2010 Summer Course on Model Theory Pete
نویسنده
چکیده
Introduction 1 0.1. Some theorems in mathematics with snappy model-theoretic proofs 1 1. Languages, structures, sentences and theories 1 1.1. Languages 1 1.2. Statements and Formulas 4 1.3. Satisfaction 5 1.4. Elementary equivalence 6 1.5. Theories 7 2. Big Theorems: Completeness, Compactness and Löwenheim-Skolem 9 2.1. The Completeness Theorem 9 2.2. Proof-theoretic consequences of the completeness theorem 10 2.3. The Compactness Theorem 12 2.4. Topological interpretation of the compactness theorem 12 2.5. First applications of compactness 14 2.6. The Löwenheim-Skolem Theorems 16 3. Complete and model complete theories 18 3.1. Maximal and complete theories 18 3.2. Model complete theories 19 3.3. Algebraically closed fields I: model completeness 20 3.4. Algebraically closed fields II: Nullstellensätze 21 3.5. Algebraically closed fields III: Ax’s Transfer Principle 23 3.6. Ordered fields and formally real fields I: background 24 3.7. Ordered fields and formally real fields II: the real spectrum 25 3.8. Real-closed fields I: definition and model completeness 26 3.9. Real-closed fields II: Nullstellensatz 26 3.10. Real-closed fields III: Hilbert’s 17th problem 29 4. Categoricity: a condition for completeness 30 4.1. DLO 31 4.2. R-modules 32 4.3. Morley’s Categoricity Theorem 34 4.4. Complete, non-categorical theories 35 5. Quantifier elimination: a criterion for model-completeness 35 5.1. Constructible and definable sets 36 5.2. Quantifier Elimination: Definition and Implications 38 5.3. A criterion for quantifier elimination 40 5.4. Model-completeness of ACF 43 5.5. Model-completeness of RC(O)F 43 5.6. Algebraically Prime Models 44
منابع مشابه
Summer 2010 Course on Model Theory: Chapter 4
By Löwenheim-Skolem, any theory in a countable language which admits infinite models admits models of every infinite cardinality, and indeed, models of any given cardinality elementarily equivalent to any fixed infinite model. Thus the next step in understanding the relation of elementary e equivalence is to consider models of a fixed cardinality. In this regard, the following definition captur...
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