Discussion on answer by Dawood ibn Kareem: Why is a set with one element distinct from the element itself?

Are you asking me to prove something from no axioms?

@DoubleKnot: If you don't have the empty set, then you can't construct the natural numbers, and if you don't have numbers, then you can't count. I would call that "useful."

It seems like a huge assumption to make - that the set of all well-defined sets is well-defined. Maybe it doesn't lead to contradictions; but I'm sure there are versions of set theory where it's not "true" (whatever "true" means).