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


Part of Lecture Notes in Logic

  • Date Published: May 2017
  • availability: Available
  • format: Hardback
  • isbn: 9781107168060

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About the Authors
  • 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.

    • Presents a new proof of Louveau's separation theorem for analytic sets
    • Suitable for graduate students and researchers in logic
    • Covers the necessary background from descriptive set theory
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    Reviews & endorsements

    '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

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    Product details

    • Date Published: May 2017
    • format: Hardback
    • isbn: 9781107168060
    • dimensions: 237 x 160 x 15 mm
    • weight: 0.36kg
    • availability: Available
  • Table of Contents

    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
    Elephant sandwiches.

  • Author

    Arnold W. Miller, University of Wisconsin, Madison
    Arnold W. Miller works in the Department of Mathematics at the University of Wisconsin, Madison.

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