A very strong set theory? (Q1580651)

The author develops a set theory \(\text{IC}^*_2\) with two \(\in\)-relations and versions of the axioms of extensionality, comprehension (of naive set theory), union and existence of complements which combine these relations. In this set theory the set-theoretic operations on the universe \(V=\{x;x=x\}\) do not depend on the particular \(\in\)-relation and, moreover, \(V\) is a Boolean algebra with respect to these operations. Reviewers Remark: Concerning the consistency question, M. Goldstern has some unpublished notes on a set theory with a universal set and the existence of complements of every set. He drops the axiom of foundation and requires power set axiom, replacement and separation (Aussonderung) only for hereditarily well-founded sets.