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.
Author(s): Arnold W. Miller
Series: Lecture Notes in Logic 4
Publisher: Cambridge University Press
Year: 2017
Language: English
Pages: 136
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
