23/10/15 15:36:11.85 saNJG4i0.net
>>193 ID:0aIpEf0M
>バナッハ-タルスキーのパラドックス 原著第2版
>URLﾘﾝｸ(www.kyoritsu-pub.co.jp)
>にも、選択公理なしにパラドックスが構成できる例が書いてあるよ。
>>234 ID:0aIpEf0M
>「あるバナッハ-タルスキー逆理」における代表系も
>非可算集合である。具体的には、ZFで構成可能な
>双曲平面内のある領域内の点集合の全体になる。
ご苦労さまです
(参考)
URLﾘﾝｸ(en.wikipedia.org)
Banach–Tarski paradox
Banach and Tarski publication
In a paper published in 1924,[6] Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the strong form of the Banach–Tarski paradox:
Connection with earlier work and the role of the axiom of choice
Banach and Tarski explicitly acknowledge Giuseppe Vitali's 1905 construction of the set bearing his name, Hausdorff's paradox (1914), and an earlier (1923) paper of Banach as the precursors to their work. Vitali's and Hausdorff's constructions depend on Zermelo's axiom of choice ("AC"), which is also crucial to the Banach–Tarski paper, both for proving their paradox and for the proof of another result:
In 1964, Paul Cohen proved that the axiom of choice is independent from ZF – that is, it cannot be proved from ZF. A weaker version of an axiom of choice is the axiom of dependent choice, DC, and it has been shown that DC is not sufficient for proving the Banach–Tarski paradox, that is,
The Banach–Tarski paradox is not a theorem of ZF, nor of ZF+DC.[8]
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