>>196
>>>187
>・「Sergiu HartのGame2（有理数の小数展開が対象）や
>　その変化形(有限小数が対象）は、（フルパワー）選択公理を使用しない」
>　は良いとして
>>>192
>>フルパワーもクソもなく、Game2では選択公理はまったく必要ない。
>>代表系がZF内で構成できるから。
>やってみて
>そしたら信用するよ

どうせ出来ないと思ったが、やっぱりｗｗ
仕方ないから、解説するよｗ

(参考)
”スレタイ 箱入り無数目を語る部屋14”
https://rio2016.5ch.net/test/read.cgi/math/1696677610/2
http://www.ma.huji.ac.il/hart/puzzle/choice.pdf
Choice Games November 4, 2013 Sergiu Hart
(抜粋)
P2
A similar result, but now without using the Axiom of Choice.2 Consider the
following two-person game game2:

By choosing i arbitrarily and » uniformly in {0, 1, ..., 9}, Player 2 can guar-
antee a win with probability 1/10. However, we have:
Theorem 2 For every ε > 0 Player 2 has a mixed strategy in game2 guaranteeing him a win with probability at least 1 - ε.

Proof. The proof is the same as for Theorem 1, except that here we do not
use the Axiom of Choice. Because there are only countably many sequences
x ∈ {0, ..., 9}^N that Player 1 may choose (namely, those x that become eventually
periodic), we can order them—say x(1), x(2), ..., x(m), ...—and then choose in
each equivalence class the element with minimal index (thus F(x) = x(m) iff m
is the minimal natural number such that x 〜 x(m)).

P1
Theorem 1 For every ε > 0 Player 2 has a mixed strategy in game1 guaran-
teeing him a win with probability at least 1 - ε.

Proof.略
Apply the Axiom of Choice to choose an element in each
equivalence class;
略
(引用終り)

つづく