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
>>> More on related concepts are found in the 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.
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