A Course in Model Theory: An Introduction to Contemporary Mathematical Logic

Bruno Poizat

Springer Science & Business Media, May 25, 2000 - Mathematics - 443 pages

Can we reproduce the inimitable, or give a new life to what has been af fected by the weariness of existence? Folks, what you have in your hands is a translation into English of a book that was first published in 1985 by its author, that is, myself, at the end of an editorial adventure about which you will find some details later. It was written in a dialect of Latin that is spoken as a native language in some parts of Europe, Canada, the U. S. A. , the West Indies, and is used as a language of communication between several countries in Africa. It is also sometimes used as a lan guage of communication between the members of a much more restricted community: mathematicians. This translation is indeed quite a faithful rendering of the original: Only a final section, on the reals, has been added to Chapter 6, plus a few notes now and then. On the title page you see an inscription in Arabic letters, with a transcription in the Latin (some poorly informed people say English!) alphabet below; I designed the calligraphy myself.

Contents

• Elementary Classes of Relations: xxxi
• 12 Examples: 3
• 13 Infinite BackandForth: 9
• 14 Historic and Bibliographic Notes: 11
• The Language Associated with a Relation: 13
• 22 Connections to the BackandForth Technique: 21
• 23 Models and Theories: 23
• Tarskis Test Löwenheims Theorem: 25
• 25 Historic and Bibliographic Notes: 27
• Extensions of the Language Structures: 29
• 32 Functions: 31
• 33 Löwenheims Theorem Revisited: 34
• 34 Historic and Bibliographic Notes: 35
• Compactness: 36
• 42 Compactness LöwenheimSkolem Theorem Theorem of Common Elementary Extensions: 40
• 43 Henkins Method: 45
• 44 Historic and Bibliographic Notes: 50
• The BackandForth Method in 𝛚Saturated Models: 53
• 52 𝛚Saturated Models: 55
• 53 Quantifier Elimination: 58
• 54 Historic and Bibliographic Notes: 61
• Examples Illustrating the BackandForth Method: 62
• 62 Differentially Closed Fields: 68
• 63 Boolean Algebras: 76
• 64 Ultrametric Spaces: 84
• 65 Modules and Existentially Closed Modules: 89
• 66 Real Closed Fields not in the original edition: 96
• 67 Historic and Bibliographic Notes: 103
• Arithmetic: 106
• 72 The Order: 108
• 73 The Sum: 109
• Coding of Finite Sets: 114
• 75 Coding of Formulas Tarskis Theorem: 120
• 76 The Hierarchy of Arithmetic Sets: 122
• 77 Some Axioms Models and Fragments of Arithmetic: 132
• 78 Nonstandard Models with Arithmetic Definitions: 139
• 79 Arithmetic Translation of Henkins Method: 140
• 710 The Notion of Proof Decidable Theories: 145
• 711 Godels Theorem: 149
• 712 A Little Mathematical Fiction: 153
• 713 Historic and Bibliographic Notes: 156
• Ordinals and Cardinals: 158
• 82 Axiom of Choice: 162
• 83 Cardinals: 169
• 84 Cofinality: 175
• 85 Historic and Bibliographic Notes: 178
• Saturated Models: 179
• 91 Svenoniuss Theorem: 181
• 92 Compact Saturated Homogeneous and Universal Models: 184
• 93 Resplendent Models: 189
• 94 Properties Preserved Under Interpretation: 193
• 95 Recursively Saturated Models: 195
• 96 Historic and Bibliographic Notes: 200
• Prime Models: 202
• The Denumerable Case: 205
• 103 Theories with Finitely Many Denumerable Models: 207
• 104 Constructed Models: 210
• 105 Minimal Models: 213
• 106 Nonuniqueness of the Prime Model: 216
• 107 Historic and Bibliographic Notes: 221
• Heirs: 223
• 112 Definable Types: 228
• 113 End Extension Types in Arithmetic: 229
• 114 Stable Types and Theories: 231
• 115 Historic and Bibliographic Notes: 234
• Special Sons Morley Sequences: 237
• 122 Coheirs: 241
• 123 Morley Sequences: 244
• 124 The Independence Property: 247
• 125 Indivisible Morley Sequences: 253
• The Theories of Chains: 260
• 127 Special Sequences: 266
• 128 Instability and Order: 268
• Ramseys Theorem: 271
• 1210 Historic and Bibliographic Notes: 273
• The Fundamental Order: 275
• 132 Stability Spectrum: 279
• 133 Some Examples: 283
• 134 Historic and Bibliographic Notes: 287
• Stability and Saturated Models: 288
• 142 Nonexistence Theorems: 289
• 143 Resplendent Models: 292
• 144 Sufficiently Saturated Extensions of a Given Model: 293
• 145 Historic and Bibliographic Notes: 296
• Forking: 297
• 151 The Theorem of the Bound: 298
• 152 Forking and Nonforking Sons: 301
• 153 Multiplicity: 303
• 154 Stable Types in an Unstable Theory: 305
• 155 Historic and Bibliographic Notes: 306
• Strong Types: 307
• 162 Spaces of Strong Types Open Mapping Theorem: 310
• 163 Morley Sequences for Strong Types Saturated Models Revisited: 312
• 164 Imaginary Elements: 316
• 165 Elimination of Imaginaries: 319
• 166 A Galois Theory for Strong Types: 326
• 167 Historic and Bibliographic Notes: 329
• Notions of Rank: 330
• 172 Shelah Rank: 334
• 173 Morley Rank: 339
• 174 Local Ranks: 343
• 175 Historic and Bibliographic Notes: 347
• Stability and Prime Models: 349
• 182 Prime Models of a Totally Transcendental Theory: 351
• 183 Galois Theory of Differential Equations: 356
• 184 Prime T+Saturated Models: 363
• 185 Ehrenfeucht Models: 365
• 186 TwoCardinal Theorem N₁Categorical Theories: 368
• 187 Historic and Bibliographic Notes: 370
• Stability Indiscernible Sequences and Weights: 372
• 192 Lascar Inequalities: 374
• 193 Weight of a Superstable Type: 379
• 194 Independence and Domination: 382
• 195 Historic and Bibliographic Notes: 390
• Dimension in Models of a Totally Transcendental Theory: 391
• 202 Dimensional Types and Theories: 400
• 203 Classification of the Models of a Dimensional Theory: 407
• 204 The Dope: 412
• 205 Depth and the Main Gap: 414
• 206 Historic and Bibliographic Notes: 415
• Bibliography: 417
• Index of Notation: 427
• Index: 431

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Bibliographic information

• Title: A Course in Model Theory: An Introduction to Contemporary Mathematical Logic; Universitext - Springer-Verlag, ISSN 0172-5939; Universitext Series
• Author: Bruno Poizat
• Edition: illustrated
• Publisher: Springer Science & Business Media, 2000
• ISBN: 0387986553, 9780387986555
• Length: 443 pages
• Subjects: Mathematics / General; Mathematics / History & Philosophy; Mathematics / Logic