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

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

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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 L÷wenheims Theorem
25
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

25 Historic and Bibliographic Notes
27
Extensions of the Language Structures
29
32 Functions
31
33 L÷wenheims Theorem Revisited
34
34 Historic and Bibliographic Notes
35
Compactness
36
42 Compactness L÷wenheimSkolem 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
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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