現代数学の系譜工学物理雑談古典ガロア理論も読む42 [無断転載禁止]©2ch.net

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618: 現代数学の系譜 工学物理雑談 古典ガロア理論も読む [sage] 2017/09/27(水) 08:33:59.06 ID:FCHsJSPl

>>593
>講釈は不要、自然数で答えて下さい。

すでに>>532で回答済み。「co-tail は、構成的に書けない」
選択公理を使うと、明示的な構成を持たない集合が存在しうることは、現代数学では常識です（＾＾
下記の、渕野昌先生、「非可測集合は存在するのか？」及び、“Axiom of choice Criticism and acceptance”（en.wikipedia）をご参照下さい

http://math.cs.kitami-it.ac.jp/~fuchino/notes.html
渕野昌 数学ノート 北見工大
http://math.cs.kitami-it.ac.jp/~fuchino/notes/nonmeasurable.tex
非可測集合は存在するのか？ 渕野昌 数学ノート 北見工大 20001018

https://en.wikipedia.org/wiki/Axiom_of_choice
Axiom of choice
(抜粋)
Criticism and acceptance

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.[6]

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.[7]
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.

Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive.[8]
One example is the Banach?Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original.
The pieces in this decomposition, constructed using the axiom of choice, are non-measurable sets.
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1505609511/618

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