INTRODUCTION 3

possible to prove that neither a proof nor a refutation of the Virtual Conjecture

is among "the collected reasonings of all mathematicians of all times." Does this

constitute a proof of Theorem 1? Almost. The assumption that all mathematics

can be formalized in set theory is an act of belief that does not lend itself to math-

ematical scrutiny. While we can be reasonably sure that set theory encompasses

essentially all correct mathematical arguments that have been used by mathemati-

cians up to this point in

history1,

there is always the somewhat remote possibility

that eventually somebody will discover an immediately recognizable mathematical

truth that transcends set theory. Thus, if it can be established that neither a proof

nor a refutation of the Virtual Conjecture is among "the collected reasonings of all

mathematicians of all times," something like the following theorem will have been

proved:

Theorem 2: The Virtual Conjecture is unsolvable, unless currently used foun-

dations of mathematics are changed.

Over the last three decades, set theorists have proved hundreds of theorems like

Theorem 2. Such theorems are called independence results. The Virtual Conjecture

does not have to be a strictly set-theoretical statement. It may be a problem in

topology, algebra, functional analysis, or measure theory.

On the other hand, the foundations of mathematics have remained remarkably

stable. Most mathematicians accept the axiomatic version ZFC2 of set theory as a

reasonably good foundation of mathematics and see little reason to exchange it for

something else.3 Thus, evidence is accumulating that many problems in set theory

and related fields may be unsolvable in an absolute sense. However, if they are, we

can never be entirely sure of this.

Set theory not only serves mathematics by providing a foundation and allowing

one to delineate the limits of the knowable. It also is good mathematics. Set-

theoretic theorems and techniques can be used in many other branches of math-

ematics much in the same way as linear algebra is used in differential equations.

This is true not only for the concepts and methods known already to Cantor, but

also for more recent results like Zorn's Lemma, the Erdos-Rado Theorem, or the

Pressing Down Lemma. Volume II of this book is devoted to a presentation of set-

theoretic techniques that are frequently applied outside of set theory. At the time

xOf

course, this becomes immediately a false statement if the deliberately vague term "set

theory" is replaced by a formal incarnation of it, like ZFC. In this case, the "immediately recog-

nizable mathematical truth that transcends ZFC" could be the assertion that ZFC is consistent.

However, if the "Virtual Conjecture" is something like the Continuum Hypothesis, then the intu-

itive picture drawn here will do for a reasonably accurate first approximation of the notion of an

independence result.

2 The letters stand for Zermelo and Fraenkel, who developed the system, and for one of the

axioms, called the Axiom of Choice.

3

Not all mathematicians share this view. Mathematics can be developed in other frameworks.

Some of these are brands of set theory similar to the version ZFC discussed in this text; others are

entirely different approaches. In this book, we concentrate almost exclusively on a presentation of

ZFC. (The only exceptions are occasional discussions of set theory without the Axiom of Choice.)

We are far from claiming superiority of ZFC over alternative foundations of mathematics. For

whatever reason, it won the competition. It does a decent job; so let us stick to it. It should be

pointed out though that, to the best of our knowledge, none of the competitors of ZFC resolves

the question of truth or falsity of CH, SH, MA, ), or of any other statement whose independence

of ZFC has been established by the method of forcing.