The Set of all Sets

I couldn't sleep last night and, like most people (I'm guessing - I haven't done a poll or anything), when faced with insomnia, I like to think about some of the great problems of analytic philosophy. "Does the set of all sets contain itself?" I wondered. Here is some of my reasoning presented in that most venerable of philosophical forms: the dialogue.

Bertrand: Does the set of all sets contain itself?

Freddie: Yes. Obviously. Otherwise it would be indistinguishable from the set of all sets except itself.

Bertrand: But, if the set of all sets contains itself, the copy of itself within itself also contains itself and so on.

Freddie: wtf! Mind = blown! 

Bertrand: I thought you'd like that.

Freddie: So the set of all sets does not contain itself?

Bertrand: But if it does not contain itself then it is at least one set short and can't really be the set of all sets.

Ludwig: It's a meaningless question. Try counting sheep.

Bertrand: Pay attention Ludwig, he tried that here.

Freddie: He was much better in those days, look it's got a nice photograph and a little poem...