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
