TITLE: Set theory without the empty set
QUESTION [12 upvotes]: Has there ever been a set theory without an empty set? Is this possible?

I ask because we usually take the empty set to exist axiomatically or obtain it through separation and a nonempty set together with the standard parameter-free predicate $X\neq X$, but it seems possible to have a 'set theory' without an axiom asserting the existence of an empty set or an axiom of separation.
I put 'set theory' in quotations because such a nonstandard axiomatization might not really deserve to be called a set theory per-se (it wouldn't prove the existence of intersections of disjoint sets), but more formally I mean

Has a theory in the language of set theory whose axioms do not prove the existence of an empty set ever been explored?

REPLY [3 votes]: I’ve explored a “set theory whose axioms do not prove the existence of an empty set,” the Incomprehensive Set Theory in my “Naive View of the Russell Paradox” (https://arxiv.org/abs/2103.00090), but for axiomatic parsimony, not with the motives I suspect you’re looking for.   I note that “Any assumptions about set existence, even that any sets at all exist, will be noted explicitly. What follows will be trivial unless a lower or an upper exists, and the main interest is when an empty set and a universal set exist.”  I imagine Incomprehensive Set Theory plus the existence of the Universal Set is consistent and does not prove the existence of the empty set, but, as noted, that is not necessarily of much interest.