## Descriptive Set Theory and Forcing: How to Prove Theorems about Borel Sets the Hard Way

Arnold W. Miller

Year: 2017

ISBN: 9781107168060

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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.

### Table of Contents

- What are the reals, anyway

**Part I. On the Length of Borel Hierarchies:**

- Borel hierarchy
- Abstract Borel hierarchies
- Characteristic function of a sequence
- Martin’s axiom
- Generic Gδ
- α-forcing
- Boolean algebras
- Borel order of a field of sets
- CH and orders of separable metric spaces
- Martin–Soloway theorem
- Boolean algebra of order ω1
- Luzin sets
- Cohen real model
- The random real model
- Covering number of an ideal

**Part II. Analytic Sets:**

- Analytic sets
- Constructible well-orderings
- Hereditarily countable sets
- Schoenfield absoluteness
- Mansfield–Soloway theorem
- Uniformity and scales
- Martin’s axiom and constructibility
- Σ12 well-orderings
- Large Π12 sets

**Part III. Classical Separation Theorems:**

- Souslin–Luzin separation theorem
- Kleen separation theorem
- Π11 -reduction
- Δ11 -codes

**Part IV. Gandy Forcing:**

- Π11 equivalence relations
- Borel metric spaces and lines in the plane
- Σ11 equivalence relations
- Louveau’s theorem
- Proof of Louveau’s theorem

- References
- Index
- Elephant sandwiches