Seminario Mundo/Lógica/Modelos: 21 de mayo. Gustavo CIPAGAUTA - ¿Teoría de la estabilidad para la lógica de la probabilidad?

[Andrés Villaveces]

Continuamos este próximo martes 21 de mayo con nuestro Seminario Mundo/Lógica/Modelos - ver además abajo la programación de charlas de nuestro seminario.

Martes 21 de mayo, 16:15 a 17:45

Google Meet Link: http://meet.google.com/iat-bdgh-qkm

Gustavo CIPAGAUTA - Universidad Nacional de Colombia

¿Teoría de la estabilidad para la lógica de la probabilidad?

Resumen: en esta lección investigamos las álgebras de Lindenbaum de la teoría de la aleatoriedad (espacios de probabilidad y procesos estocásticos) y los tipos de esta; además se explora la topología de sus espacios de tipos así como la posibilidad de derivar teoremas de omisión de tipos y de Ryll–Nardzewski en el contexto de la lógica de la probabilidad y sus extensiones. El contraste con los correspondientes teoremas para la lógica clásica de primer orden en sus versiones discreta y continua puede revelar interesantes diferencias entre los objetos algebraicos/conjuntistas, las estructuras métricas y los procesos estocásticos.

Más información: https://sites.google.com/view/bogotalogica/seminars/2024-i/seminario-mundo-l%C3%B3gica-modelos?authuser=0

[Andrés Villaveces]

Continuamos este próximo martes 28 de mayo con nuestro Seminario Mundo/Lógica/Modelos - ver además abajo la programación de charlas de nuestro seminario.

Martes 28 de mayo, 16:15 a 17:45

Google Meet Link: http://meet.google.com/iat-bdgh-qkm

Nicolás CUERVO - Universidad de los Andes (Bogotá)

The SB-Property on Randomizations

Abstract: We say that a complete theory T has the Schröder-Bernstein property, or simply, the SB-property, if for any two models M, N of T that are elementarily bi-embeddable, i.e, there exists elementary embeddings φ: M → N and ψ: N → M, we have that M and N are isomorphic.

One motivation for studying the SB-property is that if T has this property, then T would be a theory for which we have a “good understanding” of its models, in terms that they are classified by some reasonable collection of invariants. For example, by Morley’s theorem if T is countable and uncountably categorical then the models of T are classified by a single invariant cardinal number that is preserved by elementary embeddings, so T would have SB-property; however SB-property is a weaker condition than uncountable categoricity, for example the theory of an infinite set with a predicate which is infinite and co-infinite has the SB-property and it is not uncountably categorical.

The purpose of this talk is to study the Schröder-Bernstein property in the continuous context for Randomizations. Informally, a randomization of a first order (discrete) model M is a two sorted metric structure which consists of a sort of events and a sort of functions with values on the model, usually understood as random variables. More generally, given a complete first order theory T, there is a complete continuous theory known as randomized theory, T_R, which is the common theory of all randomizations of models of T. Randomizations were introduced first by Keisler and then axiomatized in the continuous setting by Ben Yaacov and Keisler in [4]. Since randomizations were introduced, many authors focused on examining which model theoretic properties of T are preserved on T_R, for example, it was shown that properties like ω-categoricity, stability and dependence are preserved. Similarly, the existence of prime models is preserved by randomization but notions like minimal models are not preserved. Following these ideas, we prove that a first order theory T with ≤ ω countable models has the SB-property for countable models if and only if T_R has the SB-property for separable randomizations. This is joint work with Alexander Berenstein and Camilo Argoty.