## Set Theory: The Third Millennium Edition, revised and expandedSet Theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The present book covers each of these areas, giving the reader an understanding of the ideas involved. It can be used for introductory students and is broad and deep enough to bring the reader near the boundaries of current research. Students and researchers in the field will find the book invaluable both as a study material and as a desktop reference. |

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### Contents

1 Axioms of Set Theory | 3 |

2 Ordinal Numbers | 16 |

3 Cardinal Numbers | 27 |

4 Real Numbers | 37 |

5 The Axiom of Choice and Cardinal Arithmetic | 46 |

6 The Axiom of Regularity | 63 |

7 Filters Ultraﬁlters and Boolean Algebras | 72 |

8 Stationary Sets | 91 |

23 The Nonstationary Ideal | 440 |

24 The Singular Cardinal Problem | 457 |

25 Descriptive Set Theory | 479 |

26 The Real Line | 510 |

Selected Topics | 542 |

27 Combinatorial Principles in L | 543 |

28 More Applications of Forcing | 557 |

29 More Combinatorial Set Theory | 573 |

9 Combinatorial Set Theory | 106 |

10 Measurable Cardinals | 125 |

11 Borel and Analytic Sets | 139 |

12 Models of Set Theory | 154 |

Advanced Set Theory | 173 |

13 Constructible Sets | 175 |

14 Forcing | 201 |

15 Applications of Forcing | 225 |

16 Iterated Forcing and Martins Axiom | 266 |

17 Large Cardinals | 285 |

18 Large Cardinals and L | 310 |

19 Iterated Ultrapowers and LU | 339 |

20 Very Large Cardinals | 365 |

21 Large Cardinals and Forcing | 388 |

22 Saturated Ideals | 409 |

30 Complete Boolean Algebras | 584 |

31 Proper Forcing | 601 |

32 More Descriptive Set Theory | 615 |

33 Determinacy | 627 |

34 Supercompact Cardinals and the Real Line | 646 |

35 Inner Models for Large Cardinals | 659 |

36 Forcing and Large Cardinals | 669 |

37 Martins Maximum | 681 |

38 More on Stationary Sets | 695 |

707 | |

Notation | 733 |

743 | |

748 | |

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### Common terms and phrases

antichain Axiom of Choice Borel set canonical closed unbounded set coﬁnal compact cardinal complete Boolean algebra construct contradiction Corollary countable chain condition deﬁnable deﬁned Deﬁnition denote dense disjoint elementary embedding elementary submodel elements Exercise exists extension ﬁlter ﬁnd ﬁnite ﬁrst follows formula function f hence ideal inaccessible cardinal indiscernibles induction inﬁnite inner model isomorphic large cardinals Lebesgue measurable Lemma let f Let G limit ordinal meager measurable cardinal model of ZFC nonempty nonprincipal normal measure notion of forcing one-to-one order-type partially ordered partition proof of Theorem prove regular cardinal satisﬁes semiproper sequence set of reals set theory Shelah singular cardinal Skolem Solovay stationary set stationary subset subalgebra supercompact Suslin tree transitive model ultraﬁlter ultrapower weakly compact well-founded well-ordering winning strategy Woodin cardinal