告菲氏微积分的徒子徒孙，无穷小不是空穴来风！

告菲氏微积分的徒子徒孙，无穷小不是空穴来风！

无穷小微积分不是空穴来风，讲科幻故事。在上世纪80年代兴起的微积分改革高潮中，基于公理化的无穷小微积分登上了历史舞台。

    说话要有凭据，要拿出真凭实据。著名的模型论专家J.Keisler在《Foundations forInfinitesimal Calculus》（无穷小微积分基础）的第15章的前言中给出了说法。

无穷小微积分的理论基础很深，需要使用数理逻辑模型论工具（超幂）来构造。在无穷小微积分基础的前言中，对此有所阐述。老翁希望（数理逻辑）圈外人士，对此不要说三道四，指手画脚。

在《无穷小微积分基础》15章前言中，J.Keisler明确表示：

Subject ofininitesimal analysis found in the research literature. To go beyondinfinitesimal calculus one should at least be familiar with some basic notionsfrom logic and model theory（模型论）. Chapter 15 introduces theconcept of a nonstandard universe, explains the use of mathematical logic,superstructures,and internal and external sets, uses ultrapowers（超幂） to build a nonstandarduniverse, and presents uniqueness（唯一性） theorems for thehyperreal number systems and nonstandard universes.

The simple set ofaxioms for the hyperreal number system given here (and in Elementary Calculus)make it possible to present infinitesimal calculus at the college freshmanlevel, avoiding concepts from mathematical logic. It is shown in Chapter 15that these axioms are equivalent to Robinson’s approach.

For additionalbackground in logic and model theory，the reader canconsult the book [CK 1990]. Section 4.4 of that book gives further results onnonstandard universes. Additional background in infinitesimal analysis can befound in the book [Goldblatt 1991].

I thank my latecolleague Jon Barwise, and Keith Stroyan of the University of Iowa, forvaluable advicein preparing the First Edition of this monograph. In the thirtyyears between the first and the present edition, I have ben geted from equallyvaluable and much appreciated advice from friends and colleagues too numerousto recount here.（全文完）

    祝大家中秋快乐！

袁萌  10月4日