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?

Answer

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


Theorem

The set of all sets does not exist.

Proof

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

Attribution
Source : Link , Question Author : Justin L. , Answer Author : BCLC

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