On 26 Oct 2011, at 05:34, Stephen P. King wrote:

On 10/25/2011 4:40 PM, Russell Standish wrote:
On Mon, Oct 24, 2011 at 04:08:38PM +0200, Bruno Marchal wrote:
On 23 Oct 2011, at 04:41, Russell Standish wrote:

On Fri, Oct 21, 2011 at 02:14:48PM +0200, Bruno Marchal wrote:
So the histories, we're agreed, are uncountable in number, but OMs (bundles of histories compatible with the "here and now") are surely
still countable.
This is not obvious for me. For any to computational states which
are in a sequel when emulated by some universal UM,there are
infinitely many UMs, including one dovetailing on the reals, leading
to intermediate states. So I think that the "computational
neighborhoods" are a priori uncoutable.
Apriori, no. The UMs dovetailing on the reals will have only executed a finite number of steps, and read a finite number of bits for a given OM. There are only a countable number of distinct UM states making up
the OM.
The 3-OM. But the first person indeterminacy depends on all the
(infinite) computations going through all possible intermediary
3-OMs states.

So does the OM I'm referring to. Does that still make is a 3 OM?

That fits with the
topological semantics of the first person logics (S4Grz, S4Grz1, X,
X*, X1, X1*). But many math problems are unsolved there.

You will need to expand on this. I don't know what you mean.
I have explained this to Stephen a long time ago, when explaining
why the work of Pratt, although very interesting fails to address
the comp mind body problem. Basically Pratt's duality is recover by
the "duality" between Bp (G) and Bp&  Dt (Z1*) or Bp&  Dt&  p (X1*).
You might serach what I said by looking at Pratt in the archive,
with some luck.

This is above my level of understanding at present. Hopefully, there
will be some quiet time soon to study this, as it sounds interesting!

Hi Russell and Bruno,,

I recommend that you read Steve Vickers' "Topology Via Logic" first.

I would not have discovered, and take seriously, the material hypostases without it, I think. I give him full credit in my publications. Abramski played some role too. Very nice book, but still quite abstract. I have already commented Pratt at large.

The other reason to use the self-reference logics is that it
distinguish automatically the quanta (sharable, communicable at
least in a first person plural way) from the qualia (not sharable,
purely individual), all this by the Gödel-Löb-Solovay proof/truth
splitting of the modal logics.
Yes - that is interesting, but is true of any modal logic (apart from
S4Grz, it would appear).
But how do you obtain the mutual orthogonality of observables on a quantum logic? We must address the relationship between orthocomplete lattices and Boolean algebras at some point!

The ortholattice are the gluing of Boolean algebraic dreams of universal machines (the boolean algebra describing their consistent histories). It gives the differentiation/fuse structure of the local and partial boolean algebras. But dually the ortholattices can be internalized as structured subsets in Boolean algebra, or by Kripkean semantics. An apparent conspiracy of nature prevent such "duality" to be algebraically interesting, in the quantum case. I guess we have to live with this. In the digital case, it is an open problem. It makes interesting to solve the digital case, just to see if such conspiracy of nature is a physical law or a geographical misfortune. This can be translated mechanically into a set of arithmetical problem, but those are *very* complex (that's the weakness of the interview of the universal machine on such question).



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