# Set of all Sets

## Theorem

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

## Proof

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.

$\blacksquare$