Translated from the French, this book is an introduction to first-order model theory. Starting from scratch, it quickly reaches the essentials, namely, the back-and-forth method and compactness, which are illustrated with examples taken from algebra. It also introduces logic via the study of the models of arithmetic, and it gives complete but accessible exposition of stability theory.
Author(s): Bruno Poizat
Series: Universitext
Edition: 1
Publisher: Springer
Year: 2000
Language: English
Pages: 476
Cover......Page 1
Series......Page 3
More books of this Series......Page 476
Title......Page 4
Copyright......Page 5
Preface to the English Edition......Page 8
History of a Publication......Page 11
Contents......Page 18
Introduction......Page 24
1.1 Local Isomorphisms Between Relations......Page 33
1.2 Examples......Page 37
1.3 Infinite Back-and-Forth......Page 43
1.4 Historic and Bibliographic Notes......Page 45
2.1 Formulas......Page 47
2.2 Connections to the Back-and-Forth Technique......Page 55
2.3 Models and Theories......Page 57
2.4 Elementary Extensions: Tarski's Test, Löwenheim’s Theorem......Page 59
2.5 Historic and Bibliographic Notes......Page 61
3.1 Multirelations, Relational Structures......Page 63
3.2 Functions......Page 65
3.3 Löwenheim’s Theorem Revisited......Page 68
3.4 Historic and Bibliographic Notes......Page 69
4.1 Ultraproducts......Page 70
4.2 Compactness, Löwenheim-Skolem Theorem, Theorem of Common Elementary Extensions......Page 74
4.3 Henkin’s Method......Page 79
4.4 Historic and Bibliographic Notes......Page 84
5.1 Spaces of Types......Page 87
5.2 ω-Saturated Models......Page 89
5.3 Quantifier Elimination......Page 92
5.4 Historic and Bibliographic Notes......Page 95
6.1 Algebraically Closed Fields......Page 96
6.2 Differentially Closed Fields......Page 102
6.3 Boolean Algebras......Page 110
6.4 Ultrametric Spaces......Page 118
6.5 Modules and Existentially Closed Modules......Page 123
6.6 Real Closed Fields (not in the original edition)......Page 130
6.7 Historic and Bibliographic Notes......Page 137
7.1 The Successor Function......Page 140
7.2 The Order......Page 142
7.3 The Sum......Page 143
7.4 Sum and Product: Coding of Finite Sets......Page 148
7.5 Coding of Formulas; Tarski’s Theorem......Page 154
7.6 The Hierarchy of Arithmetic Sets......Page 156
7.7 Some Axioms, Models, and Fragments of Arithmetic......Page 166
7.8 Nonstandard Models with Arithmetic Definitions......Page 173
7.9 Arithmetic Translation of Henkin’s Method......Page 174
7.10 The Notion of Proof; Decidable Theories......Page 179
7.11 Gödel’s Theorem......Page 183
7.12 A Little Mathematical Fiction......Page 187
7.13 Historic and Bibliographic Notes......Page 190
8.1 Well-Ordered Sets......Page 192
8.2 Axiom of Choice......Page 196
8.3 Cardinals......Page 203
8.4 Cofinality......Page 209
8.5 Historic and Bibliographic Notes......Page 212
9 – Saturated Models......Page 213
9.1 Svenonius’s Theorem......Page 215
9.2 Compact, Saturated, Homogeneous, and Universal Models......Page 218
9.3 Resplendent Models......Page 223
9.4 Properties Preserved Under Interpretation......Page 227
9.5 Recursively Saturated Models......Page 229
9.6 Historic and Bibliographic Notes......Page 234
10.1 Omitting Types Theorem......Page 236
10.2 Prime Models, Atomic Models: The Denumerable Case......Page 239
10.3 Theories with Finitely Many Denumerable Models......Page 241
10.4 Constructed Models......Page 244
10.5 Minimal Models......Page 247
10.6 Nonuniqueness of the Prime Model......Page 250
10.7 Historic and Bibliographic Notes......Page 255
11.1 Heirs......Page 257
11.2 Definable Types......Page 262
11.3 End Extension Types in Arithmetic......Page 263
11.4 Stable Types and Theories......Page 265
11.5 Historic and Bibliographic Notes......Page 268
12.1 Special Sons......Page 271
12.2 Coheirs......Page 275
12.3 Morley Sequences......Page 278
12.4 The Independence Property......Page 281
12.5 Indivisible Morley Sequences......Page 287
12.6 An Example: The Theories of Chains......Page 294
12.7 Special Sequences......Page 300
12.8 Instability and Order......Page 302
12.9 Appendix: Ramsey’s Theorem......Page 305
12.10 Historic and Bibliographic Notes......Page 307
13.1 The Fundamental Order......Page 309
13.2 Stability Spectrum......Page 313
13.3 Some Examples......Page 317
13.4 Historic and Bibliographic Notes......Page 321
14.1 Existence Theorem......Page 322
14.2 Nonexistence Theorems......Page 323
14.3 Resplendent Models......Page 326
14.4 Sufficiently Saturated Extensions of a Given Model......Page 327
14.5 Historic and Bibliographic Notes......Page 330
15 – Forking......Page 331
15.1 The Theorem of the Bound......Page 332
15.2 Forking and Nonforking Sons......Page 335
15.3 Multiplicity......Page 337
15.4 Stable Types in an Unstable Theory......Page 339
15.5 Historic and Bibliographic Notes......Page 340
16.1 The Finite Equivalence Relation Theorem......Page 341
16.2 Spaces of Strong Types; Open Mapping Theorem......Page 344
16.3 Morley Sequences for Strong Types; Saturated Models Revisited......Page 346
16.4 Imaginary Elements......Page 350
16.5 Elimination of Imaginaries......Page 353
16.6 A Galois Theory for Strong Types......Page 360
16.7 Historic and Bibliographic Notes......Page 363
17.1 Lascar Rank......Page 364
17.2 Shelah Rank......Page 368
17.3 Morley Rank......Page 373
17.4 Local Ranks......Page 377
17.5 Historic and Bibliographic Notes......Page 381
18.1 Uniqueness Theorem......Page 383
18.2 Prime Models of a Totally Transcendental Theory......Page 385
18.3 Galois Theory of Differential Equations......Page 390
18.4 Prime |T|+-Saturated Models......Page 397
18.5 Ehrenfeucht Models......Page 399
18.6 Two-Cardinal Theorem; 1א-Categorical Theories......Page 402
18.7 Historic and Bibliographic Notes......Page 404
19.1 Indiscernible Sequences......Page 406
19.2 Lascar Inequalities......Page 408
19.3 Weight of a Superstable Type......Page 413
19.4 Independence and Domination......Page 416
19.5 Historic and Bibliographic Notes......Page 424
20.1 Rudin-Keisler Order......Page 425
20.2 Dimensional Types and Theories......Page 434
20.3 Classification of the Models of a Dimensional Theory......Page 441
20.4 The Dope......Page 446
20.5 Depth and the Main Gap......Page 448
20.6 Historic and Bibliographic Notes......Page 449
A......Page 451
B-C......Page 452
D-E-F-G......Page 453
H-J-K......Page 454
L-M......Page 455
N-O-P......Page 456
R-S......Page 457
T-V-W......Page 459
Index of Notation......Page 461
A......Page 465
B-C......Page 466
D......Page 467
E-F......Page 468
G-H-I......Page 469
J-K-L-M......Page 470
N-O-P......Page 471
Q-R......Page 472
S......Page 473
T-U......Page 474
V-W-Z......Page 475
