Sesión Lógica y ComputabilidadMetric Structures and the Keisler-Shelah Theorem
Pedro Reis
Instituto de Matemática, Estatística e Ciência da Computação da Universidade de São Paulo (IME - USP), Brasil - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
In the 2000s, a continuous model theory based on metric structures emerged. This framework proves to be fruitful for geometric and analytic theories, such as Banach lattices and \(C^\ast\)-algebras, and, more broadly, within the realm of functional analysis. Unlike in classical model theory, formulae are interpreted in complete and bounded metric spaces, incorporating the metric as a binary relation that replaces equality in classical logic. From this perspective, classical model theory can be viewed as a restriction to structures with the discrete metric. In this presentation, we proffer a comprehensive introduction to the theory, culminating in some fundamental results. The main objective is to present the Keisler-Shelah theorem, which guarantees that elementarily equivalent structures have isomorphic ultrapowers. In the absence of a self-contained proof of this pivotal theorem in the literature, we provide such a demonstration within this framework. To achieve this, we follow Shelah’s construction for the classical case, making the requisite adaptations and appropriating machinery from transfinite set theory to inductively construct an ultrafilter that witnesses the existence of isomorphic ultrapowers. This communication is based on the thesis [1] presented by the author in his Master’s studies and was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.
Trabajo en conjunto con: Ricardo Bianconi (Universidade de São Paulo, Brasil).
Referencias
[1] Pedro L. S. Reis. Metric Structures and the Keisler-Shelah Theorem. 95 p. Thesis (Masters in Mathematics) – Instituto de Matemática, Estatística e Ciência da Computação, Universidade de São Paulo, São Paulo, 2026. Not Published.
[2] Itaı̈ Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov. “Model theory for metric structures”. In: Model Theory with Applications to Algebra and Analysis, vol. II. Ed. by Zoé Chatzidakis et al. London Mathematical Society Lecture Note Series 350. Cambridge University Press, 2008, pp. 315–427
[3] Saharon Shelah. “Every two elementarily equivalent models have isomorphic ultrapowers”. Israel Journal of Mathematics 10, 1971, pp. 224–233.
[4] C. Ward Henson and José Iovino. “Ultraproducts in analysis”. In: Analysis and Logic. Ed. by Catherine Finet and Christian Michaux. London Mathematical Society Lecture Note Series. Cambridge University Press, 2002, pp. 1–114