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关于超实数的寿命

(2018-09-15 15:32:17)

关于超实数的寿命

今年912,袁萌一行三人在向国家教育部高教司汇报工作时,袁萌指出:超实数有条件诞生在19世纪20年代柯西定义实数为柯西序列等价类的同一时代。然而,历史对人类开了一个大玩笑。直到20世纪60年代,在法国布尔巴基学派关于集合滤器、超幂引产婆助力之下,美国数学家鲁宾逊在1960年接生了超实数这个晚生儿。

进入本世纪,世界范围内,超实数研究方兴未艾”,来日方长。

近年来,ISAAC  GOLDBRING教授撰写了简明易懂的超实数科普文章,内容丰富,值得一读。

当前,无穷小微积分手机版投放全国高校行动已经进行三轮,少数师生对超实数也许有所耳闻,但是,对于什么滤器,什么是超幂却不甚明了。

    阅读本文附件,共计16,110,篇幅不大,即可释疑解惑。超实数的寿命究竟有多长?伴随00后宝贝一块儿成长。学无止境也。

袁萌  陈启清   915

说明:由于此文较长已经在CSDN袁萌专栏发表(限于新浪袁萌专栏容量

附:LECTURE NOTES ON NONSTANDARD ANALYSIS UCLA SUMMER SCHOOL IN LOGIC

ISAAC GOLDBRING20141110日发表)

Contents (内容目录)

1. The hyperreals 3

1.1. Basic facts about the ordered real eld 3

1.2. The nonstandard extension 4

1.3. Arithmetic in the hyperreals 5

1.4. The structure of N 7 1.5. More practice with transfer 8

1.6. Problems 9

2. Logical formalisms for nonstandard extensions 10 2.1. Approach 1: The compactness theorem 11 2.2. Approach 2: The ultrapower construction 12 2.3. Problems 16

3. Sequences and series 17 3.1. First results about sequences 17

3.2. Cluster points 19

3.3. Series 21

3.4. Problems 22

4. Continuity 23

4.1. First results about continuity 23

4.2. Uniform continuity 25 4.3. Sequences of functions 27

4.4. Problems 30

5. Dierentiation 33

5.1. The derivative 33

5.2. Continuous dierentiability 35

5.3. Problems 36

6. Riemann Integration 38 6.1. Hypernite Riemann sums and integrability 38 6.2. The Peano Existence Theorem 41

6.3. Problems 43

7. Weekend Problem Set #1 44

8. Many-sorted and Higher-Type Structures 47 8.1. Many-sorted structures 47

Date: November 10, 2014.

1

2 ISAAC GOLDBRING

8.2. Higher-type sorts 48

8.3. Saturation 51

8.4. Useful nonstandard principles 53

8.5. Recap: the nonstandard setting 54

8.6. Problems 54

9. Metric Space Topology 55 9.1. Open and closed sets, compactness, completeness 55

9.2. More about continuity 63

9.3. Compact maps 64

9.4. Problems 65

10. Banach Spaces 67

10.1. Normed spaces 67 10.2. Bounded linear maps 68

10.3. Finite-dimensional spaces and compact linear maps 69

10.4. Problems 71

11. Hilbert Spaces 73

11.1. Inner product spaces 73

11.2. Orthonormal bases and `2 75

11.3. Orthogonal projections 79

11.4. Hypernite-dimensional subspaces 82

11.5. Problems 83

12. Weekend Problem Set #2 85

13. The Spectral Theorem for compact hermitian operators 88

13.1. Problems 93

14. The Bernstein-Robinson Theorem 94

15. Measure Theory 101 15.1. General measure theory 101

15.2. Loeb measure 102 15.3. Product measure 103 15.4. Integration 104

15.5. Conditional expectation 104

15.6. Problems 105

16. Szemer´edi Regularity Lemma 106

16.1. Problems 108 References 110

Nonstandard analysis was invented by Abraham Robinson in the 1960s as a way to rescue the na¨ve use of innitesimal and innite elements favored by 

mathematicians such as Leibniz and Euler(欧拉) before the advent of the rigorous methods introduced by Cauchy and Weierstrauss. Indeed, Robinson realized that the compactness theorem of rst-order logic could be used to provide elds that “logically behaved” like the ordered real eld while containing “ideal” elements such as innitesimal and innite elements.

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