## The Axiom of Determinacy, Forcing Axioms, and the Nonstationary IdealIdentifies a canonical model in which the continuum hypothesis (CH) is false. This model is a forcing extension of L(R) and the method can be varied to produce a wide class of similar models each of which can be viewed as a reduction of this model. Analysis of these models arises from an interplay between ideas from descriptive set theory and from combinatorial set theory. The original motivation for the definition of these models resulted from the discovery that it is possible, in the presence of large cardinals, to force the effective failure of CH. Contains chapters on preliminaries, the nonstationary ideal, the Pmax-extension, application, Pmax variations, conditional variations, and condensation principles. Includes a six-page section of questions and exercises. Annotation copyrighted by Book News, Inc., Portland, OR |

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### Contents

Introduction | 1 |

Preliminaries | 18 |

The nonstationary ideal | 48 |

The Pmaxextension | 115 |

Applications | 184 |

Pmax variations | 291 |

Conditional variations | 432 |

+ principles for b | 498 |

Extensions of LI R | 617 |

Further results | 769 |

Questions | 921 |

927 | |

### Common terms and phrases

A-iterable absoluteness theorem axiom borel Chang's Conjecture class of Woodin closed under continuous codes cofinal Coll(a collapsing map contains a club continuous preimages countable elementary substructure countable iteration countable transitive model definable Definition elementary embedding exists a countable exists an iteration filter G following hold following theorem G C Pmax immediate corollary implies indiscernible iteration lemmas iteration of length k<co key point L(R)-generic large cardinals Lemma limit ordinal Martin's Maximum measurable cardinal model of ZFC nonstationary ideal normal ideal normal uniform ideal partial order pointclass closed predense prewellordering proof of Lemma proof of Theorem proper class prove Qmax R)-generic semi-generic semi-saturated sentence sequence set of reals stationary set stationary subsets structure Suppose that G Suslin tree transitive collapse transitive inner model transitive set ultrafilter ultrapower universally Baire V-generic wellfounded witnesses Woodin cardinal