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492(3): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2024/11/23(土)17:38 ID:dngn2gaF(16/22) AAS
つづき
en.wikipedia.org/wiki/Axiom_of_choice
Axiom of choice
Criticism and acceptance
A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory. For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no individual well-ordering of the reals is definable. Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proved to exist using the axiom of choice, it is consistent that no such set is definable.[8]
The axiom of choice proves the existence of these intangibles (objects that are proved to exist, but which cannot be explicitly constructed), which may conflict with some philosophical principles.[9]
Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired (as is often the case in category theory). This has been used as an argument against the use of the axiom of choice.
省2
494: 2024/11/23(土)18:28 ID:wHxaJ233(35/45) AAS
>>491-493
これだけ長々と長文書き連ねて、
>定式化してごらん
にまったく答えられてないｗ 馬鹿丸出しｗ
なんでそんなに馬鹿自慢したがるの？ どＭ？
515(1): 阿弥陀如来 ◆0t25ybzgvEX5 2024/11/24(日)07:39 ID:I9DmCuNm(1/18) AAS
>>491-492
整列定理から選択公理が導けるのは当然
選択公理から整列定理が導ける↓の証明は分かるかい？
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.
省9
527(4): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2024/11/24(日)08:46 ID:pyyDnAPQ(1/15) AAS
>>517
＞fは唯一ではない（つまり一意的でない）が、
＞少なくとも一つ存在するなら一つとれる（つまり一意化できる）
一意的でない vs 一意化できる
矛盾してないか？
そのうえで、>>492 (en.wikipedia.org/wiki/Axiom_of_choice)より
”a subset of the real numbers that is not Lebesgue measurable can be proved to exist using the axiom of choice, it is consistent that no such set is definable.[8]”
省36
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