A guide to classical and modern model theory / by Annalisa Marcja and Carlo Toffalori.

Format:

Book

Author/Creator:

Marcja, A. (Annalisa)

Contributor:

Toffalori, Carlo.

John G. Hartman Memorial Library Fund.

Series:

Trends in logic ; v. 19.

Trends in logic ; v. 19

Language:

English

Subjects (All):

Model theory.

Physical Description:

xi, 369 pages : illustrations ; 25 cm.

Place of Publication:

Dordrecht ; Boston : Kluwer Academic Publishers, [2003]

Summary:

Since its birth, Model Theory has been developing a number of methods and concepts that have their intrinsic relevance, but also provide fruitful and notable applications in various fields of Mathematics. It is a lively and fertile research area which deserves the attention of the mathematical world. This volume -is easily accessible to young people and mathematicians unfamiliar with logic; -gives a terse historical picture of Model Theory; -introduces the latest developments in the area; -provides 'hands-on' proofs of elimination of quantifiers, elimination of imaginaries and other relevant matters. A Guide to Classical and Modern Model Theory is for trainees and professional model theorists, mathematicians working in Algebra and Geometry and young people with a basic knowledge of logic.

Contents:

1 Structures 1

1.2 Sentences 5

1.3 Embeddings 9

1.4 The Compactness Theorem 18

1.5 Elementary classes and theories 20

1.6 Complete theories 30

1.7 Definable sets 35

2 Quantifier Elimination 43

2.1 Elimination sets 43

2.2 Discrete linear orders 47

2.3 Dense linear orders 52

2.4 Algebraically closed fields (and Tarski) 54

2.5 Tarski again: Real closed fields 61

2.6 pp-elimination of quantifiers and modules 68

2.7 Strongly minimal theories 76

2.8 o-minimal theories 78

2.9 Computational aspects of q. e. 79

3 Model Completeness 85

3.2 Abraham Robinson's test 88

3.3 Model completeness and Algebra 91

3.4 p-adic fields and Artin's Conjecture 96

3.5 Existentially closed structures 103

3.6 DCF[subscript 0] 109

3.7 SCF[subscript p] and DCF[subscript p] 112

3.8 ACFA 115

4 Elimination of imaginaries 121

4.1 Interpretability 121

4.2 Imaginary elements 123

4.3 Algebraically closed fields 126

4.4 Real closed fields 129

4.5 The elimination of imaginaries sometimes fails 131

5 Morley rank 133

5.2 Definable sets 133

5.3 Types 136

5.4 Saturated models 143

5.5 A parenthesis: pure injective modules 150

5.6 Omitting types 156

5.7 The Morley rank, at last 158

5.8 Strongly minimal sets 168

5.9 Algebraic closure and definable closure 172

6 [omega] -stability 181

6.1 Totally transcendental theories 181

6.2 [omega]-stable groups 184

6.3 [omega]-stable fields 192

6.4 Prime models 196

6.5 DCF[subscript 0] revisited 209

6.6 Ryll-Nardzewski's Theorem, and other things 217

7 Classifying 221

7.1 Shelah's Classification Theory 221

7.2 Simple theories 227

7.3 Stable theories 235

7.4 Superstable theories 239

7.5 [omega]-stable theories 242

7.6 Classifiable theories 261

7.7 Shelah's Uniqueness Theorem 270

7.8 Morley's Theorem 273

7.9 Biinterpretability and Zilber Conjecture 279

7.10 Two algebraic examples 286

8 Model Theory and Algebraic Geometry 291

8.2 Algebraic varieties, ideals, types 292

8.3 Dimension and Morley rank 294

8.4 Morphisms and definable functions 297

8.5 Manifolds 299

8.6 Algebraic groups 301

8.7 The Mordell-Lang Conjecture 304

9 O-minimality 313

9.2 The Monotonicity Theorem 318

9.3 Cells 320

9.4 Cell decomposition and other theorems 324

9.5 Their proofs 329

9.6 Definable groups in o-minimal structures 339

9.7 O-minimality and Real Analysis 341

9.8 Variants on the o-minimal theme 346

9.9 No rose without thorns 347.

Notes:

Includes bibliographical references (pages 351-361) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the John G. Hartman Memorial Library Fund.

ISBN:

1402013302

OCLC:

52047446