现在，00后大学生都是幸运儿，用指尖一点，问题的答案就会从互联网云端飘然而至。这是Keisler教授的首创：
实数系统的公理架构：
I. ALGEBRAIC AXIOMS FOR THE REAL NUMBERS A Closure laws 0 and 1 are real numbers. If a and b are real numbers,then so are a + b, ab, and -a. If a is a real number and a # 0, then 1/a is areal number. B Commutative laws a + b = b + a ab = ba. C Associative Jaws a +(b + c) = (a + b) + c a(bc) = (ab)c.
0 Identity Jaws
E Inverse laws
F Distributive law
DEFINITION
O+a=a
a+(-a)=O
1·a =a.
If a # 0, a ·- = 1. a a ? (b + c) = ab + ac.
The positive integers are the real numbers 1, 2 = 1 + 1, 3 = 1 + 1 + 1,4 = 1 + 1 + 1 + 1, and so on.
II. ORDER AXIOMS FOR THE REAL NUMBERS
A 0 < 1. B Transitive law If a < b andb < c then a < c. C Trichotomy law Exactly one of the relations a < b,a = b, b < a, holds. 0 Sum law If a < b, then a + c < b + c. E Productlaw If a < b and 0 < c, then ac < be. F Root axiom For every realnumber a > 0 and every positive integer n, there is a real number b > 0such that b" = a.
Ill. COMPLETENESS AXIOM
Let A be a set of real numbers such that whenever x and y are in A, anyreal number between x and y is in A. Then A is an interval.
THEOREM
An increasing sequence (S11 ) either converges or diverges to cf).
PROOF Let T be the set of all real numbers xsuch that x ::;: S" for some n. T is obviously nonempty.
Case 1 Tis the whole real line. If Hisinfinite we have x S:: SH for all real numbers x. So SH is positive infiniteand (S11 ) diverges to oo.
Case 2 T is not the whole real line. By theCompleteness Axiom, T is an interval (-w, b] or (-w, b). For each real x <b, we have
for some 11. It follows that for infinite H,SH s band SH ~ b. Therefore (S,) converges to b.
We now take up the second group of axioms,which give the properties of the hyperreal numbers. There will be two axioms,called the Extension Axiom and the Transfer Axiom, which correspond to theExtension Principle and Transfer Principle of Section 1.5. We first state theExtension Axiom.
1*. EXTENSION AXIOM
(a) The set R ofrealnumbers is a subset ofthe set R* ofhyperrea/numbers. (b) There is given a relation < * on R*, suchthat the order relation < on R is a subset of<*,<* is transitive(a< *band b <* c implies a<* c), and < * satisfies the TrichotomyLaw: for all a, b in R *, exactly one of a < * b, a = b, b <*a holds. (c)There is a hype1'1'ea/ number 8 such that 0 < * 8 and 8 < * r for eachpositive rea/number r.
(d) For each real function f, there is givena hyperreal function f* with the same number of variables, called the naturalextension off
Part (c) of the Extension Axiom states thatthere is at least one positive infinitesimal. Part (d) gives us the naturalextension for each real function. The Transfer Axiom will say that this naturalextension has the same properties as the original function. Recall that theTransfer Principle of Section 1.5 made use of the intuitive idea of a realstatement. Before we can state the Transfer Axiom, we must give an exactmathematical explanation of the notion of a real statement. This will be donein several steps, first introducing the concepts of a real expression and aformula. We begin with the concept of a real expression, or term, built up fromvariables and real constants using real functions. Real expressions can bebuilt up as follows:
(1) A real constant standing alone is a realexpression.
(2) A variable standing alone is a realexpression. (3) If e is a real expression, and f is a real function of onevariable, then f(e) is a real expression. Similarly, if e1, ... , e" arereal expressions, and g is a real function of n variables, then g(e1, ?.. ,e") is a real expression.
Step (3) can be used repeatedly to build uplonger expressions. Here are some examples of real expressions, where x and yare variables:
2, X+ y, lx- 41, g(x,f(O)), 1/0.
By a formula, we mean a statement of one ofthe following kinds, where d and e are real expressions:
(1) An equation between two real expressions,d = e. (2) An inequality between two real expressions, d < e, d s e, d >e, d :::::: e, or d =!= e.
EPILOGUE 907
(3) A statement of the form "e isdefined" or "e is undefined."
Here are some examples of formulas: X+ y = 5,1 - x2 f(x)=~,
g(x, y) < f(t)
袁萌 6月17日