I have always marveled at the difficulty Russell's paradox has caused. The ZF axiom completely solves it by saying that you can't ask whether Russell's set contains itself until you have demonstrated that Russell's set can exist.

The issue for me is whether the barber's paradox can be stated "in natural language." Isn't "How much is 5 divided by 0?" stated in natural language? Yet, we have no problem with the idea that division by zero is "undefined." Well, so is "he shaves every man who doesn't shave himself, and no one else." The words have definitions, just as 5, divided by, and 0 have definitions. But the sentence is undefined gibberish. There is no such set, end of problem.