Glossary
: the infinite cardinal whose index is the ordinal α (Chapter 10, §3)
analysis: the theory of the real numbers (Chapter 4, §1). (The adjective corresponding to ‘analysis’ is ‘analytic’. )
apeiron, to (Greek): the unlimited or unbounded, the infinite (Chapter 1, §1)
arithmetic: the theory of the natural numbers (Chapter 1, §5; see also Chapter 12, §2)
cardinal number (or cardinal): number which registers the size of a set (Chapter 10, §3). (An infinite cardinal registers the size of an infinite set. )
continuum hypothesis, the: Cantor’s hypothesis that (Chapter 10, §3). (This is equivalent to the hypothesis that the power-set of , or , is the next infinite size up from . )
countable (of a set): either finite or the smallest infinite size (Chapter 10, §3)
: the set of natural numbers (Chapter 8, §3)
natural number: non-negative whole number (Introduction, §2). (The natural numbers are 0, 1, 2, …)
Ω: the set of ordinals (Chapter 8, §4)ω: the first ordinal to succeed all the natural numbers (Chapter 8, §4)
ordinal number (or ordinal): number which registers the length, or shape, of a well-ordering (Chapter 8, §4). (The ordinals are 0, 1, 2, …, ω, ω+1, …)
peras (Greek): limit or bound (Chapter 1, §1)
power-set: set of subsets of a given set (Chapter 8, §3)

-234-

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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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