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

ISBN-13: 9781107168060 / Angielski / Twarda / 2017 / 134 str.

Arnold W. Miller

Opis

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

Recenzje

'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

Spis treści

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 Gδ; 7. α-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 ω1 ; 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. Σ12 well-orderings; 25. Large Π12 sets; Part III. Classical Separation Theorems: 26. Souslin–Luzin separation theorem; 27. Kleen separation theorem; 28. Π11 -reduction; 29. Δ11 -codes; Part IV. Gandy Forcing: 30. Π11 equivalence relations; 31. Borel metric spaces and lines in the plane; 32. Σ11 equivalence relations; 33. Louveau's theorem; 34. Proof of Louveau's theorem; References; Index; Elephant sandwiches.