On 13 Mar 2013, at 17:37, Stephen P. King wrote:

On 3/13/2013 12:22 PM, Bruno Marchal wrote:

On 12 Mar 2013, at 18:54, Stephen P. King wrote:

On 3/12/2013 12:22 PM, Bruno Marchal wrote:

On 12 Mar 2013, at 14:10, Stephen P. King wrote:

On 3/12/2013 8:58 AM, Stephen P. King wrote:
Dear Bruno,

  I have found a paper that seems to cover most of my thoughts
about the
arithmetic body problem:
Models of axiomatic theories admitting automorphisms
by A. Ehrenfeucht  A. Mostowski
http://matwbn.icm.edu.pl/ksiazki/fm/fm43/fm4316.pdf

More on related concepts are found in the Vaught conjecture:
http://en.wikipedia.org/wiki/Vaught_conjecture

"The topological Vaught conjecture is the statement that whenever a Polish group acts continuously on a Polish space, there are either countably many orbits or continuum many orbits. The topological Vaught
conjecture is more general than the original Vaught conjecture:
Given a
countable language we can form the space of all structures on the
natural numbers for that language. If we equip this with the topology
generated by first order formulas, then it is known from A.
Gregorczyk,
A. Mostowski, C. Ryall-Nardzewski, "Definability of sets of models of
axiomatic theories", Bulletin of the Polish Academy of Sciences
(series
Mathematics, Astronomy, Physics), vol. 9(1961), pp. 163–7 that the
resulting space is Polish. There is a continuous action of the
infinite
symmetric group (the collection of all permutations of the natural
numbers with the topology of point wise convergence) which gives
rise to
the equivalence relation of isomorphism. Given a complete first order
theory T, the set of structures satisfying T is a minimal, closed
invariant set, and hence Polish in its own right."



Let me refine my concerns a bit. Is there a method to consider the Vaught conjecture on finite lattice approximations of Polish spaces?

Please relate all this, as formally as in the Ehrenfeucht Mostowski
paper, to what has already been solved, in the ideal "toy" case of
simple ideally correct machine, at the propositional level (that is:
the X, Z and S4Grz1) logics.

There might be a way, but it sounds to me like a very difficult problem for expert in both provability logics and model theory. I think you will
need the diagonal algebra of Magari.

You will need to relate the work of the Italians, the Polish and the
Georgians, hmm... That is a work for the Russians (the
mathematicians!) :)


I agree! ;-) Maybe there might already exist a solution in a Russian Journal now. I am trying to re-engage Pratt on this but I may have to go
for alternatives, via Topos possibly.

You can do that. The arithmetical topos is the one you can "extract"
from the SGrz1 logic, by reversing the Boolos-Goldblatt morphism.

Again the arithmetical topos will give the (non boolean) first person
picture only. The inner god, or the universal, in Plotinus term. This is only 1/8 of the comp global picture. But it is important. As Plotinus understood, the soul as already a foot in matter, and the S4Grz1 logic
is already quantum-like.


Hi Bruno,

        Nice, so to find more people that understand what a Boolos-Goldblatt
morphism is and how to reverse it *and* how it might be parametrized is
my mission.

They need to have read my work, as S4Grz is not enough, you need S4Grz1, and thus the sigma_1 restriction, and thus UDA and the comp mind body problem.

Bruno




--
Onward!

Stephen

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http://iridia.ulb.ac.be/~marchal/



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