On arbitrary sets and ZFC

Set theory deals with the most fundamental existence questions in mathematics– questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximalist. After explaining what is meant by definability and by “arbitrariness”, a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds too fferan elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.Junta de Andalucía P07-HUM-02594Ministerio de Ciencia y Tecnología BFF2003-09579-C0

Quantitative Concept Analysis

Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the second, and vice versa. Many applications, though, provide contexts with quantitative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel intensity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially ordered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches.Comment: 16 pages, 3 figures, ICFCA 201

Linear logic for constructive mathematics

We show that numerous distinctive concepts of constructive mathematics arise automatically from an interpretation of "linear higher-order logic" into intuitionistic higher-order logic via a Chu construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of classical concepts using the choice between multiplicative and additive linear connectives. Linear logic thus systematically "constructivizes" classical definitions and deals automatically with the resulting bookkeeping, and could potentially be used directly as a basis for constructive mathematics in place of intuitionistic logic.Comment: 39 page

Probabilistic logics based on Riesz spaces

We introduce a novel real-valued endogenous logic for expressing properties of probabilistic transition systems called Riesz modal logic. The design of the syntax and semantics of this logic is directly inspired by the theory of Riesz spaces, a mature field of mathematics at the intersection of universal algebra and functional analysis. By using powerful results from this theory, we develop the duality theory of the Riesz modal logic in the form of an algebra-to-coalgebra correspondence. This has a number of consequences including: a sound and complete axiomatization, the proof that the logic characterizes probabilistic bisimulation and other convenient results such as completion theorems. This work is intended to be the basis for subsequent research on extensions of Riesz modal logic with fixed-point operators

L-Fuzzy Relations in Coq

Heyting categories, a variant of Dedekind categories, and Arrow categories provide a convenient framework for expressing and reasoning about fuzzy relations and programs based on those methods. In this thesis we present an implementation of Heyting and arrow categories suitable for reasoning and program execution using Coq, an interactive theorem prover based on Higher-Order Logic (HOL) with dependent types. This implementation can be used to specify and develop correct software based on L-fuzzy relations such as fuzzy controllers. We give an overview of lattices, L-fuzzy relations, category theory and dependent type theory before describing our implementation. In addition, we provide examples of program executions based on our framework

A Semi-Constructive Approach to the Hyperreal Line

Using an alternative to Tarskian semantics for first-order logic known as possibility semantics, I introduce an approach to nonstandard analysis that remains within the bounds of semiconstructive mathematics, i.e., does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. I define the Fr´echet hyperreal line †R as a possibility structure and show that it shares many fundamental properties of the classical hyperreal line, such as a Transfer Principle and a Saturation Principle. I discuss the technical advantages of †R over some other alternative approaches to nonstandard analysis and argue that it is well-suited to address some of the philosophical and methodological concerns that have been raised against the application of nonstandard methods to ordinary mathematics

Neutrosophic Rings

This book has four chapters. Chapter one is introductory in nature, for it recalls some basic definitions essential to make the book a self-contained one. Chapter two, introduces for the first time the new notion of neutrosophic rings and some special neutrosophic rings like neutrosophic ring of matrix and neutrosophic polynomial rings. Chapter three gives some new classes of neutrosophic rings like group neutrosophic rings,neutrosophic group neutrosophic rings, semigroup neutrosophic rings, S-semigroup neutrosophic rings which can be realized as a type of extension of group rings or generalization of group rings. Study of these structures will throw light on the research on the algebraic structure of group rings. Chapter four is entirely devoted to the problems on this new topic, which is an added attraction to researchers. A salient feature of this book is that it gives 246 problems in Chapter four. Some of the problems are direct and simple, some little difficult and some can be taken up as a research problem.Comment: 154 page

Rational and Delta expansions of the Nilpotent Minimum Logic

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona. Curs: 202x-202x. Tutor: xxx[eng] The aim of this thesis is to study some expansions of the Nilpotent minimum logic (denoted by NML), focusing on their lattices of axiomatic and finitary extensions and, additionally, exploring the structural completeness of these logics, alongside their variants (active structural completeness, passive structural completeness, ... ). The project includes research about the rational Nilpotent minimum logic (designated by RNML), which is obtained by adding rational constants to the language of NML. Moreover, we also study the Δ-core fuzzy logic obtained by expanding the language of NML with the Baaz Delta connective and examine the impact of the incorporation of rational constants to the language of this logic (which is equivalent to the addition of the Baaz Delta connective to RNML). The thesis culminates with the corresponding analysis of an extension of the later logic which is obtained by introducing bookkeeping axioms for the Δ operator, motivated by the aim for the algebra of constants to form a subalgebra. In the project, through comparative analysis, the differences and similarities between the lattices of axiomatic and finitary extensions among the previously mentioned expansions are evaluated, as well as how the structural completeness results obtained may vary from one logic to another.[spa] El objetivo de esta tesis es estudiar algunas expansiones de la lógica del Nilpotente mínimo (denotada por NML), centrándonos en sus retículos de extensiones axiomáticas y finitas y, además, explorando la completitud estructural de estas lógicas, junto con sus variantes (completitud estructural activa, completitud estructural pasiva, ...). El proyecto abarca la lógica racional del Nilpotente mínimo (designada por RNML), que se obtiene añadiendo constantes racionales al lenguaje de NML. También se estudia la lógica fuzzy Δ-core obtenida mediante la expansión del lenguaje de NML con el operador Delta de Baaz, y se examina el impacto de la incorporaci´on de constantes racionales al lenguaje de esta lógica (lo que equivale a añadir el operador Delta de Baaz a RNML). La tesis culmina con el correspondiente análisis de una extensión de la ´ultima lógica presentada, resultante de la introducción de bookkeeping axioms para el operador Δ, motivada por el objetivo de que el álgebra de constantes forme una subálgebra. En el proyecto, a través de un análisis comparativo, se evalúan las diferencias y similitudes entre los retículos de extensiones axiomáticas y finitas de las distintas expansiones mencionadas anteriormente, así como la forma en que varían los resultados de completitud estructural de una lógica a otra.[cat] L’objectiu d’aquesta tesi és estudiar algunes expansions de la lògica del Nilpotent mínim (denotada per NML), centrant-nos en els seus reticles d’extensions axiomàtiques i finites i, a més, explorant la completitud estructural d’aquestes lògiques, juntament amb les seves variants (completitud estructural activa, completitud estructural passiva, ...). El projecte abasta la lògica racional del Nilpotent mínim (designada per RNML), que s’obté afegint constants racionals al llenguatge de NML. També s’estudia la lògica fuzzy Δ-core obtinguda mitjançant l’expansió del llenguatge de NML amb l’operador Delta de Baaz, i s’examina l’impacte de la incorporació de constants racionals al llenguatge d’aquesta lògica (el que equival a afegir l’operador Delta de Baaz a RNML). La tesi culmina amb l’anàlisi corresponent d’una extensió de l'última lògica presentada, que resulta de la introducció de bookkeeping axioms per a l’operador Δ, motivada per l’objectiu que l'àlgebra de constants formi una subàlgebra. En el projecte, mitjançant una anàlisi comparativa, s’avaluen les diferències i similituds entre els reticles d’extensions axiomàtiques i finites de les diferents expansions esmentades anteriorment, així com la manera com varien els resultats de completitud estructural d’una lògica a una altra

Fibred contextual quantum physics

Inspired by the recast of the quantum mechanics in a toposical framework, we develop a contextual quantum mechanics via the geometric mathematics to propose a quantum contextuality adaptable in every topos. The contextuality adopted corresponds to the belief that the quantum world must only be seen from the classical viewpoints à la Bohr consequently putting forth the notion of a context, while retaining a realist understanding. Mathematically, the cardinal object is a spectral Stone bundle Σ → B (between stably-compact locales) permitting a treatment of the kinematics, fibre by fibre and fully point-free. In leading naturally to a new notion of points, the geometricity permits to understand those of the base space B as the contexts C — the commutative C*–algebras of a incommutative C*–algebras — and those of the spectral locale Σ as the couples (C, ψ), with ψ a state of the system from the perspective of such a C. The contexts are furnished with a natural order, the aggregation order which is installed as the specialization on B and Σ thanks to (one part of) the Priestley's duality adapted geometrically as well as to the effectuality of the lax descent of the Stone bundles along the perfect maps