## The Higher Infinite: Large Cardinals in Set Theory from Their BeginningsThe higher in?nite refers to the lofty reaches of the in?nite cardinalities of set t- ory as charted out by large cardinal hypotheses. These hypotheses posit cardinals that prescribe their own transcendence over smaller cardinals and provide a sup- structure for the analysis of strong propositions. As such they are the rightful heirs to the two main legacies of Georg Cantor, founder of set theory: the extension of number into the in?nite and the investigation of de?nable sets of reals. The investigation of large cardinal hypotheses is indeed a mainstream of modern set theory, and they have been found to play a crucial role in the study of de?nable sets of reals, in particular their Lebesgue measurability. Although formulated at various stages in the development of set theory and with different incentives, the hypotheses were found to form a linear hierarchy reaching up to an inconsistent extension of motivating concepts. All known set-theoretic propositions have been gauged in this hierarchy in terms of consistency strength, and the emerging str- ture of implications provides a remarkably rich, detailed and coherent picture of the strongest propositions of mathematics as embedded in set theory. The ?rst of a projected multi-volume series, this text provides a comp- hensive account of the theory of large cardinals from its beginnings through the developments of the early 1970’s and several of the direct outgrowths leading to the frontiers of current research. |

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### Contents

1 | |

6 | |

7 | |

Beginnings | 15 |

Partition Properties | 68 |

Forcing and Sets of Reals | 113 |

Aspects of Measurability | 208 |

Inner Models of Measurability | 261 |

25 Combinatorics of PKy | 343 |

26 Extenders | 352 |

Deterrninacy 27 Inﬁnite Games | 368 |

28 AD and Combinatorics | 383 |

29 Prewellorderings | 403 |

30 Scales and Projective Ordinals | 430 |

31 Dem111 | 437 |

32 Consistency of AD | 450 |

Embeddings 0 and 0T | 277 |

Strong Hypotheses | 297 |

22 Supercompactness | 298 |

23 Extendibility to Inconsistency | 311 |

24 The Strongest Hypotheses | 325 |

Chart of Cardinals | 472 |

Appendix | 473 |

Indexed References | 483 |

531 | |

### Other editions - View all

The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings Akihiro Kanamori Limited preview - 2003 |

The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings Akihiro Kanamori Limited preview - 2008 |

### Common terms and phrases

analysis applied argument assertion Assume Axiom basic Borel characterization closed unbounded collapse compact complete concept condition consequence considered consistency construction contradiction corresponding countable deﬁnable deﬁned deﬁnition determinacy developed direction elementary embedding equivalent established Exercise existence extended ﬁlter Finally ﬁrst follows forcing formula function given Hence hierarchy holds homogeneous hypotheses ideal implies inaccessible induction inﬁnite initial inner model iterable language large cardinals later leads least Lebesgue measurable Lemma limit Martin mathematical measurable cardinal Moreover normal ultraﬁlter observed ordinal particular partition perfect play possible principle problem projective proof Proposition recursive regular relation remarks replaced result satisfying sense sequence set of reals set theory simple Skolem Solovay stationary strategy strong structure subset sufﬁces Suppose taking Theorem transitive tree ultrapowers University weakly well-founded well-ordering winning Woodin