Saturated model theory

Description

This book contains the material for a first course in pure model theory with applications to differentially closed fields. Topics covered in this book include saturated model criteria for model completeness and elimination of quantifiers; Morley rank and degree of element types; categoricity in power; two-cardinal theorems; existence and uniqueness of prime model extensions of substructures of models of totally transcendental theories; and homogeneity of models of 1-categorical theories. Contents Introduction 1. Ordinals and Diagrams 2. Similarity Types of Structures 3. Monomorphisms and Substructures 4. First Order Languages 5. Elementary Equivalence 6. Elementary Monomorphisms 7. The Fundamental Existence Theorem 8. Model Completeness 9. Model Completeness of Algebraically Closed Fields 10. Direct Systems of Structures 11. Skolemization of Structures 12. Model Completions 13. Substructure Completeness 14. Countability Proviso with Exceptions 15. Element Types 16. Saturated Structures 17. Elimination of Quantifiers for Real Closed Fields 18. Omitting a Type 19. ω-Stable Theories 20. Homogeneous Structures 21. The Number of Countable Models 22. Vaught’s Two-Cardinal Theorem 23. Chang’s Two-Cardinal Theorem 24. Keisler’s Two-Cardinal Theorem 25. Categories and Functors 26. Inverse Systems of Compact Hausdorff Spaces 27. Towards Morley’s Analysis of l-Types 28. The Cantor–Bendixson Derivative 29. The Morley Derivative 30. Autonomous Subcategories 31. Bounds on the Ranks of 1-Types 32. Prime Model Extensions 33. Prime Extensions of Wellordered Chains 34. Order Indiscernibles 35. Indiscernibles and ω-Stability 36. Shelah’s Uniqueness Theorem 37. Categoricity in Some Uncountable Power 38. Minimal Generators and ω1-Categoricity 39. The Baldwin–Lachlan Theorem 40. Differential Fields of Characteristic 0 41. The Differential Closure 42. Some Other Reading References Notation Index Index