Hi Acw,

On 1/11/2012 1:35 PM, acw wrote:
On 1/11/2012 19:22, Stephen P. King wrote:

I have a question. Does not the Tennenbaum Theorem prevent the concept
of first person plural from having a coherent meaning, since it seems to
makes PA unique and singular? In other words, how can multiple copies of
PA generate a plurality of first person since they would be an
equivalence class. It seems to me that the concept of plurality of 1p
requires a 3p to be coherent, but how does a 3p exist unless it is a 1p
in the PA sense?



My understanding of 1p plural is merely many 1p's sharing an apparent 3p world.

OK, we could see this as an equivalence class of sorts where the relation between the 1p is a 4-diffeomorphism. The correspondence between frames of reference/coordinate systems and 1p's makes sense, but what defines its closure and compactness? There has to be something that requires the set to be finite. The demand that any one of the 1p in the set be representable as a recursively countable string might to the trick, but each must be recursively countable in some way. I think that there is a to do this and not violate the Tennenbaum theorem <http://web.mat.bham.ac.uk/R.W.Kaye/papers/tennenbaum/tennenbaum.xhtml>. I have an idea but do not know how to representing formally yet.

That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p.

If we accept Bruno's result then a 3p world must supervene on an infinite number of computations. I strongly suspect that there must be an infinity of 3p's, each a globally maximally coherent set of 1p's...

At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD.

It seems to me that there cannot be just one 3p as it could not be finite. Consider the number of Boolean Algebras that we can map (via endomorphism?) to a single orthocomplete lattice, and that would be just for one quantum mechanical system. Each of the BA would be the representation of a 1p, maybe... I am not sure...

As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. I think you might be confusing structures/relations which can be contained within PA with PA itself.

Here I part with Bruno as I do not think that a Sigma_1 sentence alone has the necessary and sufficient structure for consciousness to obtain. We can certainly see that the OMs -> Sigma_1 but there is more involved that the mere content of an experience. We have to reproduce the *appearance* of the "Cartesian theater effect" to have consciousness.



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