关于超实数的寿命
(2018-09-15 15:32:17)关于超实数的寿命
今年9月12日,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
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。
袁萌
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 GOLDBRING(2014年11月10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.
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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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