# Why is “the set of all sets” a paradox, in layman’s terms?

I’ve heard of some other paradoxes involving sets (i.e., “the set of all sets that do not contain themselves”) and I understand how paradoxes arise from them. But this one I do not understand.

Why is “the set of all sets” a paradox? It seems like it would be fine, to me. There is nothing paradoxical about a set containing itself.

Is it something that arises from the “rules of sets” that are involved in more rigorous set theory?

Let $$|S||S|$$ be the cardinality of $$SS$$. We know that $$|S|<|2S||S| < |2^S|$$, which can be proven with generalized Cantor's diagonal argument.

Theorem

The set of all sets does not exist.

Proof

Let $$SS$$ be the set of all sets, then $$|S|<|2S||S| < |2^S|$$. But $$2S2^S$$ is a subset of $$SS$$. Therefore $$|2S|≤|S||2^S| \leq |S|$$. A contradiction. Therefore the set of all sets does not exist.