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

These notes develop the theory of descriptive sets, leading up to a new proof of Louveau's separation theorem for analytic sets.

Arnold W. Miller (Author)

9781107168060, Cambridge University Press

Hardback, published 18 May 2017

134 pages
23.7 x 16 x 1.5 cm, 0.36 kg

'Miller includes interesting historical material and references. His taste for slick, elegant proofs makes the book pleasant to read. The author makes good use of his sense of humor … Most readers will enjoy the comments, footnotes, and jokes scattered throughout the book.' Studia Logica

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.

1. What are the reals, anyway
Part 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–Soloway 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
Part II. Analytic Sets: 17. Analytic sets
18. Constructible well-orderings
19. Hereditarily countable sets
20. Schoenfield absoluteness
21. Mansfield–Soloway theorem
22. Uniformity and scales
23. Martin's axiom and constructibility
24. ?12 well-orderings
25. Large ?12 sets
Part III. Classical Separation Theorems: 26. Souslin–Luzin separation theorem
27. Kleen separation theorem
28. ?11 -reduction
29. ?11 -codes
Part IV. Gandy Forcing: 30. ?11 equivalence relations
31. Borel metric spaces and lines in the plane
32. ?11 equivalence relations
33. Louveau's theorem
34. Proof of Louveau's theorem
References
Index
Elephant sandwiches.

Subject Areas: Calculus & mathematical analysis [PBK], Set theory [PBCH], Mathematical logic [PBCD]