  We present a novel treatment of set theory in a four-valued paraconsistent
and paracomplete logic, i.e., a logic in which propositions can be both true
and false, and neither true nor false. Our approach is a significant departure
from previous research in paraconsistent set theory, which has almost
exclusively been motivated by a desire to avoid Russell's paradox and fulfil
naive comprehension. Instead, we prioritise setting up a system with a clear
ontology of non-classical sets, which can be used to reason informally about
incomplete and inconsistent phenomena, and is sufficiently similar to ZFC to
enable the development of interesting mathematics.
  We propose an axiomatic system BZFC, obtained by analysing the ZFC-axioms and
translating them to a four-valued setting in a careful manner, avoiding many of
the obstacles encountered by other attempted formalizations. We introduce the
anti-classicality axiom postulating the existence of non-classical sets, and
prove a surprising results stating that the existence of a single non-classical
set is sufficient to produce any other type of non-classical set.
  Our theory is naturally bi-interpretable with ZFC, and provides a
philosophically satisfying view in which non-classical sets can be seen as a
natural extension of classical ones, in a similar way to the non-well-founded
sets of Peter Aczel.
  Finally, we provide an interesting application concerning Tarski semantics,
showing that the classical definition of the satisfaction relation yields a
logic precisely reflecting the non-classicality in the meta-theory.
