​    ​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，……

附：SUMMER COURSE ON MODELTHEORY

（作者：PETE L. CLARK，此文发表于2010年）

Contents（目录）

Introduction 2

0.1. Some theorems in mathematics with snappy model-theoretic proofs ?2

1. Languages, structures, sentences andtheories 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）andL¨owenheim-Skolem 9 2.1. The Completeness Theorem（哥德尔完备性定理） 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 L¨owenheim-SkolemTheorems 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 ?elds I: model completeness 21

3.4. Algebraically closed ?elds II: Nullstellens¨atze 22

3.5. Algebraically closed ?elds III: Ax’s Transfer Principle 24 3.6.Ordered ?elds and formally real ?elds I: background 25 3.7. Ordered ?elds andformally real ?elds II: the real spectrum 26

3.8. Real-closed ?elds I: de?nition and model completeness 26

3.9. Real-closed ?elds II: Nullstellensatz 27 3.10. Real-closed ?eldsIII: Hilbert’s 17th problem 30

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4. Categoricity: a condition for completeness30

4.1. DLO 32

4.2. R-modules 33

4.3. Morley’s Categoricity Theorem 35 4.4. Complete, non-categoricaltheories 35

5. Quanti?er elimination: a criterion formodel-completeness 36

5.1. Constructible and de?nable sets 36

5.2. Quanti?er Elimination: De?nition and Implications 39

5.3. A criterion for quanti?er elimination 41 5.4. Model-completenessof ACF 43

5.5. Model-completeness of RC(O)F 44

5.6. Algebraically Prime Models 45 1

6. Ultraproducts and ultrapowers in modeltheory 47

6.1. Filters and ultra?lters 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 involving ultraproducts 55

7. A Glimpse of the Ax-Kochen Theorem 56References ?58

Introduction

0.1. Some theorems in mathematics with snappymodel-theoretic proofs.

1) The Nullstellensatz and theR-Nullstellensatz. 2) Chevalley’s Theorem: the image of a constructible set isconstructible. 3) (Grothendieck, Ax) An injective polynomial map from Cn to Cnis surjective. 4) Hilbert’s 17th problem: a positive semide?nite rationalfunction f ∈R(t1,...,tn) is a sum ofsquares. 5) Polynomially compact operators have invariant subspaces. (Let V bea complex Hilbert space, L a bounded linear operator on H, and 0 ?= P(t) ∈ C[t] . Supposethat P(L) is a compact operator: the image of the unit ball has compactclosure. Then there exists a nontrivial, proper closed subspace W of V which isL-invariant.) 6) (Ax-Kochen) For each n ∈Z+ and allsu?ciently large primes p, a homogeneous form with coe?cients in Qp with atleast n2 + 1 variables has a nontrivial zero. 7) (Duesler-Knecht) An analogueof the Ax-Katz theorem for rationally connected varieties over the maximalunrami?ed extension of Qp. 8) (Faltings, Hrushovski) Mordell-Lang Conjecture.

Remark: So far as I know, these are inincreasing order of di?culty. However, I have barely glanced at the proof of5), so this is just a guess.

1. Languages, structures, sentences andtheories

1.1. Languages.

Denition: A language L is the supply ofsymbols that are deemed admissible when appearing in an expression. There arethree types of symbols that make up a language: for each n ∈Z+, a set ofn-ary function symbols f = f(x1,...,xn) – formally f is just a symbol which hasthe number n associated to it; for each n ∈ Z+, a set ofn-ary relation symbols R = R(x1,...,xn) – formally exactlythe same as a function – and a set of constantsymbols.

We assume that all of these sets are disjoint– i.e., that given an element a ∈L, we can sayunambiguously that it is an n-ary function symbol for a unique n ∈ Z+, an n-aryrelation symbol (for a unique n) or a constant. Otherwise the sets are quitearbitrary: any or all of them may be empty (and most of them usually will be inany given application), and the sets may be in?nite, even uncountably in?nite.

2010 SUMMER COURSE ON MODEL THEORY 3

An L-structure X is given by the data of ? anunderlying set1, which we (abusively) also denote X, ?for all n and for eachn-ary function f a map of sets n-ary function fX : Xn → X, ? for all nand for each n-ary relation R a subset RX ? Xn, ? for each constantc an element cX ∈ X. That is, to endow a s

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