On 2/26/2012 4:43 PM, Bruno Marchal wrote:
Ben Goertzel has a very nice paper discussing the use of
hypersets and consciousness here
discussion of it is here <http://s33light.org/post/17993511503>.
Yes. It is not bad, but I use combinators or lambda terms to handle
the non foundations, or the second recursion theorem, or the modal
logic (based on the use of those diagonalizations), which is natural
in the comp meta-theory.
Ben was a participant of this list years ago. We had good discussions.
It is also a not too bad material.
But polishing too much tools for solving a problem can distract from
solving the problem, or even from formulating it (or a subproblem of it).
I already told I am skeptical on the notion of sets in general. I
like very much ZF, which I have studied in deep, but I see it just as
a sort of very imaginative Löbian machine. Jean-Louis Krivine, Jech,
and recently Smullyan and Fitting wrote very nice books on set theory.
They explain the Cohen forcing technic with a nice modal construction
Have you seen any consideration of analogies to forcing using
non-standard models? I ask this as it seems that non-standard models are
defined in a way (involving extensions to PA) that is very similar to
how a forcing is defined (as "expanding the set theoretical universe
<http://en.wikipedia.org/wiki/Universe_%28mathematics%29> /V/ to a
larger universe /V/* ") . Maybe I am seeing more than is in the
language, but there is nothing to this that can inform us about how to
overcome the open problem of whether we can say that an infinite number
of physical processes are running each and every instance of a
computation just as we argue, via universality of TM, for infinite
computations running any 1p, and thereby give meaning to phrasings like
"dreams of numbers".
My idea is that *countable and recursive* is a necessary condition
for a Löbian entity (LE) to exist as "1p associated with true beliefs",
but we have to also answer to the "problem" of "Other Minds"
<http://plato.stanford.edu/entries/other-minds/>. It could be that each
and every "1p associated with true beliefs" believes that its model (of
itself as we consider self-reference) is uniquely standard - thus
countable and recursive - when it cannot know for sure. Its belief is
"justified" given all possible interviews it can conduct with versions
of itself will never yield a counterexample but not provable true since
it can not "see itself from the outside as it 'truly is' " aka "God's
point of view".
I am reminded of the game of "Blind man's bluff
the extension that enumerates the particularity (of an individual Löbian
entity cannot be seen or otherwise known by the individual) that acts as
an extension on the PA model of the Löbian entity. In this way, each and
every Löbian entity will think of itself as "centered" in the universe
of models of arithmetic (with the center defined via Kleene's fixed
point since its self-model must be countable and recursive
truthfully believe that all other Löbian entity are "situated some
distance from that center". This also makes each LE think of itself as
finite (and thus having a compact dual as its Stone space) which goes a
long way to explaining the appearance of physics by COMP.
This idea has the added nice feature that it gives us a kind of
measure (but only in the local and subjective sense) even if there is no
real 3p version of such a measure. (Bisimlarity between Löbian entities
gives us a nice measure and maybe even a metric but does not give a
distributive non-dual or monist logic! For example see
My reasoning seems to be consistent with Kripkean and possible
world semantics (allowing for an infinite number of physical worlds and
thus all possible physical systems) and ideas that Hintikka uses (in the
sense of possible two player games to discover the proof of a theorem),
but it is hard to connect them to your ideas as you use an atemporal
implicate and a temporal explicate language (to use David Bohm's
terminology). This was very confusing to me for a long time.... You talk
as if platonic objects obey physical laws but reason as if they do not.
Once I understood this, I was better able to understand your reasonings.
Do these remarks make any sense to you? I am motivated to these
extreme ideas as I do not see any other escape from the problem of other
minds <http://plato.stanford.edu/entries/other-minds/> for COMP. The
problem of zombies and fadeing qualia
<http://consc.net/papers/qualia.html> are just other forms of this
PS, The Smullyan and Fitting book on Set theory and the Continuum
looks very interesting. I have added it to my wish list of books. ;-)
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