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現代数学の系譜工学物理雑談古典ガロア理論も読む77

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>>38
> >>37 補足
> ＞応用
> ＞ZFの集合モデルは集合状かつ外延的である。
> 
> ”集合状”かｗ、これ意味わからんと思ったが（＾＾
> ”Every set model of ZF is set-like and extensional. ”の「set-like」の直訳だね（＾＾；
> 
> ＜参考引用、該当英文箇所＞ (なお、Applicationも、”応用”より”適用”が適訳かもね。微妙だが)
> https://en.wikipedia.org/wiki/Mostowski_collapse_lemma
> Mostowski collapse lemma
> (抜粋)
> Application
> Every set model of ZF is set-like and extensional.
> If the model is well-founded, then by the Mostowski collapse lemma it is isomorphic to a transitive model of ZF and such a transitive model is unique.
> 
> Saying that the membership relation of some model of ZF is well-founded is stronger than saying that the axiom of regularity is true in the model.
> There exists a model M (assuming the consistency of ZF) whose domain has a subset A with no R-minimal element,
> but this set A is not a "set in the model" (A is not in the domain of the model, even though all of its members are).
> More precisely, for no such set A there exists x in M such that A = R^-1 [x].
> So M satisfies the axiom of regularity (it is "internally" well-founded)
> but it is not well-founded and the collapse lemma does not apply to it.
> (引用終り)
> 以上

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