The relation between Cantor's view of sets and the axiomatic treatment of the
theory is similar to that between our space intuitions and Euclidean geometry. The
Euclidian axioms allow us to derive mathematical truths about points, lines and
planes deductively. This is important, because the deductive method accounts for
the high confidence mathematicians have in the truth of their theorems. However,
our "naive" space intuitions are a valuable guide in guessing these theorems and
outlines of their proofs. Also, similarities between geometric objects and features
of the real world are grasped by our intuition, not by deductive reasoning. Without
naivity, there would be no applications of mathematics.
When we say that somebody practices naive set theory, all we mean is that her
arguments are based on Cantor's definition of a set. The naive approach is quite
often the most enlightening one. If a mathematician's reasonings are also informed
by a careful analysis of how sets are being built, we say that she practices axiomatic
set theory. Frequently, this will just mean adopting the architect's point of view
without being concerned about details of the
In this book, both modes of set-theoretical thought will be practiced. We start
out with a naive treatment of some of the basics: relations, functions, equipotency,
order types and induction. As we go along, questions will arise that call for a more
careful scrutiny. Chapters 5 and 6 contain a discussion of the axiomatic method,
and in Chapter 7 we introduce the axioms. Next, we reexamine some of the material
developed in the first four chapters from an axiomatic point of view. In Volume II,
we discuss some set-theoretic results and techniques that we consider particularly
important from the point of view of other mathematical disciplines. There we shall
frequently switch from the naive to the axiomatic stance, and vice versa.
Of course, the choice of topics reflects our own biases and our desire to keep the
number of pages finite.
The quote of Hilbert is taken from an address delivered on June 4, 1925. The
text is printed in Uber das Unendliche, Math. Ann. 95 (1926), 161-190.
The quote of Poincare is taken from his article L 'Avenir des mathematiques,
Atti del IV Congresso Internazionale dei Matematici. Rome, 6-11 April 1908,
Rome, Tipografia della R. Academia dei Lincei, C. V. Salviucci, 1909, 167-182.
We thank David Fremlin for pointing out that Poincare's views were misquoted in
the first printing of this book.
Cantor's famous definition of a set is the first sentence of the article Beitrage zur
Begriindung der transfiniten Mengenlehre, Part I, Math. Ann. 46 (1895), 481-512.
Detailed accounts of the history of set theory in general, and of Cantor's work
in particular can be found in the following books:
Joseph Warren Dauben, Georg Cantor. His Mathematics and Philosophy of the
Infinite, Princeton University Press, 1990.
Michael Hallett, Cantorian Set Theory and Limitation of Size, Clarendon Press,
Gregory H. Moore, Zermelo's Axiom of Choice: Its Origins, Development and
Influence, Springer-Verlag, 1982.
The use of the phrase "axiomatic set theory" in such instances may not be entirely appro-
priate. But it is commonly used, and there is no need to further complicate the picture by naming
additional modes of practicing set theory.