A Shorter Model Theory

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Cambridge University Press, Apr 10, 1997 - Mathematics - 310 pages
2 Reviews
This is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory. Besides standard results such as the compactness and omitting types theorems, it also describes various links with algebra, including the Skolem-Tarski method of quantifier elimination, model completeness, automorphism groups and omega-categoricity, ultraproducts, O-minimality and structures of finite Morley rank. The material on back-and-forth equivalences, interpretations and zero-one laws can serve as an introduction to applications of model theory in computer science. Each chapter finishes with a brief commentary on the literature and suggestions for further reading. This book will benefit graduate students with an interest in model theory.
 

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Contents

Naming of parts
1
Structures
2
Homomorphisms and substructures
5
Terms and atomic formulas
10
Parameters and diagrams
15
Canonical models
17
Further reading
20
Classifying structures
21
Indiscernibles
152
Further reading
156
The countable case
158
Omitting types
165
Countable categoricity
171
ωcategorical structures by Fra´ssÚs method
175
Further reading
181
The existential case
182

Definable subsets
22
Definable classes of structures
30
Some notions from logic
37
Maps and the formulas they preserve
43
Translations
51
Quantifier elimination
59
Further reading
68
Structures that look alike
69
Backandforth equivalence
73
Games for elementary equivalence
82
Further reading
91
Interpretations
93
Automorphisms
94
Relativisation
101
Interpreting one structure in another
107
Imaginary elements
113
Further reading
122
The firstorder case compactness
124
Types
130
Elementary amalgamation
134
Amalgamation and preservation
141
Expanding the language
147
Existentially closed structures
183
Constructing ec structures
188
Modelcompleteness
195
Quantifier elimination revisited
201
Further reading
208
Saturation
210
The great and the good
211
Big models exist
220
Syntactic characterisations
225
Onecardinal and twocardinal theorems
233
Ultraproducts and ultrapowers
237
Further reading
248
Structure and categoricity
250
EhrenfeuchtMostowski models
251
Minimal sets
257
Totally transcendental structures
264
Stability
273
Morleys theorem
286
Further reading
296
INDEX TO SYMBOLS
298
INDEX
300
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