FOM: new Model Theory text
Dave Marker
marker at math.uic.edu
Thu Aug 22 16:55:44 EDT 2002
Springer has just published my book
Model Theory: An Introduction, Graduate Texts in Mathematics #217
This book is a modern introduction to model theory which stresses
applications to algebra throughout the text. The first half of the book
includes classical material on model construction techniques, type spaces,
prime models, saturated models, countable models, and indiscernibles and
their applications. The second half is an introduction to stability
theory beginning with Morley's Categoricity Theorem and concentrating on
omega-stable theories.
Here is a more detailed description of the contents.
Contents
Chapter 1 : Structures and Theories
1.1 Languages and Structures
1.2 Theories
1.3 Definable Sets and Interpretability
interpreting a field in the affine group,
interpreting orders in graphs
Chapter 2: Basic Techniques
2.1 The Compactness Theorem
2.2 Complete Theories
Vaught's Test,
completeness of algebraically closed fields,
Ax's Theorem
2.3 Up and Down
elementary embeddings, Lowenheim-Skolem
2.4 Back and Forth
dense linear orders,
the random graph,
Ehrenfeucht-Fraisse Games,
Scott sentences
Chapter 3: Algebraic Examples
3.1 Quantifier Elimination
quantifier elimination test,
qe for torsion free divisible abelian groups groups,
qe for divisible ordered abelian groups,
qe for Pressburger arithmetic
3.2 Algebraically Closed Fields
quantifier elimination & constructible sets
model theoretic proof of the nullstellensatz,
elimination of imaginaries,
3.3 Real Closed Fields
quantifier elimination & semialgebraic sets,
Hilbert's 17th problem,
cell decomposition
Chapter 4: Realizing and Omitting Types
4.1 Types
Stone spaces,
types in dense linear orders and algebraiclly closed fields
4.2 Omitting Types and Prime Models
Omitting types theorem,
prime and atomic models,
existence of
prime model extensions for omega-stable theories
4.3 Saturated and Homogeneous Models
saturated, homogeneous and universal models,
qe tests & application to differentially closed fields,
Vaught's two-cardinal theorem
4.4 The Number of Countable Models
aleph_0-categorical theories,
Morley's theorem on the number of countable models
Chapter 5: Indiscernibles</h4>
5.1 Partition Theorems
Ramsey's Theorem, Erdos-Rado Theorem
5.2 Order Indiscernibles
Ehrenfeucht-Mostowski models & applications,
indiscernibles in stable theories
5.3 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
5.4 An Independence Result in Arithmetic
the Kanamori-McAloon proof of the Paris-Harington theorem
Chapter 6: omega-Stable Theories
6.1 Uncountably Categorical Theories
Morley's Categoricity Theorem ala Baldwin-Lachlan
6.2 Morley Rank
Morley rank and degree,
monster models,
Morley rank in algebraically closed fields
6.3 Forking and Independence
non-forking extensions,
definability of types,
properties of independence
6.4 Uniqueness of Prime Model Extensions
6.5 Morley Sequences
saturated models in singular cardinals
Chapter 7: omega-Stable Groups
7.1 The Descending Chain Condition
7.2 Generic Types
omega-stable fields,
minimal groups
7.3 The Indepcomposability Theorem
finding a field in a solvable non-nilpotent group
7.4 Definable Groups in Algebraically Closed Fields
constructible groups are algebraic,
differential galois theory
7.5 Finding a Group
infinitely definable groups, generically presented groups
Chapter 8: Geometry of Strongly Minimal Sets
8.1 Pregeometries
8.2 Canonical Bases and Families of Plane Curves
8.3 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 on ordinals and cardinals
Appendix B: Real Algebra
basic algebra of ordered fields
More information about the FOM
mailing list