純粋・応用数学（含むガロア理論）8

[過去ﾛｸﾞ] 純粋・応用数学（含むガロア理論）8 (381ﾚｽ)
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128(3): 現代数学の系譜雑談 ◆yH25M02vWFhP 2021/05/18(火)07:18 ID:4SccZpT/(1/4) AAS
>>124 補足
(引用開始)
https://en.wikipedia.org/wiki/Well-order
Well-order
Examples and counterexamples
Reals
The standard ordering ≦ of any real interval is not a well ordering, since, for example, the open interval (0, 1) ⊆ [0,1] does not contain a least element.
From the ZFC axioms of set theory (including the axiom of choice) one can show that there is a well order of the reals.
Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals.[1]
However it is consistent with ZFC that a definable well ordering of the reals exists-for example, it is consistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.
省14

129(1): 現代数学の系譜雑談 ◆yH25M02vWFhP 2021/05/18(火)07:18 ID:4SccZpT/(2/4) AAS
>>128
つづき

P10
Thus, from 4.9 (ii) we obtain: it is consistent with Z-F, AC and GCH,that there is no set-theoretically definable well-ordering of the continuum.
(That is, it is consistent to adjoin to these axioms the statement, foreach formula ? of L with two free variables, which expresses that Fdoes not determine a well-ordering of the continuum.)
This bears onquestions dealt with by Myhill and Scott [13] (1).
(*). The following result of Scott, which will appear in that paper, is of specialinterest in this connection: it is provable in Z-F that there is a definable well-orderingA with field I a subset of the continuum, such that any other well-ordering 4, of thissort has field TiCr.
A simple explicit definition of this A can be given.
(引用終り)
以上

134(1): 現代数学の系譜雑談 ◆yH25M02vWFhP 2021/05/18(火)08:46 ID:4SccZpT/(4/4) AAS
>>132
ははは
　>>128-129
をどぞ
何のための　>>128-129だと思ったの
　>>128-129を嫁め！ｗ

161(4): 現代数学の系譜雑談 ◆yH25M02vWFhP 2021/05/19(水)07:54 ID:H7LP/xSH(2/15) AAS
>>128 補足
”選択公理⇔整列可能定理”について

（下記が分かり易いね）
http://paiotunoowari.hatenadiary.jp/entry/2015/12/03/133610
整列可能定理 2015-12-03 ぱいおつ日記
ひかるさんのアドベントカレンダー企画の3日目の記事です.
(抜粋)
Nは普通の大小関係で整列集合になってます.
Zとかも普通の大小関係は整列順序じゃないけど0<?1<1<?2<2<...<?n<n<...と並べ直したら整列集合になってます.
こんなふうに,ある順序で整列集合でないような集合でも別の順序では整列集合になっていたりします.
省26

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