This is going to be a short one today, as I have a lot of assignments due in on Thursday, and not a lot of time to do them all in.
Consider a set, I shall call it A. A is the set of all sets which are not members of themselves.
If A is not a member of itself, then by definition it is a member of itself.
But if it is a member of itself, then by definition it can't be a member of itself.
This self-contradictory definition is known as Russell's paradox.
And it makes my head hurt.
Ponder on that for a while :)