Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar

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 m...

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Autor Principal: Buitrago Díaz, Esperanza
Formato: Trabajo de grado (Bachelor Thesis)
Lenguaje:Desconocido (Unknown)
Publicado: 2014
Materias:
Acceso en línea:http://babel.banrepcultural.org/cdm/ref/collection/p17054coll23/id/322
id ir-p17054coll23-322
recordtype dspace
spelling ir-p17054coll23-3222020-09-28 Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar Buitrago Díaz, Esperanza 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. Borel cardinality; Cardinalidad de borel; Espacio de stone; Espacios topológicos; Kim-Pillay strong types; Kim-pillay types; Lascar strong types; Lascar types; Stone space; Strong types; Tipos fuertes; Topological spaces; Tipos y tipos fuertes de lascar y de Kim Pilay Ciencias naturales y matemáticas; Ciencias naturales y matemáticas / Matemáticas 2014 PDF Tesis ENG - Inglés Colfuturo © Derechos reservados del autor http://babel.banrepcultural.org/cdm/ref/collection/p17054coll23/id/322
institution Biblioteca Virtual Banco de la República - Colecciones digitales
collection Custom
language Desconocido (Unknown)
topic Borel cardinality; Cardinalidad de borel; Espacio de stone; Espacios topológicos; Kim-Pillay strong types; Kim-pillay types; Lascar strong types; Lascar types; Stone space; Strong types; Tipos fuertes; Topological spaces; Tipos y tipos fuertes de lascar y de Kim Pilay
Ciencias naturales y matemáticas; Ciencias naturales y matemáticas / Matemáticas
spellingShingle Borel cardinality; Cardinalidad de borel; Espacio de stone; Espacios topológicos; Kim-Pillay strong types; Kim-pillay types; Lascar strong types; Lascar types; Stone space; Strong types; Tipos fuertes; Topological spaces; Tipos y tipos fuertes de lascar y de Kim Pilay
Ciencias naturales y matemáticas; Ciencias naturales y matemáticas / Matemáticas
Buitrago Díaz, Esperanza
Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar
description 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.
format Trabajo de grado (Bachelor Thesis)
author Buitrago Díaz, Esperanza
author_facet Buitrago Díaz, Esperanza
author_sort Buitrago Díaz, Esperanza
title Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar
title_short Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar
title_full Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar
title_fullStr Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar
title_full_unstemmed Borel Cardinality of Lascar Strong Types = Cardinalidad de Borel de los tipos fuertes de Lascar
title_sort borel cardinality of lascar strong types = cardinalidad de borel de los tipos fuertes de lascar
publishDate 2014
url http://babel.banrepcultural.org/cdm/ref/collection/p17054coll23/id/322
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