Preface
Over many years of selecting instructional materials for my courses, I came to understand that
textbooks that are congenial to students obey a law of reciprocity that I wish to propose as an axiom:
Axiom The more effort an author puts into writing a text, the less effort is required of the reader to
understand it.
I adopted this guiding principle as a categorical imperative during the writing of this book. Even after
all the mathematics was in place, I reread and rewrote many times and tested explanatory strategies with
my students. I have dedicated this book to them because they have been my most patient and honest
critics. The book owes its present form largely to them and to their (sometimes naïve, often brilliant)
suggestions.
Mathematics has a superbly efficient language by means of which vast amounts of information can be
elegantly expressed in a few formal definitions and theorems. It is remarkable that the life work of
consecutive generations of great thinkers can often be summed up in a set of equations. The economy of
the language masks the richness and complexity of the thoughts that lie behind the symbols. Every
mathematics student has to master the conventions for using its language effectively. However, what is
far more important is that the student be initiated into the inner life of mathematics—the images, the
intuitions, the metaphors that, once grasped, make us say, “Aha! Now I understand! Now I see it!” This
inner seeing is what makes mathematics vital and exciting.
What is most unique about set theory is that it is the perfect amalgam of the visual and the abstract.
The notions of set theory, and the ideas behind many of the proofs, present themselves to the inner eye
in vivid detail. These pictures are not as overtly visual as those of geometry or calculus. You don’t see
them in the same way that you see a circle or a tangent line. But these images are the way into
abstraction. For the maturing student, the journey deeper into abstraction is a rite of passage into the
heart of mathematics.
Set theory is also the most “philosophical” of all disciplines in mathematics. Questions are bound to
come up in any set theory course that cannot be answered “mathematically”, for example with a formal
proof. The big questions cannot be dodged, and students will not brook a flippant or easy answer. Is the
continuum hypothesis a fact of the world? Is the axiom of choice a truth? If we cannot answer with a
definite yes or no, in what manner are they justified? (That one requires a long answer). What is the
meaning of the Jacob’s ladder of successive infinities so high that the very thought of it leads to a kind
of intellectual vertigo? To what extent does mathematics dwell in a Platonic realm—and if it does, then
in the words of Eugene Wigner, “how do you explain the unreasonable effectiveness of mathematics in
the natural world?” Or as Descartes wondered, where do you find the nexus between the material world
and the products of thought?
In this book I have tried, insofar as possible, not to evade these questions nor to dwell on them
excessively. Students should perceive that mathematics opens doors to far-reaching and fascinating
questions, but on the other hand they must remain anchored in mathematics and not get lost in the
narcotic haze of speculative thought. I have tried to provide the necessary background for understanding
axiomatics and the purpose and meaning of each of the axioms needed to found set theory. I have tried
to give a fair account of the philosophical problems that lie at the center of the formal treatment of
infinities and other abstractions. Above all, of course, I have endeavored to present the standard topics
of set theory with uncompromising rigor and precision, and made it clear that the formalism on the one
hand, and the intuitive explanations on the other hand, belong in two separate domains, one useful for
understanding, the other essential for doing mathematics.
