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 categorical theories.
Author(s): Gerald E Sacks
Series: Mathematics Lecture Note Series
Edition: 1st
Publisher: Benjamin
Year: 1972
Language: English
Pages: 349
Title ......Page 5
Copyright ......Page 6
Dedication ......Page 7
Epigraph ......Page 9
Contents ......Page 11
0. Introduction ......Page 13
1. Ordinals and Diagrams ......Page 20
2. Similarity Types of Structures ......Page 23
3. Monomorphisms and Substructures ......Page 26
4. First Order Languages ......Page 28
5. Elementary Equivalence ......Page 32
6. Elementary Monomorphisms ......Page 35
7. The Fundamental Existence Theorem ......Page 37
8. Model Completeness ......Page 47
9. Model Completeness of Algebraically Closed Fields ......Page 52
10. Direct Systems of Structures ......Page 58
11. Skolemization of Structures ......Page 65
12. Model Completions ......Page 69
13. Substructure Completeness ......Page 75
14. Countability Proviso with Exceptions ......Page 81
15. Element Types ......Page 83
16. Saturated Structures ......Page 88
17. Elimination of Quantifiers for Real Closed Fields ......Page 99
18. Omitting a Type ......Page 108
19. Omega-Stable Theories ......Page 116
20. Homogeneous Structures ......Page 122
21. The Number of Countable Models ......Page 130
22. Vaught's Two-Cardinal Theorem ......Page 138
23. Chang's Two-Cardinal Theorem ......Page 146
24. Keisler's Two-Cardinal Theorem ......Page 156
25. Categories and Functors ......Page 168
26. Inverse Systems of Compact Hausdorff Spaces ......Page 174
27. Towards Morley's Analysis of 1-Types ......Page 177
28. The Cantor-Bendixson Derivative ......Page 181
29. The Morley Derivative ......Page 185
30. Autonomous Subcategories ......Page 193
31. Bounds on the Rank of 1-Types ......Page 199
32. Prime Model Extensions ......Page 210
33. Prime Extensions of Wellordered Chains ......Page 224
34. Order Indiscernibles ......Page 228
35. Indiscernibles and Omega-Stability ......Page 236
36. Shelah's Uniqueness Theorem ......Page 254
37. Categoricity in Some Uncountable Power ......Page 266
38. Minimal Generators and Omega1-Categoricity ......Page 272
39. The Baldwin-Lachlan Theorem ......Page 285
40. Differential Fields of Characteristic 0 ......Page 307
41. The Differential Closure ......Page 317
42. Some Other Reading ......Page 324
43. Several References ......Page 331
44. Notation Index ......Page 335
45. Subject Index ......Page 339
