ガロア第一論文と乗数イデアル他関連資料スレ16

[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ16 (1002ﾚｽ)
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510: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/05/01(木) 16:06:21.69 ID:D1rwPzBB

>>503 >>506
>僕はキモコテハン君が実数の連続性（完備性）について何も述べてないから
>そこの正確な述べ方が身についていないのかもしれないと判断したけど、

巡回ご苦労様です
キモコテハン君です

「正確な述べ方が身についていない」は、全く正しいが （＾＾
検索すれば、すぐ見つかることを書いていると、ここに転写する必要もないし ヒマもないしｗ

おっと 院試を受ける人は、基本的な事項で 口頭試問で ツッコミありそうなことは、正確な述べ方を身につけるべし！

さて 例えば Construction of the real numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
Explicit constructions of models
抜粋
Construction from Cauchy sequences
A standard procedure to force all Cauchy sequences in a metric space to converge is adding new points to the metric space in a process called completion.
R is defined as the completion of the set Q of the rational numbers with respect to the metric |x − y| Normally, metrics are defined with real numbers as values, but this does not make the construction/definition circular, since all numbers that are implied (even implicitly) are rational numbers.[5]
Let R be the set of Cauchy sequences of rational numbers. That is, sequences
(x1, x2, x3,...)
of rational numbers such that for every rational ε > 0, there exists an integer N such that for all natural numbers m, n > N, one has |xm − xn| < ε. Here the vertical bars denote the absolute value.
略す
there exists an integer N such that for all natural numbers n > N, one has |xn − yn| < ε.
略す
The usual decimal notation can be translated to Cauchy sequences in a natural way. For example, the notation π = 3.1415... means that π is the equivalence class of the Cauchy sequence (3, 3.1, 3.14, 3.141, 3.1415, ...). The equation 0.999... = 1 states that the sequences (0, 0.9, 0.99, 0.999,...) and (1, 1, 1, 1,...) are equivalent, i.e., their difference converges to 0.

An advantage of constructing
R as the completion of Q is that this construction can be used for every other metric space.
(引用終り)

>Πが存在しなければ作れないよ

作れるよ
上記の英文の通りで ”π = 3.1415...”は あくまで ”For example”です

http://rio2016.5ch.net/test/read.cgi/math/1744899342/510

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