Equality-free saturated models

Saturated models are a powerful tool in model theory. The properties of universality and homogeneity of the saturated models of a theory are useful for proving facts about this theory. They are used in the proof of interpolation and preservation theorems and also as work-spaces. Sometimes we work with models which are saturated only for some sets of formulas, for example, recursively saturated models, in the study of models of arithmetic or atomic compact, in model theory of modules. In this article we introduce the notion of equality-free saturated model, that is, roughly speaking, a model which is saturated for the set of equality-free formulas. Our aim is to understand better the role that identity plays in classical model theory, in particular with regard to this process of saturation. Given an infinite cardinal κ, we say that a model is equality-free κsaturated if it satisfies all the 1-types over sets of parameters of power less than κ, with all the formulas in the type that are equality-free. We compare this notion with the usual notion of κ-saturated model. We prove the existence of infinite models A, which are L −-|A | +-saturated. Fro

Grades of Discrimination: Indiscernibility, Symmetry, and Relativity

There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimi- nation have recently been the subject of much philosophical and technical dis- cussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formu- las. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some detail. However, this paper also introduces grades of relativity, defined in terms of rela- tiveness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas, and draws con- nections with definability theory.This is the author accepted manuscript. It is currently under indefinite embargo pending publication by Duke University Press

Contribucions a la teoria de models de la lògica sense identitat

[spa] La tesis doctoral es un estudio de la teoria de modelos de la lógica sin identidad. Se estudia el fragmento de la lógica de primer orden compuesto por las fórmulas que no tienen el símbolo de identidad. Los conceptos fundamentales estudiados son el de "Congruencia de Leibniz" y el de "Relación de parentesco (Relative Relation)". El interés actual de estas nociones procede de los trabajos de W. Blok y de D. Pigozzi. Hemos estudiado esta lógica desde el punto de vista de la teoria de modelos clásica, desarrollando técnicas usuales en teoria de modelos: Metodo de los diagramas, sistemas de Back-and-Forth, etc. con el fin de obtener caracterizaciones algebraicas de la equivalencia elemental en esta lógica y teoremas de preservación. Una de las contribuciones más importantes de este trabajo es la caracterización de los enunciados de primer orden que son lógicamente equivalentes a un enunciado sin identidad. Hemos introducido las nociones de modelo saturado, universal y homogéneo sin identidad. Hemos estudiado sus propiedades y las hemos comparado con las de las nociones análogas en lógica de primer orden con identidad. Finalmente hemos estudiado el fragmento universal de Horn sin identidad de los lenguajes infinitarios, con Y cardinales infinitos regulares. Hemos obtenido resultados de caracterización y de preservación. Usando estos resultados hemos demostrado teoremas de interpolación y definibilidad para este fragmento