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Thread: 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 $\displaystyle Y = \{ x\in X | x\not \in x\}$. Notice that $\displaystyle Y\subseteq X$ therefore $\displaystyle Y\in \mathcal{P}(X)$. However, $\displaystyle Y\not \in X$, to show this, assume to contrary that $\displaystyle Y\in X$. Now what we have is essentially Russel's paradox, because if $\displaystyle Y\in X$ we must have either $\displaystyle Y\in Y$ or $\displaystyle Y\not \in Y$. If $\displaystyle Y\in Y$ then by construction $\displaystyle Y\not \in Y$, a contradiction. If $\displaystyle Y\not \in Y$ then by construction $\displaystyle Y\in Y$ for $\displaystyle Y\in X$ and $\displaystyle Y\not \in Y$, a contradiction. Thus, we must have $\displaystyle Y\not \in X$.
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