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
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
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
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.
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.
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