A course in model theory: an introduction to contemporary mathematical logic
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| Main Author: | |
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| Other Authors: | |
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
[2000]
|
| Series: | Universitext
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Franzoisch übersetzt |
| Physical Description: | xxxi, 443 Seiten |
| ISBN: | 9780387986555 9781441986221 |
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Record in the Search Index
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| adam_text | Contents
Preface to the English Edition vii
Introduction xxiii
1 Elementary Classes of Relations 1
1.1 Local Isomorphisms Between Relations 1
1.2 Examples 5
1.3 Infinite Back and Forth 11
1.4 Historic and Bibliographic Notes 13
2 The Language Associated with a Relation 15
2.1 Formulas 15
2.2 Connections to the Back and Forth Technique 23
2.3 Models and Theories 25
2.4 Elementary Extensions: Tarski s Test,
Lowenheim s Theorem 27
2.5 Historic and Bibliographic Notes 29
3 Extensions of the Language: Structures 31
3.1 Multirelations, Relational Structures 31
3.2 Functions 33
3.3 Lowenheim s Theorem Revisited 36
3.4 Historic and Bibliographic Notes 37
xviii Contents
4 Compactness 38
4.1 Ultraproducts 38
4.2 Compactness, Lowenheim Skolem Theorem,
Theorem of Common Elementary Extensions 42
4.3 Henkin s Method 47
4.4 Historic and Bibliographic Notes 52
5 The Back and Forth Method in w Saturated Models 55
5.1 Spaces of Types 55
5.2 w Saturated Models 57
5.3 Quantifier Elimination 60
5.4 Historic and Bibliographic Notes 63
6 Examples Illustrating the Back and Forth Method 64
6.1 Algebraically Closed Fields 64
6.2 Differentially Closed Fields 70
6.3 Boolean Algebras 78
6.4 Ultrametric Spaces 86
6.5 Modules and Existentially Closed Modules 91
6.6 Real Closed Fields (not in the original edition) 98
6.7 Historic and Bibliographic Notes 105
7 Arithmetic 108
7.1 The Successor Function 108
7.2 The Order 110
7.3 The Sum Ill
7.4 Sum and Product: Coding of Finite Sets 116
7.5 Coding of Formulas; Tarski s Theorem 122
7.6 The Hierarchy of Arithmetic Sets 124
7.7 Some Axioms, Models, and Fragments of Arithmetic . . 134
7.8 Nonstandard Models with Arithmetic Definitions .... 141
7.9 Arithmetic Translation of Henkin s Method 142
7.10 The Notion of Proof; Decidable Theories 147
7.11 Godel s Theorem 151
7.12 A Little Mathematical Fiction 155
7.13 Historic and Bibliographic Notes 158
8 Ordinals and Cardinals 160
8.1 Well Ordered Sets 160
8.2 Axiom of Choice 164
8.3 Cardinals 171
8.4 Cofinality 177
8.5 Historic and Bibliographic Notes 180
Contents xix
9 Saturated Models 181
9.1 Svenonius s Theorem 183
9.2 Compact, Saturated, Homogeneous, and Universal Models 186
9.3 Resplendent Models 191
9.4 Properties Preserved Under Interpretation 195
9.5 Recursively Saturated Models 197
9.6 Historic and Bibliographic Notes 202
10 Prime Models 204
10.1 Omitting Types Theorem 204
10.2 Prime Models, Atomic Models: The Denumerable Case . 207
10.3 Theories with Finitely Many Denumerable Models .... 209
10.4 Constructed Models 212
10.5 Minimal Models 215
10.6 Nonuniqueness of the Prime Model 218
10.7 Historic and Bibliographic Notes 223
11 Heirs 225
11.1 Heirs 225
11.2 Definable Types 230
11.3 End Extension Types in Arithmetic 231
11.4 Stable Types and Theories 233
11.5 Historic and Bibliographic Notes 236
12 Special Sons, Morley Sequences 239
12.1 Special Sons 239
12.2 Coheirs 243
12.3 Morley Sequences 246
12.4 The Independence Property 249
12.5 Indivisible Morley Sequences 255
12.6 An Example: The Theories of Chains 262
12.7 Special Sequences 268
12.8 Instability and Order 270
12.9 Appendix: Ramsey s Theorem 273
12.10 Historic and Bibliographic Notes 275
13 The Fundamental Order 277
13.1 The Fundamental Order 277
13.2 Stability Spectrum 281
13.3 Some Examples 285
13.4 Historic and Bibliographic Notes 289
14 Stability and Saturated Models 290
14.1 Existence Theorem 290
14.2 Nonexistence Theorems 291
xx Contents
14.3 Resplendent Models 294
14.4 Sufficiently Saturated Extensions of a Given Model . . . 295
14.5 Historic and Bibliographic Notes 298
15 Forking 299
15.1 The Theorem of the Bound 300
15.2 Forking and Nonforking Sons 303
15.3 Multiplicity 305
15.4 Stable Types in an Unstable Theory 307
15.5 Historic and Bibliographic Notes 308
16 Strong Types 309
16.1 The Finite Equivalence Relation Theorem 309
16.2 Spaces of Strong Types; Open Mapping Theorem .... 312
16.3 Morley Sequences for Strong Types;
Saturated Models Revisited 314
16.4 Imaginary Elements 318
16.5 Elimination of Imaginaries 321
16.6 A Galois Theory for Strong Types 328
16.7 Historic and Bibliographic Notes 331
17 Notions of Rank 332
17.1 Lascar Rank 332
17.2 Shelah Rank 336
17.3 Morley Rank 341
17.4 Local Ranks 345
17.5 Historic and Bibliographic Notes 349
18 Stability and Prime Models 351
18.1 Uniqueness Theorem 351
18.2 Prime Models of a Totally Transcendental Theory .... 353
18.3 Galois Theory of Differential Equations 358
18.4 Prime |T|+ Saturated Models 365
18.5 Ehrenfeucht Models 367
18.6 Two Cardinal Theorem; Nt Categorical Theories 370
18.7 Historic and Bibliographic Notes 372
19 Stability, Indiscernible Sequences and Weights 374
19.1 Indiscernible Sequences 374
19.2 Lascar Inequalities 376
19.3 Weight of a Superstable Type 381
19.4 Independence and Domination 384
19.5 Historic and Bibliographic Notes 392
Contents xxi
20 Dimension in Models of a Totally Transcendental Theory 393
20.1 Rudin Keisler Order 393
20.2 Dimensional Types and Theories 402
20.3 Classification of the Models of a Dimensional Theory . . 409
20.4 The Dope 414
20.5 Depth and the Main Gap 416
20.6 Historic and Bibliographic Notes 417
Bibliography 419
Index of Notation 429
Index 433
|
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| spelling | Poizat, Bruno 1946- (DE-588)1044764392 aut A course in model theory an introduction to contemporary mathematical logic Bruno Poizat New York Springer [2000] © 2000 xxxi, 443 Seiten txt rdacontent n rdamedia nc rdacarrier Universitext Aus dem Franzoisch übersetzt Modelltheorie Model theory Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Modelltheorie (DE-588)4114617-7 gnd rswk-swf Modelltheorie (DE-588)4114617-7 s DE-604 Mathematische Logik (DE-588)4037951-6 s Klein, Moses trl Erscheint auch als Online-Ausgabe 978-1-4419-8622-1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009030202&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Poizat, Bruno 1946- A course in model theory an introduction to contemporary mathematical logic Modelltheorie Model theory Mathematische Logik (DE-588)4037951-6 gnd Modelltheorie (DE-588)4114617-7 gnd |
| subject_GND | (DE-588)4037951-6 (DE-588)4114617-7 |
| title | A course in model theory an introduction to contemporary mathematical logic |
| title_auth | A course in model theory an introduction to contemporary mathematical logic |
| title_exact_search | A course in model theory an introduction to contemporary mathematical logic |
| title_full | A course in model theory an introduction to contemporary mathematical logic Bruno Poizat |
| title_fullStr | A course in model theory an introduction to contemporary mathematical logic Bruno Poizat |
| title_full_unstemmed | A course in model theory an introduction to contemporary mathematical logic Bruno Poizat |
| title_short | A course in model theory |
| title_sort | a course in model theory an introduction to contemporary mathematical logic |
| title_sub | an introduction to contemporary mathematical logic |
| topic | Modelltheorie Model theory Mathematische Logik (DE-588)4037951-6 gnd Modelltheorie (DE-588)4114617-7 gnd |
| topic_facet | Modelltheorie Model theory Mathematische Logik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009030202&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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