Model Theory: an Introduction
David Marker
Springer Graduate Texts in Mathematics 217
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, LowenheimSkolem
 Back and Forth
dense linear orders, the random graph,
EhrenfeuchtFraisse 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 omegastable theories
 Saturated and Homogeneous Models
saturated models, homogeneous and universal models, qe test &
application to differentially closed fields, Vaught's twocardinal
theorem
 The Number of Countable Models
aleph_0categorical theories, Morley's theorem on the number of
countable models

Chapter 5: Indiscernibles
 Partition Theorems
Ramsey's Theorem, ErdosRado Theorem
 Order Indiscernibles
EhrenfeuchtMostowski models & applications, indiscernibles in
stable theories
 A ManyModels Theorem
a special case of Shelah's manymodel theorem for unstable
theories in regular cardinals > aleph_1, club and stationary sets
 An Independence Result in Arithmetic
the KanamoriMcAloon proof of the ParisHarington theorem

Chapter 6: omegaStable 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
nonforking extensions, definability of types, properties of
independence
 Uniqueness of Prime Model Extensions
 Morley Sequences
saturated models in singular cardinals

Chapter 7: omegaStable Groups
 The Descending Chain Condition
 Generic Types
omegastable fields, minimal groups
 The Indepcomposability Theorem
finding a field in a solvable nonnilpotent 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, onebased 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