超实数从何而来?
(2018-09-02 15:46:25)超实数从何而来?
袁萌
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 denition of the real numbers and the basic existence and uniqueness theorems are briey 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 eld, 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 rst 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 satises the axioms. We conclude the chapter with the construction of Kanovei and Shelah [KS 2004] of a hyperreal number system which is denable 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 eld.
Axiom B R∗ is an ordered eld extension of R.
Axiom C
R∗ has a positive
innitesimal, 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 innitesimals which underlie the intuitive picture of the hyperreal line follow from Axioms A–C alone. We call R the eld of real numbers and R∗ the eld of hyperreal numbers. Definition 1.1. An element x ∈R∗ is innitesimal if |x| < r for all positive real r; nite if |x| < r for some real r; innite if |x| > r for all real r. Two elements x,y ∈ R∗ are said to be innitely close, x ≈ y, if x−y is innitesimal. (Thus x is innitesimal if and only if x ≈ 0). Notice that a positive innitesimal is hyperreal but not real, and that the only real innitesimal 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 nite}. Thus monad(0) is the set of innitesimals and galaxy(0) is the set of nite hyperreal numbers. In Elementary Calculus, the pictorial device of an innitesimal microscope is used to illustrate part of a monad, and an innite telescope is used to illustrate part of an innite galaxy. Figure 1 shows how the hyperreal line is drawn. In Section 2B we will give a rigorous treatment of innitesimal microscopes and telescopes so the instructor can use them in new situations.
1A. Structure of the Hyperreal Numbers(以下省略)