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?



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