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Math Help - Another Axiom Question

  1. #1
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    Another Axiom Question

    I'm not too sure about this one either...

    Show that P(X) is a subset of X is false for any X. In particular, P(X) does not equal X for any X.
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  2. #2
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    Construct the set Y = \{ x\in X | x\not \in x\}. Notice that Y\subseteq X therefore Y\in \mathcal{P}(X). However, Y\not \in X, to show this, assume to contrary that Y\in X. Now what we have is essentially Russel's paradox, because if Y\in X we must have either Y\in Y or Y\not \in Y. If Y\in Y then by construction Y\not \in Y, a contradiction. If Y\not \in Y then by construction Y\in Y for Y\in X and Y\not \in Y, a contradiction. Thus, we must have Y\not \in X.
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