谈逻辑与数学界线之淡化

    在近代数学研究中，逻辑与数学界线之淡化。这种现象值得关注。

    在“模型论入门”博文中，我们指出，进入二十一世纪，世界数学形式化、公理化高潮兴起。“知识共享”无穷小微积分就是在这种发展浪潮中崛起的。

    反观我们国内，数学教育界的学术认识水平似乎仍然停留在世界数学形式化、公理化发展阶段之前。

    无论如何，我们的数学教育落伍了。

    说明：本本附件，是关于模型论的长篇论文，资料详实，值得存档备查。

袁萌  陈启清   12月17日

附件：模型论（Model Theory）

Class Notes for Mathematics 571 Spring 2010

Model Theory

written by C. Ward Henson

Mathematics Department University of Illinois 1409 West Green Street Urbana, Illinois 61801 email: henson@math.uiuc.edu www: http://www.math.uiuc.edu/~henson/

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Copyright by C. Ward Henson 2010; all rights reserved.

Introduction

The purpose of Math 571 is to give a thorough introduction to the methods of model theory for rst order logic. Model theory is the branch of logic that deals with mathematical structures and the formal languages they interpret. First order logic is the most important formal language and its model theory is a rich and interesting subject with signicant applications to the main body of mathematics. Model theory began as a serious subject in the 1950s with the work of Abraham Robinson and Alfred Tarski, and since then it has been an active and successful area of research. Beyond the core techniques and results of model theory, Math 571 places a lot of emphasis on examples and applications, in order to show clearly the variety of ways in which model theory can be useful in mathematics. For example, we give a thorough treatment of the model theory of the eld of real numbers (real closed elds) and show how this can be used to obtain the characterization of positive semi-denite rational functions that gives a solution to Hilbert’s 17th Problem. A highlight of Math 571 is a proof of Morley’s Theorem: if T is a complete theory in a countable language, and T is κ-categorical for some uncountable κ, then T is categorical for all uncountable κ. The machinery needed for this proof includes the concepts of Morley rank and degree for formulas in ω-stable theories. The methods needed for this proof illustrate ideas that have become central to modern research in model theory. To succeed in Math 571, it is necessary to have exposure to the syntax and semantics of rst order logic, and experience with expressing mathematical properties via rst order formulas. A good undergraduate course in logic will usually provide the necessary background. The canonical prerequisite course at UIUC is Math 570, but this covers many things that are not needed as background for Math 571. In the lecture notes for Math 570 (written by Prof. van den Dries) the material necessary for Math 571 is presented in sections 2.3 through 2.6 (pages 24–37 in the 2009 version). These lecture notes are available at http://www.math.uiuc.edu/ vddries/410notes/main.dvi. A standard undergraduate text in logic is A Mathematical Introduction to Logic by Herbert B. Enderton (Academic Press; second edition, 2001). Here the material needed for Math 571 is covered in sections 2.0 through 2.2 (pages 67–104). This material is also discussed in Model Theory by David Marker (see sections 1.1 and 1.2, and the rst half of 1.3, as well as many of the exercises at the end of chapter 1) and in many other textbooks in model theory. For Math 571 it is not necessary to have any exposure to a proof system for rst order logic, nor to G¨odel’s completeness theorem. Math 571 begins with a proof of the compactness theorem for rst order languages, and this is all one needs for model theory.

We close this introduction by discussing a number of books of possible interest to anyone studying model theory. The rst two books listed are now the standard graduate texts in model theory; they can be used as background references for most of what is done in Math 571. David Marker, Model Theory: an Introduction. Bruno Poizat, A Course in Model Theory.

The next book listed was the standard graduate text in model theory from its rst publication in the 1960s until recently. It is somewhat out of date and incomplete from a modern viewpoint, but for much of the content of Math 571 it is a suitable reference. C. C. Chang and H. J. Keisler, Model Theory.

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