FOM: new Model Theory text

Dave Marker marker at
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.


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

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