王昆扬老师发来的材料:关于实数的构造

王昆扬老师发来的材料:关于实数的构造

我给王昆扬老师发《陶哲轩实分析》部分勘误，他来访问我的博客,看到我对实数的构造理论感兴趣,就给我发了一些他的09年写的两篇宣传材料以及他去年整理的关于实数的表示的稿子Cantor之路.（在一些研讨会上报告过）详细如下:Cantor之路, 实数的表示,实数.

然后我给他回了信.内容如下:

王老师,

谢谢您发来的材料.

当初我之所以看实数的构造理论是因为一种寻根探底的欲望,斐赫金哥尔茨写的《微积分学教程》的第一卷的最开始是用戴德金分割来介绍实数的，但是由于写的太过简略，廖廖几页，没让我明白。后来又看了我国几个学者编写的一本小册子《实数的构造理论》，看了大半也觉得不满意。直到看到了《陶哲轩实分析》这本神奇而罕见的书，从来没有一本分析书对基础阐述的这么透彻。

《陶哲轩实分析》在对待实数构造的问题上，是一贯采用非构造主义的观点的。他首先从皮亚诺公理开始不加定义地引进自然数（陶只是通过皮亚诺公理介绍了自然数满足的性质），然后引进形式符号a--b来定义整数，然后引进形式符号a//b定义有理数，然后引进形式符号LIM a_n 定义实数.和有的书不同，有的书是把实数看作一个等价类，定义为等价的有理柯西列，但是陶并没有这么做，他并没有明确的说“实数是什么”，而只是引进了一个对象“LIM a_n”,这个对象也是一个“Monster”,是个逻辑上的怪物,当初我为此困扰了很久，后来问陶，他对我进行了回复。详细如下

我问他，

Dear prof.Tao,

I finally finished learning the construction of the real number system with the help of this very rigorous book.But a big problem left:What is a real number?In your book you say a real number is $\hbox{LIM}(a_n)_{n=1}^{\infty}$ , $(a_n)_{n=1}^{\infty}$ is a Cauchy sequence of rational numbers.But I think $\hbox{LIM}(a_n)_{n=1}^{\infty}$ is nothing,it is just a notion which somewhat relates to $(a_n)_{n=1}^{\infty}$ .A few sections later,you prove that $\hbox{LIM}(a_n)_{n=1}^{\infty}$ is actually $\lim_{n\to\infty}a_n$ .But I think the definition of $\lim_{n\to\infty}a_n$ comes from $\hbox{LIM}(a_n)_{n=1}^{\infty}$ ,so one should not explain $\hbox{LIM}(a_n)_{n=1}^{\infty}$ by using $\lim_{n\to\infty}a_n$ .So what is a real number still remains a big problem for me……

I search google and find your google buzz essay https://profiles.google.com/114134834346472219368/buzz/RarPutThCJv .After seeing that essay I understand all……My only doubt is that why you didn’t define a real number as a equivalence class of Cauchy sequence of rational numbers in your text book.

That is only a step away…..It seems that you replace the equivalence class by the strange notion $\hbox{LIM}(a_n)_{n=1}^{\infty}$ .This makes me very uncomfortable because this notion seems meaningless,though it would eventually be replaced by $\lim_{n\to\infty}a_n$ ,but I think that is a “fake replace”……Because the concept of $\lim_{n\to\infty}a_n$ is based on $\hbox{LIM}(a_n)_{n=1}^{\infty}$ .So a step away makes your text book introduce me a new object $\hbox{LIM}(a_n)_{n=1}^{\infty}$ .I don’t like new objects.New objects,the fewer,the better.

陶回复:

Dear Luqing,

I think you are approaching mathematical foundations from a constructive viewpoint (focusing on what objects such as real numbers actually “are”) rather than from an axiomatic one (focusing on what properties these objects have). The distinction between the two perspectives is discussed in Remark 2.1.14 of my book. While the constructive viewpoint is initially more appealing conceptually, and is certainly of importance in foundations of mathematics, it turns out in the practice of mathematics that the axiomatic approach is much more flexible and powerful. In the end, once one leaves the foundational or logical aspects of mathematics, it doesn’t matter so much exactly which construction of the real numbers one takes as a model (whether it be equivalence classes of Cauchy sequences of rationals, formal limits of the same Cauchy sequences, Dedekind cuts of rationals, or whatever), so long as one can verify that this model obeys the basic axioms of the real numbers (e.g. the list in http://en.wikipedia.org/wiki/Real_number#Axiomatic_approach ). In particular, one is free to choose between a minimalist construction in a pure set theory in which one only works with constructions (such as tuples and equivalence classes) that were already constructed within the language of set theory, or a richer construction using an impure extension of set theory in which one adds additional formal symbols such as LIM. Both choices are equally valid for most mathematical purposes (the impure set theory with additional formal symbols is a conservative extension of pure set theory); I chose the latter for my book because it conceptually aligns the construction of the real numbers more closely with the way one usually thinks of real numbers in practice – namely, as quantities that can be approximated to arbitrary precision by rationals (or terminating decimals). (I also discuss the distinction between minimalist approaches to foundations, and rich approaches to foundations, at http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets/30231#30231 .)

Incidentally, I adopt a similar approach in previous sections of my text in constructing the integers out of formal differences $n --m$ of natural numbers, and rationals out of formal quotients $a // b$ of integers. One could, if one wished, instead adopt a “minimalist” philosophy and construct the integers as equivalence classes of pairs of natural numbers, and the rationals as equivalence classes of pairs of rationals; this is a logically equivalent approach, but is further removed from one’s conceptual intuition about what integers or rationals should actually be.

在陶眼里，那些一定要回答“实数是什么”这个问题的人,其实是采用了一种“构造主义”的观点，而陶是直接引进LIM a_n,并不是构造主义的观点，他注重的是实数满足的性质。同样，我觉得当初戴德金分割里的那个问题，

戴德金认为实数是和有理数分割在某种程度上相关连的一个东西，我们完全可以“从理智上接受它”，戴德金其实采用了和陶一样的观点，不是构造主义的（虽然他们采用了不同的方法，戴德金是采用戴德金分割，而陶是用柯西列）。而戴德金的那位朋友却一定要回答实数是什么这个问题，那位朋友指出“实数就是那个分割”，可见，那位朋友是采用了“构造主义者”的观点。还有比如，对待自然数，构造主义者们一定要阐明自然数是什么，于是他们说自然数其实就是$\emptyset$,$\{\emptyset\}$....把自然数的基础建立在集合论上，而对于非构造主义者来说，自然数是什么并不重要，重要的是有什么性质，于是非构造主义者们只注重性质(皮亚诺公理)，而只用一个形式符号来表明那个讨论对象。

这其实是不同的哲学，对待这类基础问题上，我觉得谁是谁非并不重要，因为根本就没有是非。

还有，关于用十进制表示实数，我个人还是倾向于更一般化的，可能有些人更倾向于用具体的小数，而且小数与中学生的联系也更紧密，我觉得这也是不同的倾向吧，两者间理论上基本上没有差别的。

谢谢。

我给王老师的回复里说出了自己对实数构造的感悟,放在这里希望对读者有帮助,也希望读者提出批评建议.
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2007.9.17
原文地址：https://www.cnblogs.com/yeluqing/p/3827809.html