On 10/6/2012 8:54 AM, Platonist Guitar Cowboy wrote:
Yes, but it is also in its infancy. With Aczel's work not 30 years old, and this admittedly weak analogy to consciousness only a few years old, which Aczel does not seem to be following up on himself:

http://www.cs.man.ac.uk/~petera/papers.html <http://www.cs.man.ac.uk/%7Epetera/papers.html>

My point is, this is very young, what's young is always messy and will hopefully tidy itself up, and I can feel some funky aspect, specifically the observer aspect of a non-well-founded set defining a Russell operator, hinting at quantum physics perhaps in the future.

It should be called the Cantor operator, but as Zuckerman notes: "the importance of PR and publishing makes the difference." So, knowing this, why doesn't he call it the Cantor operator...


I suspect that he named it after Russell because Russell's canonical (?) definition of the paradoxical set. I don't know that Cantor drew any attention to that set, thus he doesn't get credit for it.


On Sat, Oct 6, 2012 at 9:12 AM, Bruno Marchal <marc...@ulb.ac.be <mailto:marc...@ulb.ac.be>> wrote:

    On 06 Oct 2012, at 02:37, Stephen P. King wrote:

    Hi Folks,


      Mathematical Foundations of Consciousness

    Willard L. Miranker
    J. Zuckerman
    (Submitted on 23 Oct 2008)

        We employ the Zermelo-Fraenkel 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-Fraenkel 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.

        This is part of what I have been assuming form the beginning
    of my conversation with Bruno so many moons ago. Its nice to see
    its independent discovery.

    As the cow-boy guessed right this is assuming too much, both for
    the formalism used (which is OK), and the ontology, so it uses
    implicitly non-comp hypothesis, which is less OK, as comp is also
    assumed implicitly. IT is not uninteresting for possible progress,
    but it is unaware that matter as to be explained by statistics on
    computations "seen from inside". The role of "Russell operator" is
    played by the Kleene second recursion theorem, which encapsulates
    the "non foundation" well enough.




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