模型论简介
美国Geogia大学数论与算术几何方向P.Clark教授在2010年夏天开设“模型论夏季学习班”的参考书，60页，简短明了，读者可以下载学习、研究，就是扫一眼也好。可以根据其目录对模型论有个大致的概念。
无穷小微积分就是建立在模型论紧致性定理之上的分析理论，但是，J,Keisler巧妙地把它简化了，使其通俗易懂，适用于初学者与科普微积分。
我在想，就是看模型论的目录，让国家“教指委”了解无穷小微积分的来历，……
袁萌 7月4日
附：SUMMER COURSE ON MODEL THEORY
（作者：PETE L. CLARK，此文发表于2010年）
Contents（目录）
Introduction 2
0.1. Some theorems in mathematics withsnappy 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 （紧致性）and L¨owenheim-Skolem 9
2.1. The Completeness Theorem（哥德尔完备性定理） 9
2.2. Proof-theoretic consequences of thecompleteness theorem 11
2.3. The Compactness Theorem 13
2.4. Topological interpretation of thecompactness theorem 13
2.5. First applications of compactness 15
2.6. The L¨owenheim-Skolem Theorems 17
3. Complete andmodel complete theories 19
3.1. Maximal and complete theories 19
3.2. Model complete theories 20
3.3. Algebraically closed fields I: modelcompleteness 21
3.4. Algebraically closed fields II:Nullstellens¨atze 22
3.5. Algebraically closed fields III: Ax’sTransfer Principle 24 3.6. Ordered fields and formally real fields I: background25
3.7. Ordered fields and formally real fieldsII: the real spectrum 26
3.8. Real-closed fields I: definition andmodel completeness 26
3.9. Real-closed fields II: Nullstellensatz27
3.10. Real-closed fields III: Hilbert’s 17thproblem 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. Quantifierelimination: a criterion for model-completeness 36
5.1. Constructible and definable sets 36
5.2. Quantifier Elimination: Definition andImplications 39
5.3. A criterion for quantifier elimination41 5.4. Model-completeness of ACF 43
5.5. Model-completeness of RC(O)F 44
5.6. Algebraically Prime Models 45 1
6. Ultraproductsand ultrapowers in model theory 47
6.1. Filters and ultrafilters 47
6.2. Filters in Topology: An Advertisement 49
6.3. Ultraproducts and Los’ Theorem 51
6.4. Proof of Compactness Via Ultraproducts 54
6.5. Characterization theorems involvingultraproducts 55
7. A Glimpse ofthe Ax-Kochen Theorem 56
References 58
Introduction
（此后的内容省略，请阅读原文）