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?
Best Answer
For a good discussion of this matter, see: Kanamori, Akihiro The empty set, the singleton, and the ordered pair. Bull. Symbolic Logic 9 (2003), no. 3, 273–298.