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 >> >> -- >> 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. >> >> > -- 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.