2
INTRODUCTION
But why bother? Many mathematicians, especially those of the more applied
persuasion, never use even such basic set-theoretic tools as arithmetic of infinite
ordinals in their research. Would it not be more reasonable to study set theory
just as a separate discipline, rather than trying to fit all other disciplines into a
set-theoretic straightjacket? Or, if the topologists really cannot live without the
straightjacket, shouldn't at least the applied areas be spared?
This suggestion misses the point on two counts. First, it is one thing to claim
that set theory could in principle serve as a foundation for all of mathematics, in-
cluding, say, differential equations; and it is quite another thing to seriously propose
that the Navier-Stokes equations should be expressed in the language of set the-
ory. If you tried your luck with Exercise 1, you know that the latter is an absurdity
which nobody in his right mind would seriously consider. Second, there is a definite
advantage to having a single framework for the separate subdisciplines of mathe-
matics. Different branches of mathematics build on each other. Analytic functions
are heavily used in number theory, for example. If each branch of mathematics had
its own separate foundations, each use of a theorem from another subfield might
raise foundational issues. The existence of a common, albeit at times clumsy, frame-
work makes such mathematical cross-breeding entirely unproblematic. It is exactly
the existence of the established common framework that allows most practioners
to just do mathematics and to leave all foundational issues to the specialists: set
theorists, logicians and philosophers.
The suggestion "to spare at least the applied areas from the straightjacket of set
theory" may sound funny, but it expresses a belief that is held in earnest by many
mathematicians and science administrators: that one can draw a clear dividing
line between applied and pure mathematics. Even those who do not share this
view tend to think that there are clear-cut instances of belonging to the realm of
either the pure or the applied. And yet, the distance between the most applied and
the most abstract may be surprisingly short. Consider probability theory. This is
as applied a field as any. To a mathematician, it is just the study of probability
measures. These are functions defined on certain cr-fields of sets, for instance on the
Lebesgue measurable subsets of the unit interval. Once this framework for doing
probability theory has been established, it is very natural to ask whether there
exists a probability function defined on all subsets of the unit interval so that, as
in the case of the familiar Lebesgue measure, each individual point has probability
zero. As we shall see in Chapter 23, this is one of the deepest and most perplexing
problems in set theory. Not only is it unsolved; there are indications that it may
even be unsolvable in a very strong sense.
Let us consider a hypothetical unsolved problem. Since this will be our substi-
tute for a real problem, let us call it the Virtual Problem. Let us assume that the
problem is to prove or refute the Virtual Conjecture. How would you like this:
Theorem 1: The Virtual Problem is unsolvable.
Well, if you have been working hard on the Virtual Problem without much luck,
Theorem 1 may be a consolation prize. But could one possibly prove a theorem like
Theorem 1? Yes and no. Intuitively speaking, if all of mathematics can be formal-
ized in set theory, then also all modes of mathematical reasoning can be formalized.
Thus, "the collected reasonings of all mathematicians of all times" become a math-
ematical object, and can be studied like any other mathematical object. It may be
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