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 >>> 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!) :) >

## Advertising

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. -- Onward! Stephen -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out.