Set of all Sets

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Forming the set $\SS$ of all sets leads to a contradiction.


Let $\SS$ be the set of all sets.

Then $\SS$ must be an element of itself, in symbols, $\SS \owns \SS$.

Thus we have an infinite descending sequence of membership:

$\SS \owns \SS \owns \SS \owns \cdots$

But by the axiom of foundation, no such sequence exists, a contradiction.