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模型论是数学的新门类

(2020-02-07 11:23:34)

模型论是数学的新门类

     今后,有人会对你说:‘我是搞模型论的。”你不要感到1奇怪。

     为什么?百度一下“无穷小

  微积分”网站s 下载,下载

  

  模型论”,其中第一句话是.Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. On the one hand it is the ultimate abstraction; on the other, it has immediate applications to every-day mathematics. The fundamental tenet of Model Theory is that mathematical truth, like all truth, is relative. A statement may be true or false, depending on how and where it is interpreted. T

   阅读之后了就会明日歌曲哦了。

  

   袁萌陈启清28

   附件:

   Fundamentals of Model Theory

   William Weiss and Cherie D’Mello

   Introduction

   Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. On the one hand it is the ultimate abstraction; on the other, it has immediate applications to every-day mathematics. The fundamental tenet of Model Theory is that mathematical truth, like all truth, is relative. A statement may be true or false, depending on how and where it is interpreted. This isn’t necessarily due to mathematics itself, but is a consequence of the language that

  

   we use to express mathematical ideas. What at rst seems like a deciency in our language, can actually be shaped into a powerful tool for understanding mathematics. This book provides an introduction to Model Theory which can be used as a text for a reading course or a summer project at the senior undergraduate or graduate level. It is also a primer which will give someone a self contained overview of the subject, before diving into one of the more encyclopedic standard graduate texts. Any reader who is familiar with the cardinality of a set and the algebraic closure of a eld can proceed without worry. Many readers will have some acquaintance with elementary logic, but this is not absolutely required, since all necessary concepts from logic are reviewed in Chapter 0. Chapter 1 gives the motivating examples; it is short and we recommend that you peruse it rst, before studying the more technical aspects of Chapter 0. Chapters 2 and 3 are selections of some of the most important techniques in Model Theory. The remaining chapters investigate the relationship between Model Theory and the algebra of the real and complex numbers. Thirty exercises develop familiarity with the denitions and consolidate understanding of the main proof techniques. Throughout the book we present applications which cannot easily be found elsewhere in such detail. Some are chosen for their value in other areas of mathematics: Ramsey’s Theorem, the Tarski-Seidenberg Theorem. Some are chosen for their immediate appeal to every mathematician: existence of innitesimals for calculus, graph colouring on the plane. And some, like Hilbert’s Seventeenth Problem, are chosen because of how amazing it is that logic can play an important role in the solution of a problem from high school algebra. In each case, the derivation is shorter than any which tries to avoid logic. More importantly, the methods of Model Theory display clearly the structure of the main ideas of the proofs, showing how theorems of logic combine with theorems from other areas of mathematics to produce stunning results. The theorems here are all are more than thirty years old and due in great part to the cofounders of the subject, Abraham Robinson and Alfred Tarski. However, we have not attempted to give a history. When we attach a name to a theorem, it is simply because that is what mathematical logicians popularly call it. The bibliography contains a number of texts that were helpful in the preparation of this manuscript. They could serve as avenues of further study and in addition, they contain many other references and historical notes. The more recent titles were added to show the reader where the subject is moving today. All are worth a look. This book began life as notes for William Weiss’s graduate course at the University of Toronto. The notes were revised and expanded by Cherie D’Mello and

   2

   William Weiss, based upon suggestions from several graduate students. The electronic version of this book may be downloaded and further modied by anyone for the purpose of learning, provided this paragraph is included in its entirety and so long as no part of this book is sold for prot.

   Contents

   Chapter 0. Models, Truth and Satisfaction 4 Formulas, Sentences, Theories and Axioms 4 Prenex Normal Form 9

   Chapter 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

   3

   CHAPTER 0

   Models, Truth and Satisfaction

   We will use the following symbols: • logical symbols: – the connectives , , ¬ , , ↔ called “and”, “or”, “not”, “implies” and “i” respectively – the quantiers , called “for all” and “there exists” – an innite collection of variables indexed by the natural numbers N v0 ,v1 , v2 , ... – the two parentheses ), ( – the symbol = which is the usual “equal sign” • constant symbols : often denoted by the letter c with subscripts • function symbols : often denoted by the letter F with subscripts; each function symbol is an m-placed function symbol for some natural number m 1 relation symbols : often denoted by the letter R with subscripts; each relational symbol is an n-placed relation symbol for some natural number n 1.

  

  


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