Abstract:
Russell's paradox in naive set theory suggests the definition of what
we call the Russell operator: Rx = {y in x y is not in y}. In
Zermelo's axiomatic set theory, the Russell operator is well defined and is not
paradoxical in nature. The key property of the Russell operator is that
for any set A in the Zermelo universe, the set RA is not in A. If we drop the
axiom of foundation, and substitute Aczel's axiom of antifoundation, the Russell
operator becomes quite nontrivial, since now many sets are elements of
themselves. Thus, the original scope of the Russell paradox in naive set
theory gets transmuted to a rich and consistent theory of the Russell operator
in nonwellfounded axiomatic set theory.
In joint work with Willard Miranker of Yale Computer Science, we have
proposed a mathematical theory of consciousness based on a combination of
the standard theory of neural networks and the emerging theory of
nonwellfounded sets. The Russell operator and certain generalizations we
call consciousness operators play a central role in our proposal. We will
present many examples of nonwellfounded sets and consciousness operators,
and thus make this lecture as self contained as possible.
