# Re: Toward a solution to the Arithmetic Body problem

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On 13 Mar 2013, at 17:37, Stephen P. King wrote:```
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```On 3/13/2013 12:22 PM, Bruno Marchal wrote:
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On 12 Mar 2013, at 18:54, Stephen P. King wrote:

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```On 3/12/2013 12:22 PM, Bruno Marchal wrote:
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On 12 Mar 2013, at 14:10, Stephen P. King wrote:

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

I have found a paper that seems to cover most of my thoughts
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

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"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."

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Let me refine my concerns a bit. Is there a method to consider the Vaught conjecture on finite lattice approximations of Polish spaces?
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Please relate all this, as formally as in the Ehrenfeucht Mostowski
paper, to what has already been solved, in the ideal "toy" case of
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simple ideally correct machine, at the propositional level (that is:
```the X, Z and S4Grz1) logics.

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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.

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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!) :)

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```
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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.
```
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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.

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```
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.
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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

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```

--
Onward!

Stephen

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

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