One positive thing about Wikipedia lists: they provide a great way to explain Russell's Paradox. A lot of people have trouble with the concept of a set containi

One positive thing about Wikipedia lists: they provide a great way to explain Russell's Paradox. A lot of people have trouble with the concept of a set containing itself, but not with a web page linking to itself. So, we can imagine defining a Wikipedia page of the list of all lists which do not link to themselves, and then ask - does this list page contain a link to itself? Two cases: (1) The page does link to itself, in which case it is wrongly present in the list of lists which do not link to themselves. (2) The page does not link to itself, in which case it should be present in the list of lists which do not link to themselves. Since both cases result in a contradiction, this means we cannot define the list of all lists which do not link to themselves, and thus any formal language (now called "naive set theory") which allows us to define such a set is logically inconsistent. The solution to this paradox was to use a set of axioms (most commonly the Zermelo-Fraenkel axioms) which do not allow the construction of such a contradictory set, while also maintaining all the expressive power of naive set theory. To anyone interested in this stuff (plus Gödel) I highly recommend the graphic novel Logicomix.