ガロア第一論文と乗数イデアル他関連資料スレ12

[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12 (1002ﾚｽ)
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310: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/01/15(水) 18:43:40.92 ID:ZCTGHyhi

>>309
それ、下記のWell-ordering theorem
”The well-ordering theorem follows from the axiom of choice as follows.[9]”
とほぼ同じでしょ？

おれが、すでに どこかにアップしてあるよ

https://en.wikipedia.org/wiki/Well-ordering_theorem
Well-ordering theorem

Proof from axiom of choice
The well-ordering theorem follows from the axiom of choice as follows.[9]

Let the set we are trying to well-order be
A, and let
f be a choice function for the family of non-empty subsets of
A. For every ordinal
α, define an element
aα that is in
A by setting
aα= f(A∖{aξ∣ξ<α})
if this complement
A∖{aξ∣ξ<α}
is nonempty, or leave
aα undefined if it is. That is,
aα is chosen from the set of elements of
A that have not yet been assigned a place in the ordering (or undefined if the entirety of
A has been successfully enumerated). Then the order
< on
A defined by
aα<aβ
if and only if
α<β
(in the usual well-order of the ordinals) is a well-order of
A as desired, of order type
sup{α∣aα is defined}.

http://rio2016.5ch.net/test/read.cgi/math/1735693028/310

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