当前，我们已经把“无穷小微积分基础”教学参考书用邮件投放到全国高校微积分任课老师手中，没有收到邮件的老师可以自行下载。
此书第一章明确了超实数系统R*需要使用一组“公理”系统来引入。该公理系统一共有五条，其中最后两条属于数理逻辑模型论范畴。由此，展开了整个现代微积分学理论体系。
反观我们国内，高校基础课微积分教课书几乎都是采用原苏联菲氏微积分的“模板”，即采用非公理化体系展开微积分学。
根据国际数学发展潮流，现代数学理论几乎都是采用公理化体系来处理的，很少有人固守于非公理化处理方式，以致数学与物理学混淆不清（对此，可参阅“面向21世纪”全国高校规划教材微积分）。
我们附上该书第一章的部分内容，请大家一阅。
袁萌 陈启清 9月1日
附：
THE HYPERREAL NUMBERS
We will assume that the reader is familiar with the real number system and develop a new object, called a hyperreal number system. The definition of the real numbers and the basic existence and uniqueness theorems are briefly outlined in Section 1F, near the end of this chapter.
That section also explains some useful notions from modern algebra, such as a ring（环）, a complete ordered field（完备有序域）， an ideal（数学理想）, and a homomorphism（同构）. If any of these terms are unfamiliar, you should read through Section 1F. We do not require any knowledge of modern algebra except for a modest vocabulary. In Sections 1A–1E we introduce axioms for the hyperreal numbers and obtain some first consequences of the axioms（公理）.
In the optional Section 1G at the end of this chapter we build a hyperreal number system as an ultrapower（超幂） of the real number system.
This proves that there exists a structure which satisfies the axioms. We conclude the chapter with the construction of Kanovei and Shelah [KS 2004] of a hyperreal number system which is definable in set theory. This shows that the hyperreal number system exists in the same sense that the real number system exists.
我们要注意这句话：“……the hyperreal number system exists in the same sense that the real number system exists.”
注：不涉及公理化，谈何现代微积分？