康托尔（Georg Cantor，1845 - 1918）关于对角线论证的著名德语论文首次发表在1890年德国数学联盟（Deutsche Mathematiker-Vereinigung）的期刊上。

一，英文译文

https://web.archive.org/web/20060423090728/http://uk.geocities.com/frege%40btinternet.com/cantor/diagarg.htm

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, …, v, …, or, as I am used to saying, which do not have the power (Mächtigkeit) if the number series 1, 2, 3, …, v, ….  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 :

w₁ w₂, …, w_v­, …

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, … , xv, …)

which depend on infinitely many coordinates x1, x2, … , xv, …, 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, …, v, ….

This follows from the following proposition:

"If E1, E2, …, Ev, … 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 Ev. »

For proof, let there be

E1 = (a1.1, a1.2, … , a1,v, …)

E2 = (a2.1, a2.2, … , a2,v, …)

Eu = (au.1, au.2, … , au,v, …)

...

where the characters au,v are either m or w.  Then there is a series b1, b2, … bv,…, defined so that bv is also equal to m or w but is different from av,v.

Thus, if av,v = m, then bv = w.

Then consider the element

E0 = (b1, b2, b3, …)

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

E0 = Eu

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

bv = au,v

and so we would in particular have

bu = au,u

which through the definition of  bv 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, …, Ev, … 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.

二，中文译文

在题为“关于所有实数代数的集合的一个属性（On a property of a set of all real algebraic numbers）”（Journ. Math. Bd. 77, S. 258）中，可能第一次出现了命题的证明，即有一个无限的流形，或者如我习惯说的，数列1，2，3，...，v，....，的幂集，它不能与所有整数1，2，3，...，v，...形成一一对应的关系。 从第2节中证明的命题得出，例如，任意区间（a ... b）的所有实数的总体（Gesamtheit）不能被排列成数列：

w₁ w₂, …, w_v­, …

然而，这个命题有一个更简单的证明，它不依赖于考虑无理数。

也就是说，让m和n是两个不同的字符，并考虑一个元素的集合M

E = (x1, x2, … , xv, …)

这取决于无限多的坐标x1，x2，…，xv，...，其中每个坐标都是m或w。让M是所有元素E的总体。

在M的元素中，例如属于以下三个：

EI  = (m, m, m, m, … ),

EII = (w, w, w, w, … ),

EIII = (m, w, m, w, … ).

我现在指出，这样一个流形M不具有系列1，2，3，...，v的power，..

由以下命这题得出：

"如果E1，E2，...，Ev，...是流形M的任何简单无限元素系列，那么M中总是存在一个元素E0，它不能与任何元素Ev连接。"

为了证明这一点，让我们有

E1 = (a1.1, a1.2, … , a1,v, …)

E2 = (a2.1, a2.2, … , a2,v, …)

Eu = (au.1, au.2, … , au,v, …)

...

然后有一个系列b1, b2, ... bv, ...，定义为bv也等于m或w，但与av,v不同。

因此，如果av,v=m，那么bv=w。

然后考虑M的元素：

E0 = (b1, b2, b3, …)

那么我们可以直接看到，方程：

E0 = Eu

不能被任何正整数u所满足，否则对该u和所有v的值来说都是如此。

bv = au,v

我们将特别有：

bu = au,u

而根据bv的定义，这是不可能的。从这个命题立即可以看出，M的所有元素的整体不能被放入序列： E1, E2, ..., Ev, ...，否则我们就会产生矛盾，即一个事物E0既是M的一个元素，但也不是M的一个元素。