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INTRODUCTION

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

axiomatization.7

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.

Mathographical Remarks

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,

Oxford, 1984.

Gregory H. Moore, Zermelo's Axiom of Choice: Its Origins, Development and

Influence, Springer-Verlag, 1982.

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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.