Why Axiomatize Set Theory?

I like mathematics because I like the intellectual challenges it can offer.

Here is a good one:

Russell's paradox concerns the set R={ s | s ∉ s}. Is R∈R? If R∈R, then by the definition of R∉R. But by definition, if R∉R, then R∈R. So R is clearly not a well-defined set.

Answer? Read the rest of the original post.