Set theory with a universal set.
If we drop the axiom of foundation and require
replacement and separation (Aussonderung) only for hereditarily
sets, can we consistently postulate the existence of a universal set,
and the existence of complements of every set?
I do not know the answer, but it is not too difficult to
this is possible if we also restrict the power set axiom to
hereditarily well-founded sets.
(Available in pdf,
tex, dvi and
Note: There is a book by E.Forster that discusses several
"set theory with a universal set", most of them in the context of