# [转载]Zmn-0758 [转载] 薛问天：易129-读康托尔对角线法的原文

Zmn-0758  [转载] 薛问天：易129-读康托尔对角线法的原文

【编者按。最近Thebeater先生的文章提到康托尔对角线法的原文。刚好，我们早期的易网《评论园地》上登过薛问天先生的一篇文章。现在转载发布如下，供网友们共享。请大家关注并积极评论。另外本《专栏》重申，这里纯属学术讨论，所有发布的各种意见仅代表作者本人，不代表本《专栏》编辑部的意见。】

### 转载《易129-薛问天：读康托尔对角线法的原文》

2014-05-17 22:40:06|  分类：

【文清慧注：下面是薛问天先生投给评论园地的文章】

读康托尔对角线法的原文

G. Cantor: Uber eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Math. Vereinigung I (1890-91) 75 - 78.

ω1, ω2, ..., ων, ...

E = ( x1, x2, xν, ... ),

EI = ( m, m, m, m,... ),
EII = ( w, w, w, w, ... ),
EIII = ( m, w, m, w, ... ).

“如果E1, E2, …, Eν , …是集合M中元素的任何一个无穷序列，那么总存在M的一个元素E0，它不可能是序列中的任何一个元素Eν

E1 = ( a1,1, a1,2, ..., a1,ν, ... )
E2 = ( a2,1, a2,2, ..., a2,ν, ... )
…………
Eμ = ( aμ,1, aμ,2, ..., aμ,ν, ... )
…………

E0 = ( b1, b2, b3, ... ),

[后半部分被英文译稿省略。]

In the paper entitled "On a property of a set [Inbegriff] of all real algebraic numbers" (Journ. Math. Bd. 77, S. 258), there appeared, probably for the first time, a proof of the proposition  that there is an infinite manifold, which cannot be put into a one-one correlation with the totality [Gesamtheit] of all finite whole numbers 1, 2, 3, …, ν, …, or, as I am used to saying, which do not have the power (M?chtigkeit) if the number series 1, 2, 3, …, ν, ….  From the proposition proved in § 2 there follows another, that e.g. the totality (Gesamtheit) of all real numbers of an arbitrary interval (a ... b) cannot be arranged in the series

ω1, ω2, ..., ων, ...

However, there is a proof of this proposition that is much simpler, and which does not depend on considering the irrational numbers.

Namely, let m and n be two different characters, and consider a set [Inbegriff] M of elements

E = ( x1, x2, xν, ... ),

which depend on infinitely many coordinates x1, x2, … , xν, …, and where each of the coordinates is either m or w.  Let M be the totality [Gesamtheit] of all elements E.

To the elements of M belong e.g. the following three:

EI = ( m, m, m, m,... ),
EII = ( w, w, w, w, ... ),
EIII = ( m, w, m, w, ... ).

I maintain now that such a manifold [Mannigfaltigkeit] M does not have the power of the series 1, 2, 3, …, ν, ….

This follows from the following proposition:

"If E1, E2, …, Eν, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E0- of M, which cannot be connected with any element Eν."

For proof, let there be

E1 = ( a1,1, a1,2, ..., a1,ν, ... )
E2 = ( a2,1, a2,2, ..., a2,ν, ... )
…………
Eμ = ( aμ,1, aμ,2, ..., aμ,ν, ... )
…………

where the characters aμ,ν are either m or w.  Then there is a series b1, b2, … bν,…, defined so that bν is also equal to m or w but is different from aν,ν.

ν

Thus, if aν,ν = m, then bν = w.

Then consider the element

E0 = (b1, b2, b3, …)

of M, then one sees straight away, that the equation

E0 = Eμ

cannot be satisfied by any positive integer μ, otherwise for that μ and for all values of ν.

bν = aμ,ν ,

and so we would in particular have

bμ = aμ,μ ,

which through the definition of  bν is impossible.

From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E1, E2, …, Eν, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M.

[Second half omitted.  Cantor proves the more general theorem that the power of well-defined manifolds have no maximum].

https://blog.sciencenet.cn/blog-755313-1315261.html

## 全部精选博文导读

GMT+8, 2022-8-13 01:57