Descriptive Set Theory and Definable Forcing.

Intro -- Contents -- 1 Introduction -- 1.1 The subject of the book -- 1.2 The structure of the book -- 1.3 History and acknowledgments -- 1.4 Notation and literature -- 2 Definable forcing adding a single real -- 2.1 The factor algebras -- 2.2 Basic descriptive set theoretic considerations -- 2.3 Examples -- 2.3.1 The ideal of countable sets -- 2.3.2 The ideal of σ…bounded sets -- 2.3.3 The ideal of meager sets -- 2.3.4 The cmin ideal -- 2.3.5 Ideals generated by closed sets -- 2.3.6 The Laver ideal -- 2.3.7 Ideals associated with creature forcings -- 2.3.8 The Lebesgue null ideal -- 2.3.9 Mathias forcing -- 2.3.10 The E[sub(0)] ideal -- 2.3.11 Silver forcing -- 2.3.12 The σ…porous ideal -- 2.3.13 Steprans forcing -- 2.3.14 Hausdorff measures -- 2.3.15 Unions of ideals -- 2.3.16 Cross-products of ideals -- 2.3.17 The σ…splitting ideal -- 2.3.18 Namba forcing -- 3 The countable support iteration -- 3.1 A topological view of the iteration -- 3.2 The iterated Fubini powers of an ideal -- 3.3 A dichotomy for Π[sup(1)][sub(1)]on Σ[sup(1)][sub(1)] ideals -- 3.4 A dichotomy for almost full ideals -- 3.5 Other dichotomies -- 3.6 Cardinal invariants of the iterated ideals -- 4 Other forcings -- 4.1 Illfounded iterations -- 4.1.1 Strongly proper forcings -- 4.1.2 The ideals associated with countable length iterations -- 4.1.3 The properties of the factor ordering -- 4.1.4 The uncountable length -- 4.1.5 Sacks forcing iteration -- 4.2 Towers of ideals -- 4.2.1 Shooting a club with no infinite subset in the ground model -- 4.2.2 Shooting a club with finite intersection with every ground model ordertype w set -- 5 Applications -- 5.1 Ciesielski-Pawlikowski Axiom CPA and variations -- 5.1.1 The axioms -- 5.1.2 Absoluteness with no large cardinals -- 5.1.3 Absoluteness with large cardinals -- 5.2 Duality theorems -- 5.3 Interpolation theorems.

5.4 Preservation theorems -- 5.4.1 Ergodic ideals -- 5.4.2 Preservation and ergodicity -- 5.4.3 Uniformity of σ…ideal generated by closed sets -- A: Examples of cardinal invariants -- B: The syntax of cardinal invariants -- B.1 The covering numbers -- B.2 The tame and very tame invariants -- C: Effective descriptive set theory -- D: Large cardinals -- D.1 The absoluteness results -- D.2 The determinacy results.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.