Descriptive Set Theory And Forcing: How To Prove Theorems About Borel Sets The Hard Way (lecture Notes In Logic)

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the fourth publication in the Lecture Notes in Logic series, Miller develops the necessary features of the theory of descriptive sets in order to present a new proof of Louveau's separation theorem for analytic sets. While some background in mathematical logic and set theory is assumed, the material is based on a graduate course given by the author at the University of Wisconsin, Madison, and is thus accessible to students and researchers alike in these areas, as well as in mathematical analysis. Contents......Page 7 1 What are the reals, anyway?......Page 10 2 Borel Hierarchy......Page 12 3 Abstract Borel hierarchies......Page 16 4 Characteristic function of a sequence......Page 18 5 Martin's Axiom......Page 21 6 Generic Gδ......Page 23 7 α-forcing......Page 26 8 Boolean algebras......Page 31 9 Borel order of a field of sets......Page 35 10 CH and orders of separable metric spaces......Page 37 11 Martin-Solovay Theorem......Page 39 12 Boolean algebra of order ω1......Page 43 13 Luzin sets......Page 47 14 Cohen real model......Page 51 15 The random real model......Page 62 16 Covering number of an ideal......Page 69 17 Analytic sets......Page 73 18 Constructible well-orderings......Page 76 19 Hereditarily countable sets......Page 77 20 Shoenfield Absoluteness......Page 79 21 Mansfield-Solovay Theorem......Page 81 22 Uniformity and Scales......Page 82 23 Martin's axiom and Constructibility......Page 87 24 Σ 1 2 well-orderings......Page 89 25 Large Π 1 2 sets......Page 90 26 Souslin-Luzin Separation Theorem......Page 93 27 Kleene Separation Theorem......Page 95 28 Π 1 1-Reduction......Page 98 29 Δi-codes......Page 100 30 Π 1 1 equivalence relations......Page 103 31 Borel metric spaces and lines in the plane......Page 108 32 Σ 1 1 equivalence relations......Page 112 33 Louveau's Theorem......Page 116 34 Proof of Louveau's Theorem......Page 122 References......Page 126 Index......Page 133 Elephant Sandwiches......Page 135 These notes develop the theory of descriptive sets, leading up to a new proof of Louveau's separation theorem for analytic sets. A first course in mathematical logic and set theory is assumed, making this book suitable for advanced students and researchers.