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. Stability theory was introduced and matured in the 1960s and 1970s. Today stability theory influences and is influenced by number theory, algebraic group theory, Riemann surfaces, and representation theory of modules. There is little model theory today that does not involve the methods of stability theory. In this volume, the fourth publication in the Perspectives in Logic series, Steven Buechler bridges the gap between a first-year graduate logic course and research papers in stability theory. The book prepares the student for research in any of today's branches of stability theory, and gives an introduction to classification theory with an exposition of Morley's Categoricity Theorem.
Author(s): Steven Buechler
Series: Perspectives in Logic 4
Publisher: Cambridge University Press
Year: 2017
Language: English
Pages: 368
Contents......Page 8
Preface......Page 10
1.1 Preliminaries and Notation......Page 14
1.1.1 Elimination of Quantifiers......Page 19
2.1 Prime and Atomic Models......Page 24
2.2 Saturated and Homogeneous Models......Page 30
2.3 Countable Models of Complete Theories......Page 48
2.4 Indiscernible Sequences......Page 53
2.5 Skolem Functions......Page 56
3.1 Morley's Categoricity Theorem......Page 62
3.2 A Universal Domain......Page 83
3.3 Totally Transcendental Theories......Page 85
3.4 The Baldwin-Lachlan Theorem......Page 105
3.5 Introduction to ω — stable Groups......Page 113
3.5.1 A Group Acting on a Strongly Minimal Set......Page 127
3.5.2 ⋀ — definable Groups and Actions......Page 134
4. Fine Structure of Uncountably Categorical Theories......Page 138
4.1 T eq......Page 139
4.1.1 Totally Transcendental Theories Revisited......Page 144
4.1.2 D eq for a Strongly Minimal D......Page 149
4.2 The Pregeometries on Strongly Minimal Sets......Page 151
4.2.1 Plane Curves......Page 156
4.3 Global Geometrical Considerations......Page 163
4.3.1 1-based Theories......Page 172
4.3.2 1-based Groups......Page 177
4.4 Automorphism Groups of Constructions......Page 188
4.5 Denning a Group from a Pregeometry......Page 205
4.5.1 Germs of Definable Functions......Page 207
4.5.2 Getting a Group from an Algebraic Quadrangle......Page 217
5.1.1 Ranks and Definability......Page 226
5.1.2 Stability and the Number of Types......Page 241
5.1.3 Morley Sequences and Indiscernibles......Page 243
5.1.4 The Fundamental Order......Page 246
5.2 The Stability Spectrum and k(T)......Page 251
5.3 Stable Groups and Modules......Page 255
5.3.1 1—based Groups and Modules......Page 262
5.3.2 Modules......Page 264
5.4 Saturated Models......Page 270
5.4.1 a-models......Page 271
5.5.1 Prime Models in a t.t. Theory......Page 274
5.5.2 a-prime Models......Page 280
5.6 Orthogonality, Domination and Weight......Page 286
5.6.1 Orthogonality......Page 288
5.6.2 Domination......Page 293
5.6.3 Weight......Page 296
5.6.4 Finite Weight......Page 300
6.1 More Ranks......Page 306
6.2 Geometrical Matters: A Dichotomy Theorem......Page 314
6.3 Regular Types......Page 317
6.3.1 Rank Considerations......Page 324
6.4 Strongly Regular Types......Page 327
7.1 Bounded and Unbounded Theories......Page 336
7.1.1 Bounded ω-stable Theories......Page 340
7.1.2 Unbounded Theories......Page 348
7.2 More on Ranks......Page 350
References......Page 357
Index......Page 363
