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Do you mean x is a member of S if and only if x is a member of P(S)? I think you could use Cantor's theorem to show that such a set is impossible -- take f(x) -> x. Since this function is one-to-one (forgive me if I'm forgetting terminology here), S = P(S). But this is impossible, so your definition of a set defines a different set.
I think that's just an impossible set, and so a bad definition of a set. But I'd really feel more confident if someone who's had more set theory could confirm this. Maybe if I have time today or tonight, I'll try to refresh my memory a bit better.
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"Die Deutschen meinen, daß die Kraft sich in Härte und Grausamkeit offenbaren müsse, sie unterwerfen sich dann gerne und mit Bewunderung:[...]. Daß es Kraft giebt in der Milde und Stille, das glauben sie nicht leicht."
"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."
-- Friedrich Nietzsche
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