Descriptive set theory and forcing : how to prove theorems about Borel sets the hard way
Author(s)
Bibliographic Information
Descriptive set theory and forcing : how to prove theorems about Borel sets the hard way
（Lecture notes in logic, 4）
Springer-Verlag, c1995
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- : us
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Includes bibliographical references and index
Description and Table of Contents
Description
This advanced graduate course assumes some knowledge of forcing as well as some elementary mathematical logic, e.g. the Lowenheim-Skolem Theorem. The first half deals with the general area of Borel hierarchies, probing lines of enquiry such as the possible lengths of a Borel hierarchy in a separable metric space. The second half goes on to include Harrington's Theorem together with a proof and applications of Louveau's Theorem on hyperprojective parameters.
Table of Contents
1 What are the reals, anyway?.- I On the length of Borel hierarchies.- 2 Borel Hierarchy.- 3 Abstract Borel hierarchies.- 4 Characteristic function of a sequence.- 5 Martin's Axiom.- 6 Generic G?.- 7 ?-forcing.- 8 Boolean algebras.- 9 Borel order of a field of sets.- 10 CH and orders of separable metric spaces.- 11 Martin-Solovay Theorem.- 12 Boolean algebra of order ?1.- 13 Luzin sets.- 14 Cohen real model.- 15 The random real model.- 16 Covering number of an ideal.- II Analytic sets.- 17 Analytic sets.- 18 Constructible well-orderings.- 19 Hereditarily countable sets.- 20 Shoenfield Absoluteness.- 21 Mansfield-Solovay Theorem.- 22 Uniformity and Scales.- 23 Martin's axiom and Constructibility.- 24
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$$ sets.- III Classical Separation Theorems.- 26 Souslin-Luzin Separation Theorem.- 27 Kleene Separation Theorem.- 28
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$$
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$$ equivalence relations.- 33 Louveau's Theorem.- 34 Proof of Louveau's Theorem.- References.- Elephant Sandwiches.
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