Introduction 1
1 Structures and Theories 7
1.1 Languages and Structures . . . . . . . . . . . . . . . . . . . 7
1.2 Theories . . . . . . . . . . . . .
... [Show More] . . . . . . . . . . . . . . . . 14
1.3 Definable Sets and Interpretability . . . . . . . . . . . . . . 19
1.4 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 29
2 Basic Techniques 33
2.1 The Compactness Theorem . . . . . . . . . . . . . . . . . . 33
2.2 Complete Theories . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Up and Down . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Back and Forth . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 60
3 Algebraic Examples 71
3.1 Quantifier Elimination . . . . . . . . . . . . . . . . . . . . . 71
3.2 Algebraically Closed Fields . . . . . . . . . . . . . . . . . . 84
3.3 Real Closed Fields . . . . . . . . . . . . . . . . . . . . . . . 93
3.4 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 104
4 Realizing and Omitting Types 115
4.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 Omitting Types and PrimeModels . . . . . . . . . . . . . . 125
Contents
4.3 Saturated and HomogeneousModels . . . . . . . . . . . . . 138
4.4 The Number of CountableModels . . . . . . . . . . . . . . 155
4.5 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 163
5 Indiscernibles 175
5.1 Partition Theorems . . . . . . . . . . . . . . . . . . . . . . . 175
5.2 Order Indiscernibles . . . . . . . . . . . . . . . . . . . . . . 178
5.3 AMany-Models Theorem . . . . . . . . . . . . . . . . . . . 189
5.4 An Independence Result in Arithmetic . . . . . . . . . . . . 195
5.5 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 202
6 ω-Stable Theories 207
6.1 Uncountably Categorical Theories . . . . . . . . . . . . . . 207
6.2 Morley Rank . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.3 Forking and Independence . . . . . . . . . . . . . . . . . . . 227
6.4 Uniqueness of PrimeModel Extensions . . . . . . . . . . . . 236
6.5 Morley Sequences . . . . . . . . . . . . . . . . . . . . . . . . 240
6.6 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 243
7 ω-Stable Groups 251
7.1 The Descending Chain Condition . . . . . . . . . . . . . . . 251
7.2 Generic Types . . . . . . . . . . . . . . . . . . . . . . . . . 255
7.3 The Indecomposability Theorem . . . . . . . . . . . . . . . 261
7.4 Definable Groups in Algebraically Closed Fields . . . . . . . 267
7.5 Finding a Group . . . . . . . . . . . . . . . . . . . . . . . . 279
7.6 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 285
8 Geometry of Strongly Minimal Sets 289
8.1 Pregeometries . . . . . . . . . . . . . . . . . . . . . . . . . . 289
8.2 Canonical Bases and Families of Plane Curves . . . . . . . . 293
8.3 Geometry and Algebra . . . . . . . . . . . . . . . . . . . . . 300
8.4 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 309
A Set Theory 315
B Real Algebra 323
References 329
Index 337 [Show Less]