These notes develop the theory of descriptive sets, leading up to a new proof of Louveau's separation theorem for analytic sets. A first course in mathematical logic and set theory is assumed, making this book suitable for advanced students and researchers.

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.

'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