Classical Set Theory Research Articles

Petrification in Contemporary Set Theory: The Multiverse and the Later Wittgenstein

Abstract This paper has two aims. First, we argue that Wittgenstein’s notion of petrification can be used to explain phenomena in advanced mathematics, sometimes better than more popular views on mathematics, such as formalism, even though petrification usually suffers from a diet of examples of a very basic nature (in particular a focus on addition of small numbers). Second, we analyse current disagreements on the absolute undecidability of CH under the notion of petrification and hinge epistemology. We argue that in contemporary set theory the usage of construction techniques for set-theoretic models in which the Continuum Hypothesis holds and those in which it fails have petrified into the normative demand that CH remain undecidable. That is, the continuous and successful practices involving the construction of various set-theoretic models now act as a normative hinge shared among practitioners, i.e., have normative force in the discipline. However, not all hinges are universal, which is why we find disagreements in set theory. We will show that this is a refinement of, and partially conflicts with, the arguments presented by set theorist Joel David Hamkins.

Application of Fuzzy Mamdani Method to Predict the Number of Blood Bags Based on Demand and Supply Data Using Matlab

Fuzzy logic is a control system technique in solving problems and is applied to systems, from basic systems to difficult or complex systems. Fuzzy logic is the proper method to plan an input space into an output space using MATLAB's mathematical theory of fuzzy sets. The reason for using fuzzy logic is because it is related to uncertainty. The unstable demand for blood bags in hospitals makes the supply of blood bags excessive or lacking from demand. The lack of blood supply results in the unfulfilled demand for blood needed by the hospital, while the excess blood supply worsens the quality of blood. In this study, we will predict the number of bags produced using the Mamdani Fuzzy Inference System (FIS) method based on the minimum demand and maximum demand values and the minimum supply and maximum supply that produce output from the defuzzification process. Applying the Mamdani Fuzzy Inference System (FIS) method based on demand and supply data obtains optimal output with MATLAB in predicting the number of blood bags produced. The results of the study showed that the Mean Absolute Percentage Error (MAPE) fuzzy logic Mamdani error value was 24%, the accuracy value of the Fuzzy Inference System (FIS) Mamdani in determining the number of blood bag production was 76%, and the production output generated through the Fuzzy Inference System (FIS) Mamdani was 4,774 blood bags. The number of blood requests at the hospital is 4,443 blood bags, so the amount of blood that must be produced to meet the hospital's demand is 4,774 bags.

Unveiling Uncertainty: Exploring the Potential of Neutrosophic Statistics for Artificial Intelligence

Artificial intelligence (AI) has revolutionized various fields, but its effectiveness can hampered by limitations in handling uncertainty inherent in real-world data. Traditional statistical methods often rely on binary logic (true/false) and may not adequately capture the nuances of uncertainty present in real-world data. This paper explores neutrosophic statistics (NS) as a novel approach to address uncertainty in AI applications. NS, introduced by Smarandache (1998), generalizes classical set theory by incorporating three truth-values: truth (T), indeterminacy (I), and falsity (F). This allows for a more nuanced representation of uncertainty compared to traditional binary methods. We discuss the limitations of classical statistics and how NS can overcome them. The paper explores the potential of NS for AI in areas like machine learning, data fusion, and Explainable AI (XAI). We propose research questions and methodology to investigate the effectiveness of NS integration with AI frameworks. The expected outcome is to contribute to the development of more robust and reliable AI systems by demonstrating the effectiveness of NS in handling uncertainty. Finally, we discuss future research directions to further explore the potential of NS for AI.

Optimizing decision-making with aggregation operators for generalized intuitionistic fuzzy sets and their applications in the tech industry

Intuitionistic fuzzy sets (IFSs) represent a significant advancement in classical fuzzy set (FS) theory. This study advances IFS theory to generalized intuitionistic fuzzy sets (GIFSBs) and introduces novel operators GIFWAA, GIFWGA, GIFOWAA, and GIFOWGA, tailored for GIFSBs. The primary aim is to enhance decision-making capabilities by introducing aggregation operators within the GIFSB framework that align with preferences for optimal outcomes. The article introduces new operators for GIFSBs characterized by attributes like idempotency, boundedness, monotonicity and commutativity, resulting in aggregated values aligned with GIFNs. A comprehensive analysis of the relationships among these operations is conducted, offering a thorough understanding of their applicability. These operators are practically demonstrated in a multiple-criteria decision-making process for evaluating startup success in the Tech Industry.

Eulerian and Hamiltonian Soft Semigraphs

Soft set theory is a mathematical approach to address the challenges of handling vague or uncertain information. It is a more advanced version of classical set theory that deals with imprecise elements and enables the flexible representation of uncertain data. It involves categorizing the elements of the universe based on specific parameters. Semigraph is a generalization of a graph which is different from a hypergraph. A hypergraph extends the concept of a graph by allowing any subset of vertices to form an edge. Semigraphs, on the other hand, distinguish themselves from hypergraphs by imposing a specific order on the vertices within each edge. Soft semigraphs were developed using the principles of soft set theory applied to semigraphs. This study introduces Eulerian and Hamiltonian soft semigraphs. We establish a necessary and sufficient condition for a soft semigraph to be Eulerian, relying on parameters such as [Formula: see text]-part consecutive adjacent degree, [Formula: see text]-part end degree, and the [Formula: see text]-part consecutive adjacency graph. Additionally, we provide the conditions for a soft semigraph to be Hamiltonian. We introduce the concept of maximal non-Hamiltonian [Formula: see text]-part. Finally, we define the closure of a soft semigraph and demonstrate the relationship between a Hamiltonian soft semigraph and its closure.

On the Generalization of Neutrosophic Set Operations: Testing Proofs by Examples

This article aims to present more facts about neutrosophic set theory such as classical set theory. It includes; more connection between operations on the neutrosophic set theory of three types such as union, intersection, difference, and their generalization. further, the generalization of union, intersection, and complement with DE-Morgan's theorem. In addition, we studied the Cartesian Products and their properties. This work opens a new parallel path of a new neutrosophic structure of degree of membership functions that does not depend on intuitionistic fuzzy set and fuzzy set respectively.

Complete ranking for generalized trapezoidal fuzzy numbers and its application in supplier selection using the GTrF-CoCoSo approach

Decision-making is an indispensable tool in many real-life situations. The classical set theory is insufficient to solve problems involving uncertainty and qualitative data. In order to tackle scenarios with qualitative information and imprecise boundaries, fuzzy set theory generalizes crisp sets. Thus, ranking is an essential methodology in decision-making in a fuzzy environment. In literature, generalized trapezoidal fuzzy numbers (GTrFNs) are used to model many real-life situations with imprecise, uncertain, and incomplete information. There are various techniques available to discriminate arbitrary GTrFNs. Several researchers are seeking a standard consensus technique to distinguish between arbitrary GTrFNs. The diagonal distance right position, diagonal distance left chord, diagonal distance center position, diagonal distance, and diagonal distance right chord scores are the five score functions that are used in this work to develop a complete ranking (total ordering) principle on the class of GTrFNs to distinguish arbitrary GTrFNs. To show the potentiality of the proposed method, we compare the proposed approach with the few existing approaches suggesting to rank GTrFNs. Further, we proposed a new GTrF-CoCoSo approach by incorporating the proposed complete ordering principle. Then the efficacy of the proposed ranking principle is examined in solving multi-criteria decision-making (MCDM) using the GTrF-CoCoSo approach to select the best banana supplier for food production company. Furthermore, a sensitivity analysis is presented to evaluate the performance of the suggested ranking principle with the GTrF-CoCoSo approach.

Network intrusion detection system for IoT security using machine learning and statistical based hybrid feature selection

AbstractThe widespread adoption of Internet of Things (IoT) devices has revolutionized daily life, offering convenience and efficiency. However, this growth has also brought new security challenges. With the escalating use of Internet and network technology, the number of cyber‐attacks has increased, intensifying the focus on Intrusion Detection Systems (IDS) among researchers. Network intrusion detection (NID) plays a crucial role in securing IoT networks, becoming essential for modern security infrastructure. Recently, machine learning algorithms have shown promise in providing IDS solutions. Yet, IoT IDS systems face challenges due to their functional and physical diversity, making comprehensive feature utilization impractical. Therefore, effective feature selection becomes imperative. In this research, a novel feature selection methodology for anomaly‐based NIDS is proposed. The methodology commences by employing two filter‐based techniques, namely 1‐way ANOVA and the Pearson correlation coefficient, to meticulously identify and extract pertinent features from the dataset. These methods serve as initial filters to discern the most relevant attributes, ensuring that only the most informative features are retained for subsequent analysis. Subsequently, the most optimal features identified by both methodologies are extracted utilizing the principles of union and intersection in mathematical set theory. Using the NSL‐KDD and UNSW‐NB15 datasets, we exemplify how our model can outperform conventional ML classifiers in terms of detection rate, precision, recall. In our study, intrusion detection is carried out by SVM, kNN, Decision Tree, Logistic Regression and Random Forest using trained attack patterns. The demonstrated results highlight the exceptional performance of the proposed system, with an impressive accuracy rate of 99.6% on the NSL‐KDD dataset and a substantial 97.7% on the UNSW‐NB15 dataset, clearly surpassing the performance of contemporary methods.

MULTISET TOPOLOGICAL SPACE AS GENERALIZATION OF THE CLASSICAL TOPOLOGICAL SPACE VIA SUPPORT SET

Multiset as a genelization of the classical set has triggered the definition and introduction of some algebraic structures in classical set theory under multiset context such as multiset topological spaces. In this paper we defined a root (support) set of a multiset topological space and established that multiset topological spaces are a generalizations classical topological spaces via their root sets respectively.

Метод совершенствования планирования и управления технологическими процессами воздухообеспечения угольных шахт

Introduction. Management methods in complex technical systems of the mining and processing complex are currently considered as the main tool for improving the planning and management of technological processes in coal mines. In this regard, there is a need to develop an apparatus that allows research and design of methods on a unified basis. Methodology and research methods. During theoretical studies, methods of mathematical statistics and graph theory were used. The apparatus of mathematical logic, set theory and Gomori algorithms and their modifications were used. Discussion of research results. The article presents the results of research to develop a method for improving the planning and control of technological processes of air supply in coal mines. It has been established that the assessment and analysis of methods for improving the planning and control of technological processes of air supply (ATS) in coal mines are becoming increasingly relevant in connection with the modernization of mining production in the Russian Federation. Conclusion. Based on the research carried out, theoretical and practical problems were set and solved, an analysis of the patterns of TPV was carried out to improve the safety and prevent emergency situations of TPV in coal mines. Methods for calculating and analyzing safety for the variable structure of the system and its multifunctional states have been improved, allowing for the selection of optimal TPV parameters and performing calculations to ensure trouble-free operation of the TPV. The research results can be used to set the structure and control technological processes of air supply in coal mines. Conclusions on the article. The article presents the results of TPV studies conducted in coal mines. A method has been developed for analyzing and assessing components for controlling the air supply system of hot water supply, where its use allows for a comprehensive analysis and assessment of emergency situations of hot water supply. The research results can be useful when carrying out mining operations at all enterprises where air supply is necessary.

A COMPREHENSIVE STUDY ON SOFT BINARY PIECEWISE DIFFERENCE OPERATION

Soft set theory, developed by Molodtsov, has been applied both theoretically and practically in many fields. It is a useful mathematical tool for handling uncertainty. Numerous variations of soft set operations, which is a crucial concept for the theory, have been described and used since its introduction. In this paper, we explore more about soft binary piecewise difference operation (defined first as “difference of soft sets”) and its whole properties are examined especially in comparison with the basic properties of difference operation in classical set theory. Several striking properties of soft binary piecewise operations are obtained as analogous to the characteristic of difference operation in classical set theory. Also, we show that the collection of all soft sets with a fixed parameter set together with the soft binary piecewise difference operation is a bounded BCK-algebra.

МОДЕЛЬ ИМПУЛЬСНОГО УПРАВЛЕНИЯ УСТОЙЧИВОСТЬЮ СИСТЕМЫ ИНФОРМАЦИОННОЙ БЕЗОПАСНОСТИ

The article considers in detail issues related to organizing management of the stability in information security systems. A comparative analysis of literature sources was carried out to describe current methods, tools and algorithms for managing the stability of operation and security of information systems based on various management models. It also considers the main approaches to formalizing the control of the structural dynamics of information security systems based on the mathematical theory of reachability sets, as well as issues of ensuring comprehensive information security. The authors propose approaches to the construction of models for strategic and tactical management of the loss of stability of an information security system (ISS) under conditions of its natural wear and tear, as well as in the presence of external disturbing influences of a random impulse. There are given dynamic models that implement the principles of highly effective control of the stability in an information security system when it is upset, as well as in the presence of external disturbing influences of a random impulse nature. The basic principles of strategic management of the stability in the information security system are formed based on the criteria-based approach. Formation of tactical control of its stability was carried out based on regulation by deviation. The concept of pulse control using a regulator is proposed to maintain the dynamic stability of security information systems.

Enhancing Project Selection with Neutrosophic TOPSIS: Navigating Uncertainty in Post-Pandemic Decision-Making

This article explores the implementation of Neutrosophic TOPSIS, an advanced decision-making framework that extends classical and fuzzy set theories to handle the complexities of project selection amid uncertainty and indeterminacy. Neutrosophic sets are characterized by three parameters: truth, indeterminacy, and falsehood, which allow for a nuanced assessment of alternatives against defined criteria. Utilizing neutrosophic scales and expert evaluations, this method prioritizes projects by efficiently balancing multiple truth levels and addressing specific challenges such as judicial process optimization and labor education enhancement. The case study within the article demonstrates the application of Neutrosophic TOPSIS to select the most suitable project for improving labor relations and judicial efficiency in a post-pandemic world. The methodology proved effective in identifying the Digital Platform for Labor Education project as the optimal solution, given its alignment with strategic objectives and potential to handle identified challenges robustly. Future work could integrate Neutrosophic TOPSIS with other decision-making models and expand its application to more complex scenarios, potentially incorporating automated tools for a broader and more dynamic evaluation process.

APPLICATION OF FUZZY LOGIC IN COMPUTER SYSTEMS OF MEDICAL DIAGNOSIS

The features of medical diagnostics that affect the quality of decision-making in decision support systems are considered. The purpose of this article is to briefly introduce various soft computing methodologies and present various applications in medicine. Formalized expert information on the given sets of diagnoses and their symptoms. A method has been developed for generating a diagnostic inference during therapeutic examinations based on fuzzy logic. The goal is to demonstrate the possibilities of applying soft computing to problems related to medicine. A transition is made from fuzzy values to certain physical parameters that can serve as commands to the actuator. The result of fuzzy inference, of course, will be fuzzy, but for the executive device it means absolutely nothing, therefore, to eliminate fuzziness, special mathematical methods can be applied to obtain the exact values transmitted by a number of diseases directly to the performer in medical diagnostics. Fuzzy logic and fuzzy set theory is a branch of mathematics that generalizes classical logic and set theory. The fuzzy system copes with such a task very quickly, despite the fact that instead of complex differential equations, the entire process of movement is described in terms of natural language: “norm”, “average”, “high” - as if the operator controls the system. Keywords: medical diagnostics, fuzzy logic, decision support system, fuzzy systems.

Detecting system defects of digital devices in the course of malfunction imitation using fuzzing

Aim. The paper proposes approaches to the organisation of testing of digital systems through malfunction imitation for the purpose of ensuring compliance of international and Russian failure and fault resistance standards for the purpose of efficient (in terms of time) detection of software defects as part of mass production of products. The paper proposes a structure and operating algorithm of a hardware and software test bed for malfunction simulation intended for testing a system’s devices. The test bed collects and processes data for fuzzing, hardware error identification, as well as defines the scope of testing.Methods. The paper used basic systems approach, classical methods of the probability theory and mathematical statistics, decision theory, methods of hardware and software testing and development, mathematical theory of fuzzy sets and fuzzy logic.Results. Malfunction simulation algorithms were developed for the purpose of testing hardware and software systems using fuzzing that ensure probabilistic estimation of the termination time of testing with a specified accuracy.Conclusions. The above set of algorithms allows detecting system defects in the process of software and hardware integration into a single system that cause new malfunctions (emergence) that cannot be taken into consideration at the design stage.

Competing Roles of Aristotle’s Account of the Infinite

Abstract There are two distinct but interrelated questions concerning Aristotle’s account of infinity that have been the subject of recurring debate. The first of these, what I call here the interpretative question, asks for a charitable and internally coherent interpretation of the limited pieces of text where Aristotle outlines his view of the ‘potential’ (and not ‘actual’) infinite. The second, what I call here the philosophical question, asks whether there is a way to make Aristotle’s notion of the potential infinite coherent and rigorous with modern tools that can stand as a rival to the widely-accepted view of the infinite as characterized in a mathematical theory of sets. In this paper, I argue that the theoretical roles that Aristotle intends his account of the potential infinite to fulfill lead to a deep and irresoluble tension that can help explain the persistence of debates on both of these questions. I do so by turning to the places where Aristotle attempts to argue for or against the existence of particular infinite processes to show that he slides between different underlying notions of when changes are possible. Making these underlying notions clear can help us better understand the role of Aristotle’s account in the history of philosophy, the possible pitfalls for a contemporary theory of the potential infinite, and what each of these debates might learn from each other.

Axiomatization multisets: a comparative analysis

A multiset, unlike the classical set, allows for multiple instances of its elements. In this paper, we present a comparative analysis of theories on multisets. In particular, we examine, the first-order two-sorted multiset theory MST, and the single-sorted multiset theory MS that employs the same sort for multiplicities and the set they support. The logical strengths and significance of some axioms presented in these theories are investigated. The theory MST contains a copy of the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) but is independent of ZFC. The single-sorted multiset theory describes a stronger theory that mirrors the Zermelo-Fraenkel set theory (ZF) and is equiconsistent with ZF and antifoundation. The two-sorted multisettheory MST is a conservative extension of the classical set theory, making it a suitable theory to assume when dealing with studies that involve multisets.

Non-vanishing higher derived limits

In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups [Formula: see text] indexed by elements of [Formula: see text]. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits [Formula: see text] must vanish, for [Formula: see text]. They also proved that under the Continuum Hypothesis [Formula: see text]. The question whether [Formula: see text] vanishes, for all [Formula: see text], has attracted considerable interest from set theorists. Dow, Simon and Vaughan showed that under the Proper Forcing Axiom (PFA) [Formula: see text]. Bergfalk showed that it is consistent that [Formula: see text] does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that [Formula: see text], for all [Formula: see text]. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Hanson. We complete the picture by showing that, for any [Formula: see text], it is relatively consistent with ZFC that [Formula: see text].

CONSTRUCTION OF EXAMPLES AND COUNTEREXAMPLES IN PRIMARY SCHOOL TO VERIFY THE TRUTH OF STATEMENTS

In the modern mathematics learning process in school, the skill of creating examples and counterexamples in both familiar and new situations is emphasized. In the context of critical thinking, the construction of examples and counterexamples is an effective technique for evaluating statements and justifying arguments. This technique is suitable for students before other methods of proving general statements in mathematics are introduced.The aim of the study is to highlight the topics of the study course "Elements of Mathematical Logic and Set Theory" for the Professional Bachelor's degree in Teacher Education, which are relevant for constructing examples and counterexamples. In the study, the qualitative and quantitative analysis of the answers of the students in teacher programs to the questions of the questionnaire about the student experience related to the ability to formulate examples and counterexamples was carried out. The research shows that the creation of examples and counterexamples to the given statements does not cause great difficulties. It is much more difficult to create statements with mathematical content for grades 1-6, the truth verification of which, using the technique of constructing examples and counterexamples, would activate the formation of understanding of mathematical concepts and relationships. In order to connect the topics of the "Elements of Mathematical Logic and Set Theory" course with the planned results of primary education, a qualitative content analysis of the Latvian primary education standard and the primary education program was carried out. Based on the results obtained in the study, the authors identified several topics that are relevant in the course "Elements of Mathematical Logic and Set Theory".

Analysis of cost and profit using aggregation operators on spherical fuzzy sets with confidence level

To deal with the ambiguity in real-world problems, researchers strive to obtain extensions to classical set theory. They introduced ideas like fuzzy set theory, spherical, intuitionistic, and Pythagorean fuzzy sets. In comparison to fuzzy sets, spherical fuzzy sets are more realistic at handling uncertainty. Fundamentals are classified in Spherical Fuzzy Set according to an attribute, and each feature has a variety of criteria. In this study, we have created a new extended algebraic structure called Confidence Spherical Fuzzy Aggregation Operators by applying the idea of Confidence Levels to the already-existing Spherical Fuzzy Aggregation Operators. We have created a Confidence Spherical Fuzzy Aggregation Operators-based end-product. We demonstrated various intriguing characteristics of Confidence Spherical Fuzzy Aggregation Operators, including operational laws. The study is validated by addressing the decision-making processes.