Related papers: Set Theory and the Analyst
Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made…
Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove…
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and…
We provide an analytical argument for understanding the likely nature of parameter shifts between those coming from an analysis of a dataset and from a subset of that dataset, assuming differences are down to noise and any intrinsic…
This is a kind of introduction to some basic topics in analysis, some of which would be covered in standard graduate courses, and some not. However, an important difference is that not much in the way of prerequisites are needed, beyond…
The paper is a naive introduction to descriptive set theory. It is aimed mathematicians without a background in logic. The goal is to provide the basic facts used for applications of descriptive set theory to other areas of mathematics,…
Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…
Similarity measures play a central role in various data science application domains for a wide assortment of tasks. This guide describes a comprehensive set of prevalent similarity measures to serve both non-experts and professional.…
The theme here is category-measure duality, in the context of a topological group. One can often handle the (Baire) category case and the (Lebesgue, or Haar) measure cases together, by working bi-topologically: switching between the…
We define a notion which contains numerous basic notions of Analysis as special cases, for example limit, continuity, differential, Riemann and Lebesgue integral, root and exponential functions. Properties like additivity or linearity of…
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a…
Journal classification into subject categories is an important aspect in scholarly research evaluation as well as in bibliometric analysis. However, this classification is not standardized, resulting in several different journal subject…
Let X be a non-empty set and U a ring of subsets of X. The countable additive functions U->{0,1} are called measures. The paper gives some definitions (derivable measures, the Lebesgue-Stieltjes measures) and properties of these functions,…
The variant of calculation of functions of set and their application is offered. In particular: the new measure of system of sets generalizing classical concept of a measure is entered; the variation of set that has allowed to construct a…
The purpose of this paper is to give a survey on the notions of distance between subsets either of a metric space or of a measure space, including definitions, a classification, and a discussion of the best-known distance functions, which…
The analysis of data sometimes requires fitting many free parameters in a theory to a large number of data points. Questions naturally arise about the compatibility of specific subsets of the data, such as those from a particular experiment…
We investigate the structure common to causal theories that attempt to explain a (part of) the world. Causality implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalizing power of the…
The purpose of this book is to lay out certain aspects of descriptive set theory. After initially establishing notation and generalities we proceed to the following topics: partitions, semirings, rings, $\sigma$-rings, $\delta$-rings,…
The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In…
Evaluation metrics are a key ingredient for progress of text generation systems. In recent years, several BERT-based evaluation metrics have been proposed (including BERTScore, MoverScore, BLEURT, etc.) which correlate much better with…