S doesn't seem that interesting. Assume that S does not contain itself. Consider the set P, which can be constructed by adding P as an element to S. Note that P

S doesn't seem that interesting. Assume that S does not contain itself. Consider the set P, which can be constructed by adding P as an element to S. Note that P contains itself. Therefore, P is in S. Also note that every other element in P is also in S, and that every element in S is in P. Therefore P and S are the same set. Therefore S contains itself, contradicting our assumption that S does not contain itself. If we assume that S does contain itself, then I do not see a way at arriving at a contradiction.