Model Theory (eBook)

Front Cover 1
Model Theory 4
Copyright Page 5
Contents 16
Chapter 1. Introduction 18
1.1. What is model theory? 18
1.2. Model theory for sentential logic 21
1.3. Languages, models and satisfaction 35
1.4. Theories and examples of theories 53
1.5. Elimination of quantifiers 66
Chapter 2. Models constructed from constants 78
2.1. Completeness and compactness 78
2.2. Refinements of the method. Omitting types and interpolation theorems 94
2.3. Countable models of complete theories 113
2.4. Recursively saturated models 126
2.5. Lindström’s characterization of first order logic 144
Chapter 3. Further model-theoretic constructions 153
3.1. Elementary extensions and elementary chains 153
3.2. Applications of elementary chains 164
3.3. Skolem functions and indiscernibles 180
3.4. Some examples 195
3.5. Model completeness 203
Chapter 4. Ultraproducts 228
4.1. The fundamental theorem 228
4.2. Measurable cardinals 244
4.3. Regular ultrapowers 265
4.4. Nonstandard universes 279
Chapter 5. Saturated and special models 309
5.1. Saturated and special models 309
5.2. Preservation theorems 323
5.3. Applications of special models to the theory of definability 340
5.4. Applications to field theory 359
5.5. Application to Boolean algebras 389
Chapter 6. More about ultraproducts and generalizations 401
6.1. Ultraproducts which are saturated 401
6.2. Direct products, reduced products, and Horn sentences 422
6.3. Direct products, reduced products, and Horn sentences (continued) 437
6.4. Limit ultrapowers and complete extensions 464
6.5. Iterated ultrapowers 480
Chapter 7. Selected topics 499
7.1. Categoricity in power 499
7.2. An extension of Ramsey’s theorem and applications some two-cardinal theorems
7.3. Models of large cardinality 551
7.4. Large cardinals and the constructible universe 575
Appendix A. Set theory 596
Appendix B. Open problems in classical model theory 614
Historical notes 620
References 640
Additional references 651
Index of definitions 658
Index of symbols 666