INTRODUCTION

5

To see its flaw, let us reexamine the process by which the set M of our example

was constructed. M was the collection of certain definite objects of our thought,

distinguished by a property V But wait a minute, how "definite" are these

objects? If M might or might not be one of those, isn't M a bit indefinite? So,

maybe, we should disqualify M as a possible element of M on the grounds of its

indefiniteness?

EXERCISE 3(R): Try this and convince yourself that you get a similar paradox;

this time with the added clause of some vaguely understood "definiteness" in the

defining property of M.

But perhaps Cantor's definition could be salvaged by giving a precise meaning

to the word "definite?" Think of the elements of a set as building blocks, and the

formation of a set as assembling these building blocks into a whole. It is reasonable

to require that at the moment a given set M is being formed, all its building blocks

must have already attained their final shape; in this sense they should be "definite."

Let us call this stance the architect's view of set theory. It stipulates that although

it is possible to contemplate all sets at once, each set has to be formed at some

moment in an abstract "time," and at that moment, all its building blocks must

already have been available in their final shape.5 Also, once a set is formed, one

should be able to use it as a building block of other sets.

This view solves Russell's Paradox in an unexpected way: M is not a set, because

it could never have been assembled! At no moment in set-theoretic time do all the

building blocks for the construction of M exist.

How can the architect's view of set theory be expressed with sufficient math-

ematical precision? The approach concentrates not on what sets are, but on how

sets are being formed. At the beginning of set-theoretic time, the only set that can

be formed is the empty set, since no previous building blocks exist. Once this set is

formed, it can be used as a building block for further sets. The modern alternative

to Cantor's definition is to describe precisely by which operations new sets can be

built from existing ones, and then to apply these operations successively to the

empty set.

Can we get all sets in this way? Perhaps not, but we can construct a universe

of sets rich enough to encompass all known mathematics. This will do for starters.

The architect's view of set theory can be formalized by axioms, similar to the way

in which our space intuitions were formalized by Euclid more than two thousand

years ago. The axiom system ZFC that will be studied in this book was proposed by

E. Zermelo and A. Praenkel early in this century. Once an axiom system has been

formulated, one can ask whether a given mathematical statement or its negation

follows from the axioms. The answer may be a "yes," a "no," or an independence

result.

One often talks about "naive" versus "axiomatic" set theory. This may suggest

a much deeper partition than there actually is.

EXERCISE 4(R): Does the Continuum

Hypothesis6

belong to naive or to ax-

iomatic set theory?

5

Note that this view is a synthesis of Platonist and Aristotelian elements. We shall see in

Chapter 12 how the "timeline" of the set-theoretic universe looks.

6If

you do not know what the Continuum Hypothesis is, ignore this exercise.