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 October 16, During fall 2004 I expect to update this document often. In case your copy is more than couple of months old, please distroy it and contact me for a current version. rami c Rami Grossberg

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3 Contents Preface 11 Acknowledgments 15 Course outlines 17 Part 1. The basics 19 Chapter 1. Fundamentals 21 Introduction Structures and languages The basic concepts On existence of models and elementary submodels Abstract classes The Erdős-Rado Theorem Applications of the compactness theorem Some extensions of first-order logic Model complete-theories Skolemization The filter of closed unbounded sets Ultraproducts Models of weak set theory Ehrenfeucht-Fraïssé games Two applications to algebra Non-standard analysis* A mathematical introduction to the book 195 3

4 4 CONTENTS 17. When does a class have a structure theory? Shelah s thesis 201 Chapter 2. Types and indicsernibles 203 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 Shelah s Generalization of Eherenfeucht-Mostowski models D(T ) as a topological space* The topology 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* 324 Chapter 3. Morley s Theorem 331 Introduction Dimension in model theory A rank function Existence of indiscernibles, non-splitting and cohiers ℵ 0-stability Primary models 355

5 CONTENTS 5 6. Every model is saturated Strongly minimal sets Some properties of T + -categorical theories Some properties of ℵ 0-stable theories The Baldwin Lachlan proof Keisler s proof* 381 Chapter 4. Basics of Stability 383 Introduction Basics The order dichotomy Sequences of indiscernibles Noetherian topological spaces 401 Part 2. Stability and Simplicity 405 Chapter 5. Stability 407 Introduction Ranks revisited Characterizations of stable theories Definability of types On the function D[θ(x; a),,µ + ] The finite cover property Simple theories 444 Chapter 6. Stability in algebra 445 Introduction Definable groups Superstable fields are algebraically closed Algebraic and model-theoretic dimensions The indecomposability theorem Model Theory of algebraically closed fields 451

6 6 CONTENTS 6. An application to differentially closed fields* 454 Chapter 7. Strong splitting and averages 459 Introduction The independence and strict order properties Strong splititing and the stability spectrum 466 Chapter 8. Splitting, dividing and forking in arbitrary theories 469 Introduction Forking Indiscenible sequences based on a set* 482 Chapter 9. Forking calculus in simple theories 493 Introduction General notion of independence Forking in Simple Theories Ranks and Simple Theories Shelah s Boolean Algebra Semi simple theories 529 Chapter 10. Forking in stable theories 531 Introduction Finite equivalence relations theorem The stability spectrum theorem Chains of saturated models Canonical bases and C eq 539 Chapter 11. Orthogonality calculus 541 Introduction Regular types Unidimensional theories 541 Chapter 12. Morley s theorem for uncountable theories 543

7 CONTENTS 7 Introduction Weakly minimal formulas Another proof A third proof? 544 Chapter 13. Prime models 545 Introduction Isolation notions and existence Uniqueness 548 Chapter 14. The Hrushovski-Zilber group configuration 551 Introduction Basics Unidimensional theories are superstable Laskowski s proof of categoricity 551 Part 3. Classification theory 553 Chapter 15. Classification theory for non-elementary classes 555 Introduction 555 Chapter 16. Classification theory for Abstract Elementary Classes Fundamentals K-embeddings A substitute for saturation Abstract classes are PC-classes The amalgamation property and the weak diamond On pseudo elementary classes Galois types Minimal types 629 Chapter 17. Excellent classes 631 Introduction 631

8 8 CONTENTS 1. The basic framework and concepts Examples and applications Good sets Basic stability of the class of atomic models Transfering results up and down Categoricity Non-excellence gives many models 646 Chapter 18. The main gap 647 Introduction stable systems otop and dop good systems Tree decomposition theorem 648 Chapter 19. Non structure theory 649 Introduction Unstable theories Unsuperstable theories and Boolean algebras Combinatorial theorems on trees The generalized order property 673 Part 4. Survey and History 681 Chapter 20. Survey 683 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 685

9 CONTENTS 9 7. Abstract model theory Finite model theory Non standard analysis 687 Chapter 21. A miniguide to the literature 689 Chapter 22. Open Problems 691 Introduction Classification theory for non-elementary classes Shelah s categoricity conjecture Main Gap for uncountable theories Other problems 692 Chapter 23. Historical comments 695 Part 5. Appendix(s) 703 Appendix A. Elementary set theory 705 Introduction Cardinal numbers ordinals More set theory 709 Appendix B. Combinatorial geometry 711 Introduction Pregeometries (or Matroids) Abstract dependence Projective geometries 718 Bibliography 719 Index 729