Related papers: Set Theory and the Analyst
The paper provides an introduction to the field of Algebraic Set Theory (AST). AST is a flexible categorical framework for studying different kinds of set theories: both classical and constructive, predicative and impredicative. We discuss…
Attribute weighting and differential weighting, two major mechanisms for computing context-dependent similarity or dissimilarity measures are studied and compared. A dissimilarity measure based on subset size in the context is proposed and…
Comparing the top $k$ elements between two or more ranked results is a common task in many contexts and settings. A few measures have been proposed to compare top $k$ lists with attractive mathematical properties, but they face a number of…
This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems…
This is a non-standard paper, containing some problems, mainly in model theory, which I have, in various degrees, been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me,…
This report consists of two parts. The first part is a brief exposition of classical descriptive set theory. This part introduces some fundamental concepts, motivations and results from the classical theory and ends with a section on the…
The central focus is on clarifying the distinction between sets and proper classes. To this end we identify several categories of concepts (surveyable, definite, indefinite), and we attribute the classical set theoretic paradoxes to a…
Computational measures of semantic similarity between geographic terms provide valuable support across geographic information retrieval, data mining, and information integration. To date, a wide variety of approaches to geo-semantic…
Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a…
It is shown to be consistent with set theory that the uniformity invariant for Lebesgue measure is strictly greater than the corresponding invariant for Hausdorff r-dimensional measure where 0<r<1.
L^p spaces of mappings taking values in arbitrary metric spaces, which we call nonlinear Lebesgue spaces, play an important role in several fields of mathematics. For instance, membership in these spaces is typically required for transport…
Finding all the mutually unbiased bases in various dimensions is a problem of fundamental interest in quantum information theory and pure mathematics. The general problem formulated in finite-dimensional Hilbert spaces is open. In the…
Similarity is a core notion that is used in psychology and two branches of linguistics: theoretical and computational. The similarity datasets that come from the two fields differ in design: psychological datasets are focused around a…
The main aim of this paper is to make a remark about the relation between (i) dualities between theories, as `duality' is understood in physics and (ii) equivalence of theories, as `equivalence' is understood in logic and philosophy. The…
This paper argues that mathematical objects are constructions and that constructions introduce a flexibility in the ways that mathematical objects are represented (as sets of binary sequences for example) and presented (in a particular…
A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer…
Both algebraic and computational approaches for dealing with similarity spaces are well known in generalized rough set theory. However, these studies may be said to have been confined to particular perspectives of distinguishability in the…
This article focuses on the importance of the precise calculation of similarity factors between papers and reviewers for performing a fair and accurate automatic assignment of reviewers to papers. It suggests that papers and reviewers'…
Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice DC rather than the…
In this work we suggest the use of a set-theoretical interpretation of semantic tableaux for teaching propositional logic. If the student has previous notions of basic set theory, this approach to semantical tableaux can clarify her the way…