A Course in Model Theory I:

Transcription

1 A Course in Model Theory I: Introduction 1 Rami Grossberg DEPARTMENT OFMATHEMATICAL SCIENCES, CARNEGIE MELLON UNI- VERSITY, PITTSBURGH, PA This preliminary draft is dated from August 15, The book will be published by Cambridge University Press. The book is approximately 96.75% complete, I expect that the final version will have about 800 pages, many sections of the current version will be revised and few will be added. I hope to have a stable version of this volume soon. This version is made only for students studying model theory with me and not for distribution outside CMU. If you have a copy not received directly from me, it is an illegal copy and I request that you will not share with others. Exercise #=749 rami c Rami Grossberg

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3 Contents Preface 7 1. About this book 7 2. A mathematical introduction to the book 15 Course outlines 22 Part 1. Definability 25 Chapter 1. Fundamentals 27 Introduction Structures and languages The basic concepts On existence of models and elementary submodels The Erdős-Rado Theorem Applications of the compactness theorem Joint embedding and the Amalgamation properties in first-order logic Types and the diagram of T Some extensions of first-order logic Countable models and Henkin s omitting types theorem Models of weak set theory Absoluteness Two cardinal theorems, by Vaught, Chang, Keisler and Morley Model complete-theories Skolemization 219 Chapter 2. Abstract Elementary Classes 225 Introduction Abstract Classes Abstract Elementary Classes The major open questions concerning AEC Shelah s presentation Theorem and 2-categories Basic Examples PC-classes and omitting types I(@ 0, K) =I(@ 1, K) =1 =) 6= ; Categoricity 1 for AECs is not absolute Random power set in higher order Ext 1 Z(G, Z) Weak amalgamation Few models imply the amalgamation property 323 Chapter 3. More Fundamentals 329 Introduction The filter of closed unbounded sets Ultraproducts 342 3

4 4 CONTENTS 3. Ehrenfeucht-Fraissé games Two applications to algebra Non-standard analysis* When does a class have a structure theory? Shelah s thesis 365 Part 2. Galois Theory 367 Chapter 4. Complete types and indiscernibles 369 Introduction Saturated models The monster model, homogenous and special models Indiscernibles and Ehrenfeucht-Mostowski models Galois types and monster models in Abstract Elementary Classes 426 Chapter 5. More on Types 443 Introduction Definability and the Lascar group Using models of set theory to establish consistency of a first-order theory Game theoretic characterization of elementary embedding and isomorphism Saturation of ultraproducts Keisler-Shelah s theorem* More on model complete theories* 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* More on two cardinal theorems* Chang s conjecture and Jónsson algebras* 486 Chapter 6. Morley s Theorem 493 Introduction Dimension in model theory A rank function stability Existence of indiscernibles, non-splitting and cohiers Prime, primary and atomic models Every model is saturated Chang s Conjecture is true 0-stable theories Quasi-minimal formulas and an omitting types Theorem Strongly minimal sets and the Baldwin-Lachlan proof Some properties of T + -categorical theories The Baldwin Lachlan proof Keisler s rank-free proof of Morley s theorem Morely s rank and the local rank Some properties 0-stable theories 581 Chapter 7. Basics of Stability 583 Introduction Local Types Infinitely many Rank Functions Characterizations of stability by rank and '-types Definability of types is equivalent to stablity The order dichotomy 613

5 CONTENTS 5 6. Sequences and sets of indiscernibles Towards Los conjecture for uncountable first-order theories The independence and strict-order properties Superstable theories Simple Theories Noetherian topological spaces 649 Chapter 8. Forking calculus 653 Introduction Basics of Forking Stability spectrum theorem Forking in Simple theories is symmetric and transitive Applications of forking Forking is canonical 688 Chapter 9. Applications 689 Introduction Harnik s theorem Uniqueness and characterization of prime models Uniqueness of prime models Stability spectrum 691 Chapter 10. Survey 693 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 697 Chapter 11. A miniguide to the literature 699 Chapter 12. Open Problems 701 Introduction Classification theory for non-elementary classes Shelah s categoricity conjecture Main Gap for uncountable theories Other problems 702 Chapter 13. Historical comments 705 APPENDIX 715 Chapter 14. Some set theory 717 Introduction Sets, functions and relations Cardinal numbers Ordinals Martin s Axiom On weak diamonds The small subsets of + is a normal ideal 737

6 6 CONTENTS 7. Kuratowski s and Hajnal s free subset theorems The building-stones of many models 744 Chapter 15. Combinatorial geometry 747 Introduction Pregeometries (or Matroids) Abstract dependence Projective geometries 756 Chapter 16. Plato: The Allegory of the Cave, from book VII The Republic 757 Bibliography 761 Index 773