00后大学生：必须懂得实数系统的公理化，据此现微积分的公理化

    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 a real 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 laws

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 and b < 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 Product law If a < b and 0 < c, then ac < be. F Root axiom For every real number a > 0 and every positive integer n, there is a real number b > 0 such that b" = a.

Ill. COMPLETENESS AXIOM

Let A be a set of real numbers such that whenever x and y are

in A, any real number between x and y is in A. Then A is an interval.

袁萌  6月17日