Lecture Notes in Logic, 39

Simple Theories and Hyperimaginaries

Enrique Casanovas

Year: 2011

ISBN-13: 9780521119559
184 pages. Hardcover.
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In the 1990s Kim and Pillay generalized stability, a major model theoretic idea developed by Shelah twenty-five years earlier, to the study of simple theories. This book is an up-to-date introduction to simple theories and hyperimaginaries, with special attention to Lascar strong types and elimination of hyperimaginary problems. Assuming only knowledge of general model theory, the foundations of forking, stability, and simplicity are presented in full detail. The treatment of the topics is as general as possible, working with stable formulas and types and assuming stability or simplicity of the theory only when necessary. The author offers an introduction to independence relations as well as a full account of canonical bases of types in stable and simple theories. In the last chapters the notions of internality and analyzability are discussed and used to provide a self-contained proof of elimination of hyperimaginaries in supersimple theories.

Table of Contents

  1. Preliminaries
  2. ∂-types, stability and simplicity
  3. ∆-types and the local rank D(π∆k)
  4. Forking
  5. Independence
  6. The local rank CB∆(π)
  7. Heirs and coheirs
  8. Stable forking
  9. Lascar strong types
  10. The independence theorem
  11. Canonical bases
  12. Abstract independence relations
  13. Supersimple theories
  14. More ranks
  15. Hyperimaginaries
  16. Hyperimaginary forking
  17. Canonical bases revisited
  18. Elimination of hyperimaginaries
  19. Orthogonality and analysability
  20. Hyperimaginaries in supersimple theories