元旦学习模型论有感

  首先，袁萌与陈启清（此刻在上海出差）向大家预祝元旦快乐   ！

  过去几年，袁萌经常学习本文附件关于模型论的文章，每次都有所心得。

  我们希望，数学同行们学点儿模型论知识。

  请见本文附件。

袁萌  陈启清  2020年元旦

附件;

Fundamentals of Model Theory

Contents

Chapter  0.

Models, Truth and Satisfaction 4 Formulas, Sentences, Theories and Axioms 4

Prenex Normal Form 9

Chapte   1.

Notation and Examples    11

Chapter 2. Compactness and Elementary Submodels   14

The Compactness Theorem 14 Isomorphisms, elementary equivalence and complete theories 15 The Elementary Chain Theorem 16 The L¨owenheim-Skolem Theorem    19

The L o´s-Vaught Test 20

Every complex one-to-one polynomial map is onto 22

Chapter 3.

Diagrams and Embeddings 24 Diagram Lemmas 25 Every planar graph can be four coloured 25

Ramsey’s Theorem 26

The Leibniz Principle and innitesimals 27 The Robinson Consistency Theorem 27

The Craig Interpolation Theorem  31

Chapter   4.

Model Completeness 32

Robinson’s Theorem on existentially complete theories 32 Lindstr¨om’s Test 35 Hilbert’s Nullstellensatz 37

Chapter   5.

The Seventeenth Problem 39

Positive denite rational functions are the sums of squares   39

Chapter   6.

Submodel Completeness 45 Elimination of quantiers 45

The Tarski-Seidenberg Theorem  48

Chapter   7.

Model Completions 50

Almost universal theories 52 Saturated models 54 Blum’s Test  55

Bibliography   60

Index    61