康托尔论对角线法的论文 - “关于所有实数代数集合的一个属性”

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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, …

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.

w1 w2, …, w, …

E = (x1, x2, … , xv, …)

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, … ).

"如果E1E2...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。"

E1 = (a1.1, a1.2, … , a1,v, …)

E2 = (a2.1, a2.2, … , a2,v, …)

Eu = (au.1, au.2, … , au,v, …)

...

E0 = (b1, b2, b3, …)

E0 = Eu

bv = au,v

bu = au,u

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