of this writing, the use of elementary submodels seemingly is becoming popular
with general topologists. In order to promote this trend, we decided to include a
section on elementary submodels.
The history of set theory has not always been a smooth ride. In fact, the
start was rather bumpy. While some mathematicians embraced set theory eagerly,
others were openly hostile. The two quotes at the beginning of this Introduction
are soundbites of rhetoric from the early days of set theory.
The power of the set concept lies in the possibility of treating collections of in-
finitely many objects m as a single entity M. Many contemporaries of Cantor felt
uneasy about this approach. The question as to whether infinity actually exists,
or is just an abstraction, a remote possibility that can be considered and approxi-
mated, but never attained, is as old as philosophy itself. For the Greek philosopher
Plato, infinity was as
as any finite object. His disciple Aristotle took the
opposite stand: Infinity exists only as a potential that is never actually attained.
The chasm between the Platonist and the Aristotelian approaches has permeated
philosophical thought ever since. In essence, Cantor's treatment of infinity followed
Plato, whereas his opponents espoused Aristotelian thinking. Nihil novi ... , —and
yet, for a brief moment in history the Platonist case seemed lost. In 1901, Bertrand
Russell discovered that Cantor's definition of a set leads to a contradiction.
Let us say that an object x has property V, if x is a set, but x is not an element
of itself, which will be denoted by x £ x. Let us collect all objects x with property
V into a set M. Does M have property VI Well, suppose M does not have property
V. Then it is not the case that M £ M, i.e., M e M. But, as an element of M, the
set M must have property V, which contradicts our assumption. If we assume that
M has property V, then M would have to be an element of M, i.e., M would have
to be an element of itself, which contradicts the assumption that M has property
V. Thus, it is impossible for M to have property V, and it is impossible for M to
not have this property. This dilemma is called Russell's Paradox.
EXERCISE 2(PG) (a) Go carefully over the above argument and see whether
our reasoning is perhaps wrong or incomplete.
(b) Should you find a gap in the reasoning, fill in the missing details, and see
whether you still get a paradox.
Can one resolve Russell's Paradox, or do we have to accept it as a refutation of
set theory? Of course it can be resolved; otherwise this book would not exist. Let
us go back to Exercise 2(a). As you noticed, we forgot to check whether M is a set.
Property V has two clauses. If M is a set, then M has property V iff M £ M; but
if M is not a set, then M does not have property Vy period. In the latter case, the
paradox would disappear.
But is M a set? Let us consider Cantor's definition. In the spirit of his theory,
collections of definite, distinct objects of our thought are sets. M satisfies the
criterion of distinctness, since its members are distinguished from its nonmembers
by a certain property. Thus, Cantor's definition implies that M is a set, and we
get Russell's Paradox. (If you postponed Exercise 2(b), do it right now.)
We have seen that Russell's Paradox disappears if M is not a set. We have also
seen that Cantor's definition implies that M i s a set. Is there perhaps something
wrong with Cantor's definition?
In a sense, infinity was even more real for Plato than the finite objects of our perception.