Set of all Sets
Forming the set $\mathcal S$ of all sets leads to a contradiction.
Let $\mathcal S$ be the set of all sets.
Then $\mathcal S$ must be an element of itself, in symbols, $\mathcal S \owns \mathcal S$.
Thus we have an infinite descending sequence of membership:
- $\mathcal S \owns \mathcal S \owns \mathcal S \owns \cdots$
But by the axiom of foundation, no such sequence exists, a contradiction.