The point of departure for Positive theories of sets is
the observation that all definitions of paradoxical sets
exploit negation. The proposal is that - for a suitable
notion of positive - every positive formula should determine
Olivier Esser starts with an existing positive theory enriched
with an axiom scheme that endows each class with a "closure".
He shows that the resulting theory interprets the theory
of Kelley-Morse together with the axiom "the class
of ordinals is ramifiable", and conversely.
His results show clearly that the orthodox theories have
excluded too many sets. By eschewing solely what causes
problems in the paradoxes - namely the use of negation in
comprehension axioms - he recovers not only the sets of
the usual theories but also lots of others that one had
no reasonable motive to exclude.