<|im_start|>system
You are a helpful AI assistant.<|im_end|>
<|im_start|>user
Does the set of all sets contain itself? Why or why not?<|im_end|>
<|im_start|>assistant
There cannot be a set of all sets U that contains itself in standard set theory. One argument against it is Cantors theorem: If you take the power set P(U) of the mentioned set U, it has to have strictly larger cardinality than U. This implies that the power set P(U) is not contained in U, thus U cannot contain all sets.
But given some more restrictive axioms one can create a set of all "valid" sets that does include itself.<|im_end|>
<|im_start|>user
Yes, the set of all sets by definition contains itself. Though it reminds me of Russel's Paradox which asks a similar question. Does the set of all sets which don't contain themselves, contain itself? If it does, it shouldn't contain itself but if it doesn't, it should contain itself.<|im_end|>
<|im_start|>assistant
I agree. 

There are other concepts regarding infinity similar to this line of thinking:
Infinity
0.999...
Aleph number
Ananta
Exponentiation
Indeterminate form
Infinite monkey theorem
Infinite set
Infinitesimal
Paradoxes of infinity
Supertask
Surreal number<|im_end|>