ガロア第一論文と乗数イデアル他関連資料スレ13

[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13 (1002ﾚｽ)
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367: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/02/08(土) 11:19:17.08 ID:23ITt7NX

>>360
>>順序数全体の集まりは集合でない。
>順序数全体のクラスOを集合と仮定する。
>このときOも順序数だからO∈O。正則性公理に反するから仮定は偽、すなわちOは集合でない。

アホなおサルと>>7-10、 10分議論をする暇があったら
下記のen.wikipedia Ordinal number を、3分黙読する方が、よほど有益だわｗ ；ｐ）
（日wikipediaには、順序数のクラスの記述はないけどね （＾＾）

(参考)
https://ja.wikipedia.org/wiki/%E9%A0%86%E5%BA%8F%E6%95%B0
順序数（じゅんじょすう、英: ordinal number）とは、整列集合同士の“長さ”を比較するために、自然数[1]を拡張させた概念である。

https://en.wikipedia.org/wiki/Ordinal_number
Ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets.[1]

Definitions
Well-ordered sets
Essentially, an ordinal is intended to be defined as an isomorphism class of well-ordered sets: that is, as an equivalence class for the equivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be the order type of any set in the class.

Definition of an ordinal as an equivalence class
The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).

Von Neumann definition of ordinals
See also: Set-theoretic definition of natural numbers and Zermelo ordinals
Rather than defining an ordinal as an equivalence class of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.

http://rio2016.5ch.net/test/read.cgi/math/1738367013/367

609: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/02/10(月) 20:14:10.08 ID:fq1QO0q/

>>551-553
おっちゃん、ご苦労さまです
下記 e (mathematical constant) 、皆さんの参考に貼ります ；ｐ）

(参考)
ja.wikipedia.org/wiki/%E3%83%8D%E3%82%A4%E3%83%94%E3%82%A2%E6%95%B0
ネイピア数（ネイピアすう、英: Napier's constant）は、数学定数の一つであり、自然対数の底である。ネーピア数、ネピア数とも表記する。記号として通常は e が用いられる。

en.wikipedia.org/wiki/E_(mathematical_constant)
e (mathematical constant)
Properties
Number theory
The real number e is irrational. Euler proved this by showing that its simple continued fraction expansion does not terminate.[38] (See also Fourier's proof that e is irrational.)

Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any non-zero polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873.[39] The number e is one of only a few transcendental numbers for which the exact irrationality exponent is known (given by
μ(e)=2.[40]

An unsolved problem thus far is the question of whether or not the numbers e and π are algebraically independent. This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.[41][42]

It is conjectured that e is normal, meaning that when e is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).[43]

In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. The constant π is a period, but it is conjectured that e is not.[44]

（google訳）
実数 e は無理数です。オイラーは、単純な連分数展開が終了しないことを示してこれを証明した。[38] (e が無理数であるというフーリエの証明も参照してください。)

さらに、リンデマン・ワイエルシュトラスの定理によれば、e は超越数であり、有理係数を持つ非ゼロ多項式方程式の解ではないことを意味します。これは、特にこの目的のために構築されることなく超越数であることが証明された最初の数でした（リウヴィル数と比較してください）。この証明は1873年にシャルル・エルミートによってなされた。[39] eは、正確な無理数指数が知られている数少ない超越数のうちの1つです（
μ(e)=2.[40]

つづく

http://rio2016.5ch.net/test/read.cgi/math/1738367013/609

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