# Re: Toward a solution to the Arithmetic Body problem

```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
>>>>> 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
> ```
```
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.

--
Onward!

Stephen

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