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 >>>>> 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!) :) >>> >> >> 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 > is already quantum-like. >

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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 -- 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.