Contents

• Chapter 1 : Structures and Theories

  • Languages and Structures
  • Theories
  • Definable Sets and Interpretability
    interpreting a field in the affine group, interpreting orders in graphs
  
  

• Chapter 2: Basic Techniques

  • The Compactness Theorem
  • Complete Theories
    Vaught's Test, completeness of algebraically closed fields, Ax's Theorem
  • Up and Down
    elementary embeddings, Lowenheim-Skolem
  • Back and Forth
    dense linear orders, the random graph, Ehrenfeucht-Fraisse Games, Scott sentences
  
  

• Chapter 3: Algebraic Examples

  • Quantifier Elimination
    quantifier elimination test, qe for torsion free divisible abelian groups groups, qe for divisible ordered abelian groups, qe for Pressburger arithmetic
  • Algebraically Closed Fields
    quantifier elimination & constructible sets model theoretic proof of the nullstellensatz, elimination of imaginaries,
  • Real Closed Fields
    quantifier elimination & semialgebraic sets, Hilbert's 17th problem, cell decomposition
  
  

• Chapter 4: Realizing and Omitting Types

  • Types
    Stone spaces, types in dense linear orders and algebraiclly closed fields
  • Omitting Types and Prime Models
    Omitting types theorem, prime and atomic models, existence of prime model extensions for omega-stable theories
  • Saturated and Homogeneous Models
    saturated models, homogeneous and universal models, qe test & application to differentially closed fields, Vaught's two-cardinal theorem
  • The Number of Countable Models
    aleph_0-categorical theories, Morley's theorem on the number of countable models
  
  

• Chapter 5: Indiscernibles

  • Partition Theorems
    Ramsey's Theorem, Erdos-Rado Theorem
  • Order Indiscernibles
    Ehrenfeucht-Mostowski models & applications, indiscernibles in stable theories
  • A Many-Models Theorem
    a special case of Shelah's many-model theorem for unstable theories in regular cardinals > aleph_1, club and stationary sets
  • An Independence Result in Arithmetic
    the Kanamori-McAloon proof of the Paris-Harington theorem
  
  

• Chapter 6: omega-Stable Theories

  • Uncountably Categorical Theories
    Morley's Categoricity Theorem
  • Morley Rank
    Morley rank and degree, monster models, morley rank in algebraically closed fields
  • Forking and Independence
    non-forking extensions, definability of types, properties of independence
  • Uniqueness of Prime Model Extensions
  • Morley Sequences
    saturated models in singular cardinals
  
  

• Chapter 7: omega-Stable Groups

  • The Descending Chain Condition
  • Generic Types
    omega-stable fields, minimal groups
  • The Indepcomposability Theorem
    finding a field in a solvable non-nilpotent group
  • Definable Groups in Algebraically Closed Fields
    constructible groups are algebraic, differential galois theory
  • Finding a Group
    infinitely definable groups, generically presented groups
  
  

• Chapter 8: Geometry of Strongly Minimal Sets

  • Pregeometries
  • Canonical Bases and Families of Plane Curves
  • Geometry and Algebra
    finding a group on a nontrivial locally modular strongly minimal set, one-based groups, Zariski geometries, outline of applications to diophantine geometry
  
  

• Appendix A: Set Theory

  basics of cardials and ordinals
  
  

• Appendix B: Real Algebra

  basic algebra of ordered fields