2010 Summer Course on Model Theory
نویسنده
چکیده
Introduction 2 0.1. Some theorems in mathematics with snappy model-theoretic proofs 2 1. Languages, structures, sentences and theories 2 1.1. Languages 2 1.2. Statements and Formulas 5 1.3. Satisfaction 6 1.4. Elementary equivalence 7 1.5. Theories 8 2. Big Theorems: Completeness, Compactness and Löwenheim-Skolem 9 2.1. The Completeness Theorem 9 2.2. Proof-theoretic consequences of the completeness theorem 11 2.3. The Compactness Theorem 13 2.4. Topological interpretation of the compactness theorem 13 2.5. First applications of compactness 15 2.6. The Löwenheim-Skolem Theorems 17 3. Complete and model complete theories 19 3.1. Maximal and complete theories 19 3.2. Model complete theories 20 3.3. Algebraically closed fields I: model completeness 21 3.4. Algebraically closed fields II: Nullstellensätze 22 3.5. Algebraically closed fields III: Ax’s Transfer Principle 24 3.6. Ordered fields and formally real fields I: background 25 3.7. Ordered fields and formally real fields II: the real spectrum 26 3.8. Real-closed fields I: definition and model completeness 26 3.9. Real-closed fields II: Nullstellensatz 27 3.10. Real-closed fields III: Hilbert’s 17th problem 30 4. Categoricity: a condition for completeness 30 4.1. DLO 32 4.2. R-modules 33 4.3. Morley’s Categoricity Theorem 35 4.4. Complete, non-categorical theories 35 5. Quantifier elimination: a criterion for model-completeness 36 5.1. Constructible and definable sets 36 5.2. Quantifier Elimination: Definition and Implications 39 5.3. A criterion for quantifier elimination 41 5.4. Model-completeness of ACF 43 5.5. Model-completeness of RC(O)F 44 5.6. Algebraically Prime Models 45
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