Why is "the set of all sets" a paradox, in Layman's terms? I've heard of some other paradoxes involving sets (i.e., &quot;the set of all sets that do not contain themselves&quot;) and I understand how paradoxes arise from them.  But this one I do not understand.
Why is &quot;the set of all sets&quot; 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 &quot;rules of sets&quot; that are involved in more rigorous set theory?
Let $|S|$ be the cardinality of $S$. We know that $|S| &lt; |2^S|$, 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 $|S| &lt; |2^S|$, but $2^S$ is a subset of $S$, because every set in $2^S$ is in $S$. Therefore $|2^S| \leq |S|$. A contradiction. Therefore the set of all sets does not exist.
I think this paradoxon is not so complicated and can be explained simply.

If you put all existing sets S(1) to S(k) together in a set S(all), you create a new set S(k+1)={S(1),...,S(k)} and this set has to be in the set of all sets as well (because it should contain all sets). Thus, you put the new set S(k+1) in the set of all sets S(all). But this in turn creates a new set S(k+2)={S(1),...,S(k+1)} wich has to be inserted in the set of all sets as well. Putting S(k+2) in S(all), however, creates a new set and so on and so on.

In conclusion, everytime you put the new set into the set of all sets you create a new set. As a consequence, you can never build a set S(all) that contains all sets without creating a set which is not contained in S(all).