无穷小：历史与应用

上世纪60年代，鲁宾逊意识到，借助现代数理逻辑新进展（模型论紧致性定理）可以把无穷小概念严谨化，后来就有了我们的故事。

美国数学史专家Joel A. Tropp教授对于无穷小的演化历史很有研究。现将其研究论文引用如下，供读者参阅、浏览。

袁萌 陈启清  10月29日

附：

Innitesimals: History & Application

Joel A. Tropp

Plan II Honors Program, WCH 4.104, The University of Texas at Austin, Austin, TX 78712

Abstract.

An innitesimal is a number whose magnitude exceeds zero but somehow fails to exceed any nite, positive number. Although logically problematic, innitesimals are extremely appealing for investigating continuous phenomena. They were used extensively by mathematicians until the late 19th century, at which point they were purged because they lacked a rigorous foundation. In 1960, the logician Abraham Robinson revived them by constructing a number system, the hyperreals, which contains innitesimals and innitely large quantities. This thesis introduces Nonstandard Analysis (NSA), the set of techniques which Robinson invented. It contains a rigorous development of the hyperreals and shows how they can be used to prove the fundamental theorems of real analysis in a direct, natural way. (Incredibly, a great deal of the presentation echoes the work of Leibniz, which was performed in the 17th century.) NSA has also extended mathematics in directions which exceed the scope of this thesis. These investigations may eventually result in fruitful discoveries.

Contents

Introduction: Why Innitesimals? vi

Chapter 1. Historical Background 1

1.1. Overview 1

1.2. Origins 1

1.3. Continuity 3

1.4. Eudoxus and Archimedes 5

1.5. Apply when Necessary 7

1.6. Banished 10

1.7. Regained 12

1.8. The Future 13

Chapter 2. Rigorous Innitesimals 15 2.1. Developing Nonstandard Analysis 15

2.2. Direct Ultrapower Construction of ∗R 17

2.3. Principles of NSA 28

2.4. Working with Hyperreals 32

Chapter 3. Straightforward Analysis 37 3.1. Sequences and Their Limits 37

3.2. Series 44

3.3. Continuity 49

3.4. Dierentiation 54

3.5. Riemann Integration 58

Conclusion 66（容量限制，全文可查阅“无穷小微积分”网站）