By Willard L. Miranker, Gregg J. Zuckerman
in Journal of Applied Logic, Volume 7, Issue 4 (2009)
We employ the Zermelo–Fränkel Axioms that characterize sets as mathematical
primitives. The Anti-foundation Axiom plays a significant role in our
development, since among other of its features, its replacement for the Axiom
of Foundation in the Zermelo–Fränkel Axioms motivates Platonic interpretations.
These interpretations also depend on such allied notions for sets as pictures,
graphs, decorations, labelings and various mappings that we use. A syntax and
semantics of operators acting on sets is developed. Such features enable
construction of a theory of non-well-founded sets that we use to frame
mathematical foundations of consciousness. To do this we introduce a
supplementary axiomatic system that characterizes experience and consciousness
as primitives. The new axioms proceed through characterization of so-called
consciousness operators. The Russell operator plays a central role and is shown
to be one example of a consciousness operator. Neural networks supply striking
examples of non-well-founded graphs the decorations of which generate
associated sets, each with a Platonic aspect. Employing our foundations, we
show how the supervening of consciousness on its neural correlates in the brain
enables the framing of a theory of consciousness by applying appropriate
consciousness operators to the generated sets in question.
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