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