## Understanding the InfiniteHow can the infinite, a subject so remote from our finite experience, be an everyday working tool for the working mathematician? Blending history, philosophy, mathematics and logic, Shaughan Lavine answers this question with clarity. An account of the origins of the modern mathematical theory of the infinite, his book is also a defense against the attacks and misconceptions that have dogged this theory since its introduction in the late 19th century. |

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### Contents

Introduction | 1 |

II Infinity Mathematics Persistent Suitor | 11 |

1 Incommensurable Lengths Irrational Numbers | 12 |

2 Newton and Leibniz | 15 |

3 Go Forward and Faith Will Come to You | 22 |

4 Vibrating Strings | 26 |

5 Infinity Spurned | 32 |

6 Infinity Embraced | 37 |

Knowing the Infinite 1 What Do We Know? | 154 |

2 What Can We Know? | 162 |

3 Getting from Here to There | 181 |

4 Appendix | 203 |

Leaps of Faith 1 Intuition | 213 |

2 Physics | 218 |

3 Modality | 221 |

4 SecondOrder Logic | 224 |

Sets of Points 1 Infinite Sizes | 42 |

2 Infinite Orders | 44 |

3 Integration | 49 |

4 Absolute vs Transfinite | 51 |

5 Paradoxes | 57 |

What Are Sets? 1 Russell | 63 |

2 Cantor | 76 |

Letter from Cantor to Jourdain 9 July 1904 | 98 |

On an Elementary Question of Set Theory | 99 |

1 The Axiom of Choice | 103 |

2 The Axiom of Replacement | 119 |

3 Definiteness and Skolems Paradox | 123 |

4 Zermelo | 134 |

5 Go Forward and Faith Will Come to You | 141 |

1 Who Needs SelfEvidence? | 241 |

2 Picturing the Infinite | 246 |

3 The Finite Mathematics of Indefinitely Large Size | 267 |

4 The Theory of Zillions | 288 |

Extrapolations 1 Natural Models | 309 |

2 Many Models | 314 |

3 One Model or Many? Sets and Classes | 316 |

4 Natural Axioms | 320 |

5 Second Thoughts | 322 |

6 Schematic and Generalizable Variables | 325 |

Bibliography | 329 |

349 | |

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### Common terms and phrases

actually analysis argument Axiom of Choice Axiom of Replacement axiomatic axiomatic set theory axioms of set background set theory Cantor Cantorian cardinal combinatorial collections commitment consistent construction Continuum Hypothesis counted defined definition denumerable discussed epistemic example extrapolation fact Fin(T Fin(ZFC finitary finitary mathematics finite mathematics finite set theory finitist first-order logic formal Foundation Fourier Fraenkel Frege given Godel hfpsets Hilbert idea indefinitely large infinitary infinite sets infinity intuition intuitionist irrational numbers isomorphic iterative conception limit mathematical objects mathematicians natural model natural numbers Neumann normal domain notion ordinal numbers paradoxes Peano arithmetic philosophy of mathematics possible Power Set predecessors primitive recursive principle problem proof propositional functions prove quantifiers quasi-categoricity rational numbers real numbers relation Replacement result Russell Russell's schematic variables second-order logic second-order set theory self-evident sense sentence sequence set-theoretic Skolem subset symbols theorem tion transfinite true truth well-ordered set Zermelo Zillion