Russell's Paradox

I recommend scrolling down to the discussion of stratification and New Foundations, which over the years has felt more and more like the correct resolution to me. ZFC uses a mighty cop-out: "welp, guess there's no set of all sets". But NF's resolution basically boils down to "you fool, 'x is a member of itself' isn't well-typed; being a member of a thing means you have to be somehow smaller than the thing; why would you even think that this was a subset selection criterion you could try and use".

More concretely, it introduces a (very weak and permissive!) type system onto the subset selections you can make, and declares that you can only select "the subset of X which satisfies property P" if P is well-typed. No more selecting the set of all x which are not members of themselves, because the "member of" relation can only be formed when there exists an assignment of types such that the thing on the left has a "strictly smaller" type than the thing on the right. Of course, x has the same type as itself, so "x is a member of x" is forbidden.

This is thought to result in a consistent set theory, Quine's New Foundations, although Randall Holmes's proof of this fact is subtle and is not generally agreed to be solid.

A very cute factoid about New Foundations: the first proof of the existence of an infinite set in NF was by the curious route of disproving the Axiom of Choice. Since Choice is true over finite sets, that means there must be an infinite set.

Gotta love ZFC