On 2/26/2012 4:43 PM, Bruno Marchal wrote:
[SPK]
Ben Goertzel has a very nice paper discussing the use of hypersets and consciousness here <goertzel.org/consciousness/consciousness_paper.pdf>. Craig's 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 in S4.

Dear Bruno,

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 <http://en.wikipedia.org/wiki/Blind_man%27s_bluff_%28poker%29>" where 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 <http://en.wikipedia.org/wiki/Kleene%27s_recursion_theorem>!) and 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 www.cs.ru.nl/B.Jacobs/PAPERS/iandc-solutions.pdf) 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 problem...


Onward!

Stephen

PS, The Smullyan and Fitting book on Set theory and the Continuum problem <http://books.google.com/books/about/Set_Theory_and_the_Continuum_Problem.html?id=ZoKoPgAACAAJ> looks very interesting. I have added it to my wish list of books. ;-)

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