[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)
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618(2): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/07(土)08:43 ID:H2e5WMAT(2/14) AAS
>>617
つづき

For once 0 is a common element of all elements Z1 of T, and on the other hand, if a is a common element of all these Z1, then also {a} is common to all and therefore also an element of Z0.
If Z 'is any other quantity of the nature required in the axiom, then in the same way as Z0 it corresponds to Z for a smallest subset Z0' of the property under consideration.
Now, however, the average [Z0, Z0 '], which is a common subset of Z and Z', must have the same properties as Z and Z and, as a subset of Z, the constituent Z0 and, as a subset of Z ', the constituent Z0 ' contain.
After I it follows that [Z0, Z0 '] = Z0 = Z0', and that Z0 is therefore the common component of all possible quantities, such as Z, although these do not need to form the elements of a set.
The set Z0 contains the elements 0, {0}, {{0}}, and so on, and may be called a "series of numbers" because their elements can represent the location of the numerals.
省3

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619(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/07(土)08:44 ID:H2e5WMAT(3/14) AAS
>>618
つづき

(ドイツ語原文)
P263
Axiom I. Ist jedes Element einer Menge M gleichzeitig Element von N und umgekehrt, ist also gleichzeitig M =E N und N =E M, so ist immer M = N. Oder kurzer: jede Menge ist durch ihre Elemente bestimmt.

P266
Um aber die Existenz "unendlicher" Mengen zu sichern, bedurfen wir noch des folgenden, seinem wesentlichen Inhalte von Herrn R. Dedekind**) herruhrenden Axiomes.
省6

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621(2): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/07(土)08:49 ID:H2e5WMAT(5/14) AAS
>>618 補足

(引用開始)
The set Z0 contains the elements 0, {0}, {{0}}, and so on, and may be called a "series of numbers" because their elements can represent the location of the numerals.
It is the simplest example of a "countless infinite" set (Nos. 36).
注：36節(Nos. 36 P280)で、ZERMELOは無限（"unendliche"）について論じている。
(引用終り)

ってことね
省4

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