XIV

HOW TO READ THIS BOOK

new to you, and begin reading right there. If things start to look new only around

Chapter 22 or so, you probably do not want to waste your time reading this book,

but go to the more advanced literature on the subject. In the "mathographical

remarks" at the end of most chapters, you will find ample suggestions for further

reading.

If things look new to you right from the beginning, check whether you know

most of the concepts and symbols listed under "Basic Notations". If so, read

Chapter 1, where some of the prerequisite material is reviewed. If Chapter 1 is

pleasant, easy reading, then you are probably ready for this book. If more than

two concepts listed under "Basic Notations" are entirely new to you, or if Chapter

1 feels challenging, then you may want to read one of the more elementary texts

listed in the mathographical remarks at the end of Chapter 1.

Roughly speaking, the only prerequisite for this book is that you are at ease

with set-theoretic notation. However, some knowledge of mathematical logic and

(for Volume II) general topology is indispensible. Therefore, to make the exposition

somewhat self-contained, we included a minicourse in mathematical logic in Chap-

ters 5 and 6, and also an Appendix on general topology at the end of Volume II.

Once you have determined your point of entrance, it is best to read the rest

of the book line by line. Much of this book is written like a dialogue between

the authors and the reader. This is intended to model the practice of creative

mathematical thinking, which more often than not takes on the form of an inner

dialogue in a mathematician's mind. You will quickly notice that this text contains

many question marks. This reflects our conviction that in the mathematical thought

process it is at least as important to have a knack for asking the right questions at

the right time as it is to know some of the answers.

You will benefit from this format only if you do your part and actively participate

in the dialogue. This means in particular: Whenever we pose a rhetorical question,

pause for a moment and ponder the question before you read our answer. Sometimes

we put a little more pressure on you and call our rhetorical questions EXERCISES.

Not all exercises are rhetorical questions that will be answered a few lines later.

Sometimes, the completion of a proof is left as an exercise. We also may ask you

to supply the entire proof of an interesting theorem, or an important example.

Nevertheless, we recommend that you attempt the exercises right away, especially

all the easier ones. Most of the time it will be easier to digest the ensuing text if

you have worked on the exercise, even if you were unable to solve it.

Here is a well-kept secret: All mathematical research papers contain plenty

of exercises. These usually appear under the disguise of seemingly unnecessary

assumptions, missing examples, or phrases like: "It is easy to see." One of the

most important steps in becoming a mathematician is to learn to recognize hidden

EXERCISES, and to develop the habit of tackling them right away.

We often make references to solutions of exercises from earlier chapters. Some-

times, the new material will make an old and originally quite hard exercise seem

trivial, and sometimes a new question can be answered by modifying the solution

to a previous problem. Therefore, it is a good idea to collect your solutions and

even your failed attempts at solutions in a folder where you can look them up later.

The level of difficulty of our exercises varies greatly. To help the reader save

time, we rated each exercise according to what we perceive as its level of difficulty.

The rating system is the same as used by American movie theatres. Everybody

should attempt the exercises rated G (general audience). Beginners are encouraged