On 3/12/2013 9:27 AM, Richard Ruquist wrote:
> Steve,
> Does not the wiki ref imply "that the number of countable models of a
> first-order complete theory in a countable language is finite" or Xo?
> Richard

Hi Richard,

        Yes, "the number of countable models of a first-order complete theory
in a countable language is finite or ℵ0 or 2ℵ0"

        That is, finite, countable (integers Aleph_0) or uncountable (2^aleph_0
or aleph_1 as in the Real numbers).

> 
> On Tue, Mar 12, 2013 at 8:58 AM, Stephen P. King <stephe...@charter.net> 
> 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."
>>
>>
>> --
>> Onward!
>>
>> Stephen
>>
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> 


-- 
Onward!

Stephen

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