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!) :)



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