>>156
「選択公理、選択公理、・・」か
まるで、念仏だね

さて、「Sergiu Hart氏のPDF November 4, 2013」（下記）というのがあって
game2：A similar result, but now without using the Axiom of Choice.
で
Proof. The proof is the same as for Theorem 1, except that here we do not use the Axiom of Choice.
となっております

この話は、１年前にもした記憶があるんだが、再度引用しよう
選択公理を連呼する人は、game2 ”without using the Axiom of Choice”を、どう考えているのかね？

選択公理なしで、類似のgame2が考えられると書いてあるぜ
はてさて？

スレ47 https://rio2016.5ch.net/test/read.cgi/math/1512046472/41 （2017/11/30(木) ） より
http://www.ma.huji.ac.il/hart/puzzle/choice.pdf
Sergiu Hart氏のPDF November 4, 2013
(抜粋)
A similar result, but now without using the Axiom of Choice.*2
Consider the following two-person game game2:

・ Player 1 chooses a rational number in the interval [0, 1] and writes down
its infinite decimal expansion *3 0.x1x2...xn..., with all xn ∈ {0, 1, ..., 9}.

・ Player 2 asks (in some order) what are the digits xn except one, say xi;
then he writes down a digit ξ ∈ {0, 1, ..., 9}.

・ If xi = ξ then Player 2 wins, and if xi ≠ ξ then Player 1 wins.

By choosing i arbitrarily and ξ 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 guaran-
teeing 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.

Note
*2 Due to Phil Reny.
*3 When there is more than one expansion, e.g., 0.1000000... = 0.0999999..., Player 1 chooses
which expansion to use.
(引用終り)
以上