Title: Descriptive Set Theory
and Forcing;
How to prove theorems about Borel sets the hard way.
Author: A.Miller
Email: miller@math.wisc.edu
Abstract:
These are lecture notes from a course I gave at the University of
Wisconsin during the Spring semester of 1993. Part 1 is concerned with
Borel hierarchies. Section 13 contains an unpublished theorem of Fremlin
concerning Borel hierarchies and MA. Section 14 and 15 contain new results
concerning the lengths of Borel hierarchies in the Cohen and random real
model. Part 2 contains standard results on the theory of Analytic sets.
Section 25 contains Harrington's Theorem that it is consistent to have
$\Pi^1_2$ sets of arbitrary cardinality. Part 3 has the usual separation
theorems. Part 4 gives some applications of Gandy forcing. We reverse
the usual trend and use forcing arguments instead of Baire category. In
particular, Louveau's Theorem on $\Pi^0_\alpha$ hyp-sets has a simpler
proof using forcing.
Table of Contents
1 What are the reals, anyway?
Part 1. 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_\delta $
7 $\alpha $-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 $\omega _1$
13 Luzin sets
14 Cohen real model
15 The random real model
16 Covering number of an ideal
Part 2. 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 $\Sigma ^1_2$ well-orderings
25 Large $\Pi ^1_2$ sets
Part 3. Classical Separation Theorems
26 Souslin-Luzin Separation Theorem
27 Kleene Separation Theorem
28 $\Pi ^1_1$-Reduction
29 $\Delta ^1_1$-codes
Part 4. Gandy Forcing
30 $\Pi ^1_1$ equivalence relations
31 Borel metric spaces and lines in the plane
32 $\Sigma ^1_1$ equivalence relations
33 Louveau's Theorem
34 Proof of Louveau's Theorem
References
Index