%0 Trabajo de grado (Bachelor Thesis)
%A Buitrago Díaz, Esperanza
%D 2014
%G Desconocido (Unknown)
%T Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar
%U http://babel.banrepcultural.org/cdm/ref/collection/p17054coll23/id/322
%X A Strong type is a class of a bounded equivalence relation (i.e. the quotient is a proper set) on tuples of the monster model of a complete theory T. Today, there are three different notions of strong types: a Shelah’s strong type is a class under the smallest definable equivalence relation on the monster model C with finite classes; a Kim-Pillay strong type is a class of the least bounded type-definable equivalence relation kp, in fact, this type may be characterized as the finest notion of strong type for which the corresponding quotient is a compact Hausdorff space when it is equipped with the so-called logic topology; finally, a Lascar strong type is simply a class of the smallest equivalence relation ls, which is bounded and invariant under automorphisms.
For some years, it was an open problem to find an example of a theory for which the Kim-Pillay and the Lascar strong type do not coincide. After finding such example, it was suggested that Lascar strong types could be described from the point of view of Decriptive Set Theory, in particular of quotient of Polish spaces by Borel equivalence relations. Furthermore, it was conjectured that for every tuple a of the monster model, if [a]ls = [a]kp, then [a]kp restricted to [a]ls is nonsmooth, using the so-called Silver dichotomy.
Despite the Stone space is not always Polish, the authors proved such conjecture distinguishing the case when the theory is countable -in which case for any countable model M of T, S(M) is Polish- and when it is not. My Master’s thesis consists in filling up all the details for both cases, specially for the case when T is countable. Given a complete first order theory T, its monster model C has none topological structure. Nevertheless, given a model M of T, the Stone space S(M) has it, this is, it is well known that it is Hausdorff, that given any formula ', the set of [ ] determines a basis of clopens, etc.