# The Infinite

By A. W. Moore | Go to book overview

CHAPTER 10

Transfinite Mathematics 1

Thus quantum impels itself beyond itself; this other which it becomes is in the first place itself a quantum; but it is quantum as a limit which does not stay, but which impels itself beyond itself. The limit which again arises in this beyond is, therefore, one which simply sublates itself again and impels itself beyond to a further limit, and so on to infinity.

(G. W. F. Hegel)

It is often said that mathematics is the science of the infinite. 2 And yet, before the advent of Cantor’s work at the end of the nineteenth century, few mathematicians even looked upon the infinite as a serious object of mathematical study. Many still do not. This situation is not as crazy as it sounds. Even when the infinite is not itself serving as an object of mathematical study, mathematicians can still be said to be exploring the infinite insofar as what they are studying presupposes an infinite framework. (This was a point that first arose when we were looking at early Greek mathematics (see above, Chapter 1, §5). ) When the infinite does become an object of mathematical study, as in contemporary set theory (which is the modern development of Cantor’s pioneering work on the infinite), it is as if mathematicians have chosen to step back and scrutinize the framework itself. If mathematics is the science of the infinite, then set theory is self-conscious mathematics.

My aim in the next three chapters is to look further into that self-conscious mathematics and to explore other technical work that bears directly on the infinite. This work will serve as a useful peg on which to hang a number of more general ideas. It will help to crystallize many of the puzzles and conundrums that beset any inquiry into the infinite. Later in Part Two the discussion will be extended to broader, non-mathematical issues.

1 The iterative conception of a set. The paradox of the Set of ail Sets

In Chapter 8, §6, I sketched an intuitive picture of what sets are like, the picture which informs contemporary set theory. (The nine axioms of ZF

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The Infinite

• Title Page iii
• Contents vii
• Preface to the Second Edition xi
• Preface xx
• Introduction: Paradoxes of the Infinite 1
• Part One - The History 15
• Chapter 1 - Early Greek Thought 17
• Chapter 2 - Aristotle 34
• Chapter 3 - Medieval and Renaissance Thought 45
• Chapter 4 - The Calculus 57
• Chapter 5 - The Rationalists and the Empiricists 75
• Chapter 6 - Kant 84
• Chapter 7 - Post-Kantian Metaphysics of the Infinite 96
• Chapter 8 - The Mathematics of the Infinite, and the Impact of Cantor 110
• Chapter 9 - Reactions 131
• Part Two - Infinity Assessed 145
• Chapter 10 - Transfinite Mathematics 147
• Chapter 11 - The Löwenheim-Skolem Theorem 159
• Chapter 12 - Gödel's Theorem 172
• Chapter 13 - Saying and Showing 186
• Chapter 14 - Infinity Assessed. the History Reassessed 201
• Chapter 15 - Human Finitude 218
• Glossary 234
• Bibliography 250
• Index 261
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