30fb766f-8d42-44ac-b00f-cb0697a6369c
prompter
Does the set of all sets contain itself? Why or why not?
545fff5b-6b2a-4142-8f81-5b15e2de89cd
30fb766f-8d42-44ac-b00f-cb0697a6369c
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.
8dab357c-c4f1-46fb-aa66-3181d9d41f78
545fff5b-6b2a-4142-8f81-5b15e2de89cd
prompter
In what sens can we speak of "valid" sets?