## Descriptive Set Theory and Definable Forcing, Volume 793 (Google eBook)The subject of the book is the relationship between definable forcing and descriptive set theory. The forcing serves as a tool for proving independence of inequalities between cardinal invariants of the continuum. The analysis of the forcing from the descriptive point of view makes it possible to prove absoluteness theorems of the type ``certain forcings are the provably best attempts to achieve consistency results of certain syntactical form'' and others. There are connections to such fields as pcf theory, effective descriptive set theory, determinacy and large cardinals, Borel equivalence relations, abstract analysis, and others. |

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### Contents

1 | |

The countable support iteration | 47 |

Other forcings | 71 |

Applications | 91 |

A Examples of cardinal invariants | 117 |

Effective descriptive set theory | 125 |

Large cardinals | 133 |

### Common terms and phrases

A C R a-ideal Adam's analytic set antichain arbitrary argument axioms basic open set Borel equivalence relation Borel function Borel perfect set Borel perfect subset Borel positive set Borel positive subset Borel subset cardinal invariants Claim closed sets Corollary countable elementary submodel countable ordinal countable set countable support iteration covering number defined definition dense set determinacy dichotomy disjoint dom(B element ergodic everg filter g forcing extension function f ground model coded homogeneous infinite integer isomorphic large enough structure Laver ideal Lemma maximal antichain modulo finite node open dense set open dense subsets open set partial order poset poset Pi positive Borel set preimages Proof proves rgen Sacks forcing sequence f set B C set Bn set of reals small sets Souslin structure containing Suppose theorem tree universally Baire set universallyBaire vertical sections winning strategy Woodin cardinals ZFC+LC

### Popular passages

Page vii - The subject of the book is the relationship between definable forcing and descriptive set