recursion theory. However,
just adding a section on descriptive set theory to a text like this would look artificial
and do no justice to the area. It seems to us that one should either cover a lot of
descriptive set theory, or none at all.
At the end of most sections, 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.
This book owes its existence as much to our students and colleagues as it does
to its authors. Parts of the original version of Martin Weese's lecture notes were
1 In German.