## Handbook of Set TheoryMatthew Foreman, Akihiro Kanamori Numbers imitate space, which is of such a di?erent nature —Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century. |

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### Contents

93 | |

129 | |

Coherent Sequences | 214 |

Borel Equivalence Relations | 297 |

Proper Forcing | 333 |

Combinatorial Cardinal Characteristics of the Continuum | 395 |

Invariants of Measure and Category | 491 |

Constructibility and Class Forcing | 556 |

Cardinal Arithmetic | 1148 |

Successors of Singular Cardinals | 1229 |

PrikryType Forcings | 1351 |

Beginning Inner Model Theory | 1448 |

The Covering Lemma | 1497 |

An Outline of Inner Model Theory | 1595 |

A Core Model Toolbox and Guide | 1685 |

Structural Consequences of AD | 1752 |

Fine Structure | 605 |

Sigma Fine Structure | 657 |

Elementary Embeddings and Algebra | 737 |

Iterated Forcing and Elementary Embeddings | 775 |

Ideals and Generic Elementary Embeddings | 885 |

Determinacy in LR | 1877 |

Large Cardinals from Determinacy | 1951 |

2121 | |

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### Common terms and phrases

antichain apply argument Aronszajn tree assume axioms Boolean algebra Borel equivalence relations bounded closed unbounded club cofinal Cohen collapse combinatorial consistency construction Corollary countably complete cov(B defined Definition denote dense disjoint elementary embedding elements example exists extension filter finite forcing poset function f ground model Hence holds hypothesis implies inaccessible induction infinite initial segment inner model intersection isomorphic Jensen large cardinal Lemma Let G limit ordinal Logic Magidor maximal antichain measurable cardinal morphism n e w node nonstationary ideal normal order type P-generic partial ordering poset preserves proof proper forcing Proposition prove recursion regular cardinal result Saharon Shelah satisfies Sect sequence set theory singular cardinal stationary set stationary subset structure successor supercompact Suppose Theorem tree ultrafilter ultrapower uncountable weakly compact Woodin cardinal ωω