微积分手机版作者j.Keisler的学术贡献

    2014年8月，K.Gannon发表文章，介绍J,Keisler在模型轮方面的原创学术成就，并且指出其在证明“p = t”（自然数与实数一样多！）时的重要作用。

    实际情况是，由于种种原因，国人对数理逻辑模型轮缺乏研究，对于J.Keisler的学术贡献更是“不甚了了”。

     我们建议，由于K.Gannon的这篇介绍文章并不长，有兴趣的读者可以快速浏览一下，以便对Keisler的学术贡献有个大致的了解。

袁萌   陈启清   8月21日

附：INRODUCTION TO THE KEISLER ORDER

By KYLE GANNON

Abstract. In this paper, we introduce the basic denitions and concepts necessary to dene the Keisler Order. We will prove the order is well-dened as well as the existence of a maximal class with respect to the order.

Contents

1. Introduction 1

2. Notation and Basic Denitions 2

3. Ultrapowers 3

4. Saturation and Satisfaction 6

5. An Early Application 7

6. The Order 9

7. Existence of a Maximal Class 10 Acknowledgments 12

References 12

1. Introduction

The Keisler Order was rst introduced by H. Jerome Keisler in 1967. Currently, this order is known to be a pre-order on (countable) rst-order theories which, broadly speaking, ranks classes of theories by complexity. Stronger theorems have been proven for stable theories (e.g. the Keisler Order on stable theories is linear [5]), while the complete structure of the Keisler Order is still an open problem. The classication of rst-order theories is both a classic and modern program in model theory. Shelah’s stability program, the most famous type of classication framework, organizes theories relative to the number of denable types over subsets of a model. While the stability program has had great success, the program also leaves unstable theories in some unclassiable purgatory. Work on the Keisler Order has shed light on dividing lines between classes of unstable theories. Additionally, one of the major results in a paper by Malliaris and Shelah [4] shows that theories, which have the SOP2−property, are maximal. This result was important in proving p = t,  （p = t，这是21世纪世界数学的最大进展之一！ 以下省略，有兴趣者，请查阅原文）