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A Translation of G. Cantor’s “Ueber eine elementare Frage der Mannigfaltigkeitslehre”

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A Translation of G. Cantor’s “Ueber eine elementare Frage der Mannigfaltigkeitslehre” which contains the diagonal argument.
A Translation of G. Cantor’s “Ueber eine elementare Frage der
Mannigfaltigkeitslehre”.
Google TranslateTM,1DeepLTM,2and Peter P. Jones
1https: // translate. google. com
2https: // www. deepl. com
(Dated: August 23, 2019)
An English translation of G. Cantor’s “Ueber eine elementare Frage der Mannigfaltigkeitslehre”[1]
article: “On an elementary question of the theory of manifolds.”
Translation Note: We have translated “Inbegriff” as collection, and “M¨achtigkeit” as power. Apart from these
adjustments and a few other specific edits the bulk of this English language text was obtained directly from the
machine translators acknowledged as the main authors.
I.
In the essay titled ”On a Property of the Collection of All Real Algebraic Numbers” (Journ. Math. Vol. 77,
p.258), it is probable that for the first time there is proof for the proposition that there are infinite manifolds which
do not relate to each other uniquely to the set of all finite integers 1,2,3, . . . , ν, . . . or, as I say, that do not have the
power of the 1,2,3, . . . , ν, . . . series.
As shown there in Sec.2, it follows without further ado that, for example, the totality of all real numbers of any
interval (a, b) can not be imagined in the series form:
ω1, ω2, . . . , ων, . . .
But it is possible to provide a much simpler proof of that theorem, which is independent of the consideration of
irrational numbers.
If mand ware any two mutually exclusive characters, then we consider a collection Mof elements, E=
(x1, x2, . . . , xν, . . .), which depend on an infinite number of coordinates, x1, x2, . . . , xν, . . ., where each of these coor-
dinates is either mor w.
Mis the totality of all elements E.
The elements of Minclude, for example, the following three:
EI= (m, m, m, m, . . .),
EII = (w, w, w, w, . . .),
EII I = (m, w, m, w, . . .).
I now claim that such a manifold Mdoes not have the power of the series 1,2,3, . . . , ν, . . ..
This follows from the following sentence:
If E1, E2, . . . , Eν, . . . are any simply infinite series of elements of the manifold M, then there is always
an element E0of Mthat does not agree with any Eν.
senoj.p.p@gmail.com
2
To prove it:
E1= (a1,1, a1,2, . . . , a1, . . .),
E2= (a2,1, a2,2, . . . , a2, . . .),
. . .
Eµ= (aµ,1, aµ,2, . . . , aµ,ν , . . .).
. . .
Here the aµ,ν are in a certain way mor w. Let us now define a series b1, b2, . . . , bν, . . . such that bνis also only
equal to mor wand different from aν,ν.
So if aν,ν =m, then bν=w, and if aν,ν =w, then bν=m.
If we then consider the element:
E0= (b1, b2, b3, ...)
of M, we can easily see that the equation:
E0=Eµ
for no positive integer value of µ, can be satisfied, otherwise for the given µand for all integer values of ν:
bν=aµ,ν ,
so also in particular,
bµ=aµ,µ,
which would be excluded by the definition of bν.
From this theorem follows immediately that the totality of all elements of Mcan not be put into the series form:
E1, E2, . . . , Eν, . . ., otherwise we would be faced with the contradiction that a thing E0is both an element of Mas
well as not being element of M.
This proof is noteworthy not only for its great simplicity, but also for the reason that the principle followed therein
can be readily extended to the general proposition that the powers of well-defined manifolds have no maximum or,
what is the same, that for any given manifold Lanother Mcan be put on the side, which is of greater power than L.
For example, let Lbe a linear continuum, for example, the collection of all real number numbers z, which are
>0 and 61.
Under Mwe understand the collection of all unique functions f(x) which only take the two values 0 or 1, while
xpasses through all real values which are >0 and 61.
The fact that Mhas no smaller power than L, follows from the fact that subsets of Mcan be specified, which
have the same power as L, for example, the subset consisting of all the functions of xthat have the value 1 for a
single value x0of x, and 0 for all other values of x.
But also Mdoes not have the same power with L, because otherwise the manifold Mcould be brought into a
mutually unambiguous relation to the variable z, and Mcould be thought of in the form of an unambiguous function
of the two variables xand z:φ(x, z), so that through each specialization of zan element f(x) = φ(x, z) of Mis
obtained and also vice versa each element f(x) of Memerges from φ(x, z) through a single particular specialization
of z. However, this leads to a contradiction.
3
For if we understand g(x) to be that unique function of xwhich takes only the values 0 or 1 and is different for
every value of xof φ(x, x), then on the one hand g(x) is an element of M, on the other hand, g(x) can not result from
φ(x, z) by any specialization z=z0because φ(z0, z0) is different from g(z0).
If, therefore, the power of Mis neither smaller nor equal to that of L, it follows that it is greater than the power
of L. (Cf. Crelle’s Journal, vol. 84, p. 242).
Already in the “Foundations of a General Theory of Manifolds” (Leipzig, 1883, Math. Annalen, Vol. 21), I have
shown by very different means that the powers have no maximum; there it was even proved that the collection of all
powers, if we think of the latter in order of size, forms a “well-ordered crowd”, so that in nature there is one next
greater in every power, but also a next greater one follows every infinitely increasing set of powers.
The “powers” represent the only and necessary generalization of the finite “cardinal numbers”, they are nothing
else than the actual infinite-sized cardinal numbers, and they have the same reality and certainty as those; only that
the lawful relations among them, the “number theory” related to them is partly different, than in the area of the
finite.
The further development of this field is the task of the future.
[1] Georg Cantor. Ueber eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-
Vereinigung, 1:75–78, 1891.
... We will not dwell on the mixture of human weaknesses and historical contingencies that can make this process of improvement somewhat inefficient. In the following, we present a set of arguments exposing key flaws in the construction commonly known as Cantor's Diagonal Argument (CDA) found in his paper [3] [7]. CDA is indeed beguiling, and its persistence for so long within an orthodox canon of pure mathematics is excusable to that degree. ...
... We assume that the reader is familiar with CDA [3] (if not from the original German text then from a translation [7] or equivalents in recent texts, e.g. Theorem 2.11, in [15], or the version in [1]). ...
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We examine Cantor’s Diagonal Argument (CDA). If the same basic assumptions and theorems found in many accounts of set theory are applied with a standard combinatorial formula a contradiction is obtained concerning the Powerset Theorem. In consequence, one of either 20>0 2^{\aleph_0} > \aleph_0 or the idea that 0 \aleph_0 has the properties of a full infinity must be abandoned. In both cases, under the respective entailed modifications of the model of the number system, Cantor’s Diagonal Argument (CDA) can no longer support the idea of higher infinities. Hence, concerning CDA, claims for higher infinite cardinality cannot be used to justify claims of the uncountability of the real numbers, nor vice versa. However, it is also found that CDA has a structure that is inadequately justified and does not count properly, which further undermines its claim to support the uncountability of the real numbers in general. Therefore, we recommend that CDA should no longer be used in mathematics. On the basis of a different argument it remains possible to independently assume that the property of uncountability only refers to an order type of certain infinite sets under specific pairwise relations.