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
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
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
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
belong to naive or to ax-
iomatic set theory?
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.
you do not know what the Continuum Hypothesis is, ignore this exercise.