Classical Set Theory Research Articles

Free functor from the category of G-nominal sets to that of 01-G-nominal sets

The category $$\mathbf{Nom}$$ , of all finitely supported G-sets, called G-nominal sets, where $$G=\mathrm{Perm}_{\mathrm{f}}({\mathbb {D}})$$ is the group of all finitary permutations over a countable infinite set $${\mathbb {D}}$$ , is a subject of interest by both set theorists and computer scientists. The category 01-Nom, of all G-nominal sets equipped with the source and the target operations, was introduced by Pitts. He has shown that this category is isomorphic to the category of sets whose elements have a finite support property with respect to an action of the monoid Cb of name substitutions. The latter category is a coreflective subcategory of the category $$\mathbf{Set}^{Cb}$$ , of sets with the action of the monoid Cb. For a functorial relation between the categories $$\mathbf{Nom}$$ and $$\mathbf{Set}^{Cb}$$ , we study the existence of the free objects in the category 01-Nom. More precisely, we construct the left adjoint to the forgetful functor from the category of 01-G-nominal sets to the category of G-nominal sets, where G is a suitable subgroup of the group of all permutations over a countable infinite set $${\mathbb {D}}$$ .

哲学的存在論・宗教的無限論・数学的集合論(宗教における「語りえぬもの」と「示しうるもの」,パネル, 第六十五回学術大会紀要)

哲学的存在論・宗教的無限論・数学的集合論(宗教における「語りえぬもの」と「示しうるもの」,パネル, 第六十五回学術大会紀要)

A complete ranking of trapezoidal fuzzy numbers and its applications to multi-criteria decision making

The problem (or scenario) involving qualitative or imprecise information is not solvable by classical set theory. To overcome the shortcoming of classical set theory, Zadeh (Inf Control 8(3):338–356, 26) introduced the concept of fuzzy sets that generalizes the concept of classical sets. Fuzzy set theory allows modelling and handling of imprecise information in an effective way. As a special class of fuzzy sets, fuzzy numbers (FN) which are very much important in decision making was introduced by Dubois and Prade (Int J Syst Sci 9:631–626, 12). The available methods for solving multi-criteria decision making problems (MCDM) are problem dependent in nature due to the partial ordering on the class of FN. Total ordering on the class of FN by countable number of real-valued parameters was achieved by Wang and Wang (Fuzzy Sets Syst 243:131–141, 21). A complete ranking on the class of trapezoidal fuzzy numbers (TrFNs) using finite number of score functions is achieved in this paper. In this paper, a new ranking procedure (complete) on the class of TrFNs using the concepts of mid-point, radius, left and right fuzziness of TrFN is proposed and further we introduce a method for solving fuzzy multi-criteria decision making (Fuzzy MCDM) problem. Finally, comparisons of our proposed method with familiar existing methods are listed.

Attribute Reduction Based on Consistent Covering Rough Set and Its Application

As an important processing step for rough set theory, attribute reduction aims at eliminating data redundancy and drawing useful information. Covering rough set, as a generalization of classical rough set theory, has attracted wide attention on both theory and application. By using the covering rough set, the process of continuous attribute discretization can be avoided. Firstly, this paper focuses on consistent covering rough set and reviews some basic concepts in consistent covering rough set theory. Then, we establish the model of attribute reduction and elaborate the steps of attribute reduction based on consistent covering rough set. Finally, we apply the studied method to actual lagging data. It can be proved that our method is feasible and the reduction results are recognized by Least Squares Support Vector Machine (LS-SVM) and Relevance Vector Machine (RVM). Furthermore, the recognition results are consistent with the actual test results of a gas well, which verifies the effectiveness and efficiency of the presented method.

The formal-logical analysis of the foundation of set theory

The critical analysis of the foundation of set theory is proposed. The unity of formal logic and rational dialectics is the correct methodological basis of the analysis. The analysis leads to the following results: (1) the mathematical concept of set should be analyzed on the basis of the formal-logical clauses “Definition of concept”, “Logical class”, “Division of concept”, “Basis of division”, “Rules of division”; (2) the standard mathematical theory of sets is an erroneous theory because it does not contain definition of the concept “element (object) of set”; (3) the concept of empty set (class) is a meaningless, erroneous, and inadmissible one because the definition of the concept “empty set (class)” contradicts to the definition of the logical class. (If the set (class) does not contain a single element (object), then there is no feature (sign) of the element (object). This implies that the concept of empty set (class) has no content and volume (scope). Therefore, this concept is inadmissible one); (4) the standard mathematical operations of union, intersection and difference of sets (classes) are meaningless, erroneous and inadmissible operations because they do not satisfy the following formal-logical condition: every separate element (object) of the set (class) must be in only one some set (class) and cannot be in two sets (classes). Thus, the results of formal-logical analysis prove that the standard mathematical theory of sets is an erroneous theory because it does not satisfy the criterion of truth.

Elements of a New Set Theory Based Quantum Mechanics with Applications in High Energy Quantum Physics and Cosmology

We outline the quintessence of a fairly new quantum theory based on transfinite set theory representing a synthesis of the work of J. von Neumann’s pointless continuous geometry, A. Connes’ noncommutative quantum geometry, Menger-Urysohn dimensional theory, Sir R. Penrose’s fractal tiling universe, ‘tHooft’s renormalon as well as D. Shectman quasi crystals and Einstein-Kaluza-Klein spacetime theories. We conclude that the Aether is a real non-materialistic material and that is essentially what we commonly call spacetime which can be accurately described as the empty five dimensional set in a Penrose quasi crystal universe. Seen in the light of the above, empty spacetime can be the source of infinite clean energy.

Principles of inclusion and exclusion for interval-valued fuzzy sets and IF-sets

The principle of inclusion and exclusion is one of basic tools of classical set theory. Many previous papers have presented its generalizations to fuzzy sets or IF-sets (Atanassov's intuitionistic fuzzy sets) with Gödel or product operations. In our previous work we clarified these questions for fuzzy sets with intersections and unions based on continuous triangular norms and conorms. Here we extend these results to interval-valued fuzzy sets and IF-sets.

SPACE PHILOSOPHY

INSPIRED by a dynamist Naturphilosophie and looking for a mathematics of the natura naturans, the founders of modern mathematics in Germany made some lasting contributions in the attempt to go beyond perceptible space (“Anschauungsraum”). Hermann Grassmann’s extension theory, Johann Benedict Listing’s topology, Bernhard Riemann’s non-Euclidean manifold theory, Carl Gustav Jacob Jacobi’s approach to non-mechanistic theory and last but not least Georg Cantor’s transfinite set theory were all influenced by the tradition of Naturphilosophie. One central motivation for the new mathematics was to decipher the so-called “inner side of nature” which was thought to be invisible but nonetheless physical. As a result, mathematics was understood not only in terms of quantity but of morphogenetic quality and as a tool to recognize the intrinsic possibilities of structuring and organizing nature. This essay focuses on the philosophical viewpoints of Christian Samuel Weiss, Justus and Hermann Grassmann, Bernhard Riemann and their reception of Schelling’s philosophy. Riemann deconstructed empiricist theories of space so as to glean the fundamental constitution of space. One result was that a given, fixed and static metric is not necessary for the understanding of space. The underlying topological substratum (Hans Freudenthal’s term) has no specific metric but is capable of exhibiting any possible metric. Riemann’s claims are compared with Schelling’s theory of space in his “Darstellung des Naturprocesses” (1843–44). My thesis is that Schelling’s philosophy of the emergence of space can be read in Riemannian terms and that it is an interesting model in the context of modern questions.

Reducing the Cold-Start Problem by Explicit Information with Mathematical Set Theory in Recommendation Systems

The explosive growth of enormous information makes our world as a global village. Recommendation systems are widely used in many different kinds of commercial web sites. A key challenge is how to provide recommendations when historical data for a user is missing and mining relationship between user and recommender system, a problem generally known as cold-start recommendation. This issue is a well-known problem for recommender systems and much research has been done to find the best way to overcome this. The proposed idea is querying user through an initial interview to get explicitly information, this process proposed as new user preferences to make user profile. In this paper, we present mathematical set theory to solve cold-start recommendation problem. Experiment will perform on Movie-lens dataset and user preferences can be formed into mathematical set. The combinations of multiple preferences represent as subsets and the largest subset will be the first choice as input for the recommender system. User Item matrix strategy will be used to get predicted rating for cold users. Experimental results on mathematical set approach for recommendation on the Movie-lens dataset demonstrate that the proposed approach significantly outperforms existing methods for cold-start recommendation.

Maximality Principles in Set Theory

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification of maximality principles.

Towards a descriptive theory of cb0-spaces

The paper tries to extend some results of the classical Descriptive Set Theory to as many countably basedT0-spaces (cb0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we also consider the more general case ofk-partitions. In particular, we investigate the difference hierarchy ofk-partitions and the fine hierarchy closely related to the Wadge hierarchy.

An intuitionistic fuzzy graded covering rough set

Exploring rough sets from the perspective of covering represents a promising direction in rough set theory, where concepts are approximated by substituting of an equivalent relation in classical rough set theory with a covering in covering-based rough set theory. By combining intuitionistic fuzzy (IF) β-neighborhoods induced by an IF β-covering with IF rough sets, this study develops a new rough set model, which is a generalization of the β-neighborhood fuzzy covering rough sets and IF rough sets. First, we present the concepts of IF graded covering and IF graded neighborhood, namely, IF β-covering and IF β-neighborhood, respectively. We propose an IF graded approximation space on the basis of IF graded neighborhood and discuss its uncertainty measures, namely, information entropy and rough entropy. Second, we generalize an IF rough set based on IF relation to one based on the presented IF graded neighborhood and use the distance between two IF sets to characterize the latter. Third, we present the matrix computation method for the upper and lower approximations of the presented IF rough sets based on the IF graded neighborhood. Fourth, from the multi-granulation perspective, we examine the IF graded approximation space, uncertainty measures, IF rough sets, and computation methods for its reducts. Finally, we discuss several generalizations similar to the presented IF covering rough sets.

Demystifying Ambiguity in Legislative Writing

Ambiguity in legislative writing causes confusion and misunderstanding. Sometimes, the ambiguity comes from conflicts between the drafter’s intended relationship of words and the reader’s perceived relationship of those words. At other times, ambiguity comes from the drafter’s failure to define important terms clearly. This paper (i) examines reasons for ambiguity in legislative writing as well as some of the problems that can be created by that ambiguity and (ii) offers strategies for avoiding ambiguity while drafting legislation, including the use of mathematical set theory to examine potentially ambiguous sentences.

A Survey on Hesitation Information Mining

tremendous advancement in technology has given rise to an increasing requirement for the storage of data in files, databases and other data repositories. As a result, decision- makers are required to use new and powerful automated tools for the purpose of the analysis and interpretation of the stored data, as well as for the extraction of interesting patterns in data. Since the stored data is not always exact and precise, some means are required to handle this aspect of data and extract the useful information (e.g. hesitation information) arising from such uncertainties. For this purpose vague set theory has been applied for efficiently modeling the uncertainties that occur in datasets. Vague sets are an extension of the classical set theory. They extend the application of set theory to vague and uncertain problems. Vague set theory has emerged as a new tool to deal with the uncertainties of the data and the parameters attached to the data. This theory exhibits a very promising approach to analyzing uncertain data and deriving some interesting results suitable for use in various applications. This paper discusses the notions of vague sets and vague association rules. Also, optimization techniques have been introduced that would be helpful in optimizing the outcomes. When both are implemented together, a new approach will be created which is expected to generate much improved results. This paper emphasizes on the study of vague sets for a series of applications especially in decision making problems.

A space of lexicographic preferences

There are many lexicographic probability systems (LPS’s) that represent the same lexicographic expected utility (LEU) preference relation (Blume et al., 1991). The space of all LPS’s on a Polish space therefore contains redundant representations of preferences. We show that there exists a Polish subspace of LPS’s that represents all LEU preference relations without such redundancies. Our approach is novel in that it frames the question as what is called a Borel section problem in classical descriptive set theory. The results are motivated by conceptual issues relevant to applications in epistemic game theory.

A new non-specificity measure in evidence theory based on belief intervals

In the theory of belief functions, the measure of uncertainty is an important concept, which is used for representing some types of uncertainty incorporated in bodies of evidence such as the discord and the non-specificity. For the non-specificity part, some traditional measures use for reference the Hartley measure in classical set theory; other traditional measures use the simple and heuristic function for joint use of mass assignments and the cardinality of focal elements. In this paper, a new non-specificity measure is proposed using lengths of belief intervals, which represent the degree of imprecision. Therefore, it has more intuitive physical meaning. It can be proved that our new measure can be rewritten in a general form for the non-specificity. Our new measure is also proved to be a strict non-specificity measure with some desired properties. Numerical examples, simulations, the related analyses and proofs are provided to show the characteristics and good properties of the new non-specificity definition. An example of an application of the new non-specificity measure is also presented.

Using qualitative comparative analysis in a systematic review of a complex intervention.

BackgroundSystematic reviews evaluating complex interventions often encounter substantial clinical heterogeneity in intervention components and implementation features making synthesis challenging. Qualitative comparative analysis (QCA) is a non-probabilistic method that uses mathematical set theory to study complex phenomena; it has been proposed as a potential method to complement traditional evidence synthesis in reviews of complex interventions to identify key intervention components or implementation features that might explain effectiveness or ineffectiveness. The objective of this study was to describe our approach in detail and examine the suitability of using QCA within the context of a systematic review.MethodsWe used data from a completed systematic review of behavioral interventions to improve medication adherence to conduct two substantive analyses using QCA. The first analysis sought to identify combinations of nine behavior change techniques/components (BCTs) found among effective interventions, and the second analysis sought to identify combinations of five implementation features (e.g., agent, target, mode, time span, exposure) found among effective interventions. For each substantive analysis, we reframed the review’s research questions to be designed for use with QCA, calibrated sets (i.e., transformed raw data into data used in analysis), and identified the necessary and/or sufficient combinations of BCTs and implementation features found in effective interventions.ResultsOur application of QCA for each substantive analysis is described in detail. We extended the original review findings by identifying seven combinations of BCTs and four combinations of implementation features that were sufficient for improving adherence. We found reasonable alignment between several systematic review steps and processes used in QCA except that typical approaches to study abstraction for some intervention components and features did not support a robust calibration for QCA.ConclusionsQCA was suitable for use within a systematic review of medication adherence interventions and offered insights beyond the single dimension stratifications used in the original completed review. Future prospective use of QCA during a review is needed to determine the optimal way to efficiently integrate QCA into existing approaches to evidence synthesis of complex interventions.Electronic supplementary materialThe online version of this article (doi:10.1186/s13643-016-0256-y) contains supplementary material, which is available to authorized users.

Heideggerian Mathematics

This paper provides a historical redescription and reinterpretation of Alain Badiou’s major work, Being and Event. The work is approached historically, as a text that uses Heideggerian metaphysics to perform an allegorical exegesis of mathematical set theory and does so as a means of fashioning a supremacist spiritual pedagogy for a philosophical elite in the context of a national intellectual subculture.

Extended rough set model based on known same probability dominant valued tolerance relation

Classical rough set theory is based on the conventional indiscernibility relation. It is not suitable for analyzing incomplete information. Some successful extended rough set models based on different non-equivalence relations have been proposed. The data-driven valued tolerance relation is such a non-equivalence relation. However, the calculation method of tolerance degree has some limitations. In this paper, known same probability dominant valued tolerance relation is proposed to solve this problem. On this basis, an extended rough set model based on known same probability dominant valued tolerance relation is presented. Some properties of the new model are analyzed. In order to compare the classification performance of different generalized indiscernibility relations, based on the category utility function in cluster analysis, an incomplete category utility function is proposed, which can measure the classification performance of different generalized indiscernibility relations effectively. Experimental results show that the known same probability dominant valued tolerance relation can get better classification results than other generalized indiscernibility relations.

A roughness measure of fuzzy sets from the perspective of distance

Covering generalized rough set theory is an important extension of classical rough set theory. To characterize a fuzzy set in a given covering approximation space, a pair of fuzzy sets, called covering rough fuzzy lower and upper approximations, were introduced, but they do not describe well how much uncertainty is induced by the granularity of knowledge. In this paper, we first discuss the relationship between uncertainty and granularity of knowledge. Then we examine several commonly used distance measures, and indicate that some of them exhibit some limitations. Next we propose a roughness measure based on Minkowski distance, and examine some important properties of this measure. Finally, an illustrative example is provided to demonstrate the application of the roughness measure to incomplete information systems with fuzzy decision.