Model Theory

Contents......Page 5
Introduction......Page 9
Note on notation......Page 12
1 Naming of parts......Page 15
1.1 Structures......Page 16
1.2 Homomorphisms and substructures......Page 19
1.3 Terms and atomic formulas......Page 25
1.4 Parameters and diagrams......Page 29
1.5 Canonical models......Page 32
History and bibliography......Page 35
2 Classifying structures......Page 37
2.1 Definable subsets......Page 38
2.2 Definable classes of structures......Page 47
2.3 Some notions from logic......Page 54
2.4 Maps and the formulas they preserve......Page 61
2.5 Classifying maps by formulas......Page 68
2.6 Translations......Page 72
2.7 Quantifier elimination......Page 80
2.8 Further examples......Page 89
History and bibliography......Page 96
3.1 Theorems of Skolem......Page 101
3.2 Back-and-forth equivalence......Page 108
3.3 Games for elementary equivalence......Page 116
3.4 Closed games......Page 125
3.5 Games and infinitary languages......Page 133
3.6 Clubs......Page 138
History and bibliography......Page 142
4 Automorphisms......Page 145
4.1 Automorphisms......Page 146
4.2 Subgroups of small index......Page 154
4.3 Imaginary elements......Page 162
4.4 Eliminating imaginaries......Page 171
4.5 Minimal sets......Page 177
4.6 Geometries......Page 184
4.7 Almost strongly minimal theories......Page 192
4.8 Zil\'ber\'s configuration......Page 202
History and bibliography......Page 211
5 Interpretations......Page 215
5.1 Relativisation......Page 216
5.2 Pseudo-elementary classes......Page 220
5.3 Interpreting one structure in another......Page 226
5.4 Shapes and sizes of interpretations......Page 233
5.5 Theories that interpret anything......Page 241
5.6 Totally transcendental structures......Page 251
5.7 Interpreting groups and fields......Page 262
History and bibliography......Page 274
6 The first-order case: compactness......Page 278
6.1 Compactness for first-order logic......Page 279
6.2 Boolean algebras and Stone spaces......Page 285
6.3 Types......Page 291
6.4 Elementary amalgamation......Page 299
6.5 Amalgamation and preservation......Page 308
6.6 Expanding the language......Page 314
6.7 Stability......Page 320
History and bibliography......Page 332
7.1 Fraisse\'s construction......Page 337
7.2 Omitting types......Page 347
7.3 Countable categoricity......Page 355
7.4 cocategorical structures by Fraisse\'s method......Page 362
History and bibliography......Page 371
8 The existential case......Page 374
8.1 Existentially closed structures......Page 375
8.2 Two methods of construction......Page 380
8.3 Model-completeness......Page 388
8.4 Quantifier elimination revisited......Page 395
8.5 More on e.c. models......Page 405
8.6 Amalgamation revisited......Page 414
History and bibliography......Page 423
9 The Horn case: products......Page 426
9.1 Direct products......Page 427
9.2 Presentations......Page 434
9.3 Word-constructions......Page 444
9.4 Reduced products......Page 455
9.5 Ultraproducts......Page 463
9.6 The Feferman-Vaught theorem......Page 472
9.7 Boolean powers......Page 480
History and bibliography......Page 487
10 Saturation......Page 492
10.1 The great and the good......Page 493
10.2 Big models exist......Page 503
10.3 Syntactic characterisations......Page 510
10.4 Special models......Page 520
10.5 Definability......Page 529
10.6 Resplendence......Page 536
10.7 Atomic compactness......Page 541
History and bibliography......Page 546
11 Combinatorics......Page 549
11.1 Indiscernibles......Page 550
11.2 Ehrenfeucht-Mostowski models......Page 559
11.3 EM models of unstable theories......Page 569
11.4 Nonstandard methods......Page 581
11.5 Defining well-orderings......Page 590
11.6 Infinitary indiscernibles......Page 600
History and bibliography......Page 608
12 Expansions and categoricity......Page 613
12.1 One-cardinal and two-cardinal theorems......Page 614
12.2 Categoricity......Page 625
12.3 Cohomology of expansions......Page 638
12.4 Counting expansions......Page 646
12.5 Relative categoricity......Page 652
History and bibliography......Page 663
A.1 Modules......Page 667
A.2 Abelian groups......Page 676
A.3 Nilpotent groups of class 2......Page 687
A.4 Groups......Page 702
A.5 Fields......Page 709
A.6 Linear orderings......Page 720
References......Page 730
Index to symbols......Page 769
Index......Page 771