Sets are clearly defined. The usual formalism for defining a set is ZFC set theory. In fact to a first approximation all of modern mathematics is sufficiently w

Sets are clearly defined. The usual formalism for defining a set is ZFC set theory. In fact to a first approximation all of modern mathematics is sufficiently well-defined to be entirely formalized in a computer. This is an approximation primarily because currently the burden to do so is immensely high. With today's technology formalizing a proof can easily take several orders of magnitude more time (if not more!) than writing the informal, but commonly accepted as rigorous, proof. There are philosophical questions around what it means to "define" something. But if you want to go there, well then nothing is clearly defined.