CHAPTER 11

The Löwenheim-Skolem Theorem 1

What is that thing which does not give itself, and which if it were to give itself would not exist? It is the infinite! (Leonardo da Vinci)

We say: but that isn’t how it is!—it is like that though! and all we can do is keep repeating these antitheses. (Ludwig Wittgenstein)


1 An introduction to the Löwenheim-Skolem theorem. Reactions and counter-reactions

At the end of Chapter 8, I mentioned two results in mathematical logic, those established by Skolem and Gödel, which both have a direct bearing on the infinite. Each is a rich supply of material lying ready to be woven into our understanding of the infinite. Each can be used to strengthen our grasp of the basic issues and problems that have begun to arise. I shall devote this chapter to a study of Skolem’s result, the next to Gödel’s.

Before I begin, I should emphasize that the two results are as far beyond controversy as any piece of pure mathematics can be. It is true that I shall be taking for granted certain methods of proof that have been challenged (for example, by intuitionists). But for current purposes we do best to take the results as a kind of datum. We can think of the really interesting philosophical dialectic as beginning at the point where their import is being probed and they are being used to illustrate, defend, or challenge non-mathematical ideas.

I have referred in various different ways to what it was that Skolem proved, but since in fact he was embellishing a result that had earlier been established by the mathematician Löwenheim, his theorem is usually referred to as the Löwenheim-Skolem theorem. It is beyond the scope of this book to go into the details of the theorem, but I shall try to present its essence. 2

Suppose the iterative conception of a set which was outlined in the last chapter either to be, or somehow to have been made, fully determinate. In particular suppose that every sentence in the language in which ZF is

-159-

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The Infinite
Table of contents

Table of contents

  • 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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