[spekcular]

They don't do it for good reason: set theory is basically a strictly better foundation for mathematics, and you have to ape all the set theoretic constructions when doing various things anyway (e.g. constructing the real numbers), so it doesn't buy you anything.

This issue has been litigated extensively, and in my view successfully, by Harvey Friedman on the Foundations of Mathematics mailing list, if you want to check its archives.