Paradoxes – Why ‘The Set of All Sets’ is a Paradox in Layman’s Terms

Question

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?

Answer 1

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

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Theorem

The set of all sets does not exist.

Proof

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