CHAPTER 12

Gödel’s Theorem

The human mind is incapable of formulating…all its mathematical intuitions, ie., if it has succeeded in formulating some of them, this very fact yields new intuitive knowledge, eg., the consistency of this formalism. This fact may be called the ‘incompletability’ of mathematics.

(Kurt Gödel)


1 Introduction: the Euclidean paradigm

Gödel’s theorem is one of the most profound results in pure mathematics. When it was first published, in 1931, it had a devastating impact. On the one hand, it laid waste a variety of firmly held convictions and initiated a struggle that has been going on ever since to come to terms with its mathematical and philosophical implications. On the other hand, it took the breath away for its sheer beauty. My aim in this chapter is to present an outline of the theorem and to say what some of its implications are for our own enquiry.

In a nutshell, it concerns the Euclidean paradigm—the paradigm of axiomatization. It is possible, we know, to devise a finite stock of fundamental principles or axioms from which all of the infinitely many truths of Greek geometry can be derived: this is the Euclidean paradigm. 1 Prior to 1931 many people had assumed that what was possible in geometry must be possible anywhere else in mathematics (and perhaps in non-mathematical contexts too); the paradigm must represent the very essence of mathematical method. 2 One of the reasons for this relates back to our discussion in the last chapter. Suppose we grant that the meaning of a mathematical expression has to be grasped in terms of how it figures in the truths of a formal theory. Then must there not be some way of ‘capturing’ these truths and providing them with a finite characterization—precisely what an axiomatization (and that alone?) can supply? How else could anyone assimilate the truths and grasp the expression’s meaning? Again, relatedly, do we have any sense of mathematical truth apart from mathematical provability? When we say that a given mathematical state-

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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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