A Course in Model Theory

Transcription

1 A Course in Model Theory Author address: Rami Grossberg 1 DEPARTMENT OF MATHEMATICAL SCIENCES, CARNEGIE MELLON UNI- VERSITY, PITTSBURGH, PA address: rami@cmu.edu 1 This preliminary draft is dated from January 28, rami c Rami Grossberg

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3 Contents Preface 11 Acknowledgments 15 Course outlines 16 Part 1. The basics 17 Chapter 1. Fundamentals 19 Introduction Structures and languages The basic concepts On existence of models and elementary submodels The Erdős-Rado Theorem Applications of the compactness theorem Some extensions of first-order logic Skolemization The filter of closed unbounded sets Ultraproducts Models of weak set theory Ehrenfeucht-Fraïssé games Model complete-theories Two applications to algebra* Non-standard analysis* A mathematical introduction to the book When does a class have a structure theory? Shelah s thesis 183 3

4 4 CONTENTS Chapter 2. Types and indicsernibles 185 Introduction D(T ) the diagram of the theory T Saturated models More on saturated models and the monster model Definability and the Lascar group Game theoretic characterization of elementary embedding and isomorphism Saturation of ultraproducts Keisler-Shelah s theorem* More on model complete theories* Indiscernibles and Ehrenfeucht-Mostowski models Countable models and Henkin s omitting types theorem PC-classes and more on omitting types D(T ) as a topological space* The toplogy of Lascar s groups More on existence, omitting types, and the completeness theorem The Paris Harrington s theorem* Basics of two cardinal theorems More on two cardinal theorems* Chang s conjecture and Jónsson algebras* 281 Chapter 3. Morley s Theorem 287 Introduction Dimension in model theory A rank function Existence of indiscernibles, non-splitting and cohiers ℵ 0-stability Primary models Every model is saturated Strongly minimal sets 320

5 CONTENTS 5 8. Some properties of T + -categorical theories Some properties of ℵ 0-stable theories The Baldwin Lachlan proof Keisler s proof* 335 Chapter 4. Basics of Stability 337 Introduction Basics The order dichotomy Sequences of indiscernibles Noetherian topological spaces 354 Part 2. Stability and Simplicity 357 Chapter 5. Stability 359 Introduction Ranks revisited Characterizations of stable theories Definability of types On the function D[θ(x; a),,µ + ] The independence and strict order properties The finite cover property Simple theories Basics of Forking 402 Chapter 6. Stability in algebra 405 Introduction Definable groups Superstable fields are algebraically closed Algebraic and model-theoretic dimensions The indecomposability theorem Model Theory of algebraically closed fields 411

6 6 CONTENTS 6. An application to differentially closed fields* 414 Chapter 7. Forking calculus in simple theories 419 Introduction General notion of independence Dividing and forking Ranks and Simple Theories Forking in Simple Theories Shelah s Boolean Algebra Semi simple theories 461 Chapter 8. More stability 465 Introduction Finite equivalence relations theorem The stability spectrum theorem Chains of saturated models Canonical bases and C eq 467 Chapter 9. Prime models 469 Introduction Isolation notions and existence Uniqueness 472 Chapter 10. Orthogonality calculus 475 Introduction Regular types Unidimensional theories 475 Chapter 11. Morley s theorem for uncountable theories 477 Introduction Weakly minimal formulas Another proof 477

7 CONTENTS 7 Chapter 12. The Hrushovski-Zilber group configuration 479 Introduction Basics Unidimensional theories are superstable Laskowski s proof of categoricity 479 Part 3. Classification theory 481 Chapter 13. Classification theory for non-elementary classes 483 Introduction Fundamentals K-embeddings A substitute for saturation Abstract classes are PC-classes The amalgamation property and the weak diamond On pseudo elementary classes Types Minimal types 556 Chapter 14. Excellent classes 557 Introduction 557 Chapter 15. The main gap 559 Introduction stable systems otop and dop good systems Tree decomposition theorem 559 Chapter 16. Non structure theory 561 Introduction Unstable theories 565

8 8 CONTENTS 2. Unsuperstable theories and Boolean algebras Combinatorial theorems on trees The generalized order property 586 Part 4. Survey and History 595 Chapter 17. Survey 597 Introduction The main gap (Shelah s great theorem) Classification theory for non-elementary classes Geometric stability (or the fine structure theory) Lang-Mordell Ax and Kochen o-minimal theories Abstract model theory Finite model theory Non standard analysis 601 Chapter 18. A miniguide to the literature 603 Chapter 19. Historical comments 605 Part 5. Appendix(s) 607 Appendix A. Elementary set theory 609 Introduction Cardinal numbers ordinals More set theory 614 Appendix B. Combinatorial geometry 615 Introduction Pregeometries (or Matroids) 615

9 CONTENTS 9 2. Abstract dependence Projective geometries 622 Bibliography 623 Index 635

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