## Cantorian Set Theory and Limitation of SizeCantor's ideas formed the basis for set theory and also for the mathematical treatment of the concept of infinity. The philosophical and heuristic framework he developed had a lasting effect on modern mathematics, and is the recurrent theme of this volume. Hallett explores Cantor's ideas and, in particular, their ramifications for Zermelo-Frankel set theory. |

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### Contents

The Cantorian origins of set theory | 1 |

Cantors theory of infinity | 12 |

The ordinal theory of powers | 49 |

Cantors theory of number | 119 |

The origin of the limitation of size idea | 165 |

The limitation of size argument and axiomatic set theory | 195 |

The completability of sets | 214 |

The Zermelo system | 240 |

Von Neumanns reinstatement of the ordinal theory of size | 270 |

Conclusion | 299 |

Bibliography | 307 |

321 | |

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### Common terms and phrases

1ndeed absolute collections abstraction actual infinite aleph antinomies argument assumed assumption attempt axiom of choice axiom of infinity axiom of replacement axiom of separation axiomatic set theory Borel Burali-Forti Cantor's theory Cantorian certainly clear completed comprehension construction continuum hypothesis continuum problem contradiction countable counting crucial Dedekind defined definition doctrine domain elements enumeral example explain extension finite numbers follows Fraenkel Frege function Godel Hausdorff Hessenberg heuristic hierarchy idea important impredicative infinite sets infinity intuition Jourdain Kant Kant's large cardinal later limitation limitation of size logical Mirimanoff natural numbers Neumann notion number sequence number-class order-type ordinal numbers ordinal theory paradoxes point-sets possible power-set axiom precisely principle proof properties proved real numbers reductionism reductionist remarks Russell Russell's seems sense set theory set-theoretic Shoenfield simple stage subset suggests theory of cardinality theory of number tion transfinite numbers transfinite ordinal uncountable universe well-ordered set well-ordering theorem