今年8月31日，我们开始向全国高校投放J.Keisler为担任微积分手机版课程的数学教员撰写的教学参考书（也是手机版）。
该参考书的第一章，共计6个小节，由浅入深，集中讲解了超实数从何而来的问题。
涉及超实数的公理组共计有6条，由此公理组系统展开整个微积分理论体系，完全与物理实践无关。
为培养00后新一代大学生，深入学习、研讨数学基础问题，大家共勉之。
袁萌 陈启清 9月2日
附：超实数公理组
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, a complete ordered field, an ideal, and a homomorphism. 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. In the optional Section 1G at the end of this chapter we build a hyperreal number system as an ultrapower 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.
1A. Structure of the Hyperreal Numbers (§1.4, §1.5)
In this and the next section we assume only Axioms A, B, and C below.
Axiom A R is a complete ordered field.
Axiom B R∗ is an ordered field extension of R.
Axiom C R∗ has a positive infinitesimal, that is, an element ε such that 0 < ε and ε < r for every positive r ∈R. In the next section
we will introduce two powerful additional axioms which are needed for our treatment of the calculus. However, the algebraic facts
2 1. The Hyperreal Numbers
about infinitesimals which underlie the intuitive picture of the hyperreal line follow from Axioms A–C alone. We call R the field of real numbers and R∗ the field of hyperreal numbers. Definition 1.1. An element x ∈R∗ is infinitesimal if |x| < r for all positive real r; finite if |x| < r for some real r; infinite if |x| > r for all real r. Two elements x,y ∈ R∗ are said to be infinitely close, x ≈ y, if x−y is infinitesimal. (Thus x is infinitesimal if and only if x ≈ 0). Notice that a positive infinitesimal is hyperreal but not real, and that the only real infinitesimal is 0. Definition 1.2. Given a hyperreal number x ∈R∗, the monad of x is theset monad(x) = {y ∈R∗: x ≈ y}. The galaxy of x is the set galaxy(x) = {y ∈R∗: x−y is finite}. Thus monad(0) is the set of infinitesimals and galaxy(0) is the set of finite hyperreal numbers. In Elementary Calculus, the pictorial device of an infinitesimal microscope is used to illustrate part of a monad, and an infinite telescope is used to illustrate part of an infinite galaxy. Figure 1 shows how the hyperreal line is drawn. In Section 2B we will give a rigorous treatment of infinitesimal microscopes and telescopes so the instructor can use them in new situations.
1A. Structure of the Hyperreal Numbers（以下省略）