DOCSLIB.ORG

New Trends in Fuzzy Set Theory and Related Items

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
New Trends in Fuzzy Set Theory and Related Items axioms New Trends in Fuzzy Set Theory and Related Items Edited by Esteban Induráin, Humberto Bustince and Javier Fernandez Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms New Trends in Fuzzy Set Theory and Related Items New Trends in Fuzzy Set Theory and Related Items Special Issue Editors Esteban Indur´ain Humberto Bustince Javier Fernandez MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors Esteban Indurain´ Humberto Bustince Universidad Publica´ de Navarra Universidad Publica´ de Navarra Spain Spain Javier Fernandez Universidad Publica´ de Navarra Spain Editorial Office MDPI St. Alban-Anlage 66 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) from 2017 to 2018 (available at: http://www.mdpi.com/journal/axioms/special issues/fuzzy set theory) For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03897-123-8 (Pbk) ISBN 978-3-03897-124-5 (PDF) Articles in this volume are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book taken as a whole is c 2018 MDPI, Basel, Switzerland, distributed under the terms and conditions of the Creative Commons license CC BY-NC-ND (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents About the Special Issue Editors ..................................... vii Preface to ”New Trends in Fuzzy Set Theory and Related Items” .................. ix Humberto Bustince, Javier Fernandez and Esteban Indur´ain Introduction to Special Issue: New Trends in Fuzzy Set Theory andRelated Items Reprinted from: Axioms 2018, 7, 37, doi: 10.3390/axioms7020037 ................... 1 Mar´ıa Jesus ´ Campi´on, Edurne Falc´o, Jos´e Luis Garc´ıa-Lapresta and Esteban Indur´ain Assigning Numerical Scores to Linguistic Expressions Reprinted from: Axioms 2017, 6, 19, doi: 10.3390/axioms6030019 ................... 3 Pablo Hern´andez, Susana Cubillo, Carmen Torres-Blanc and Jos´e A. Guerrero New Order on Type 2 Fuzzy Numbers Reprinted from: Axioms 2017, 6, 22, doi: 10.3390/axioms6030022 ................... 20 Leire Legarreta, Inmaculada Lizasoain and Iraide Mardones-P´erez Orness For Idempotent Aggregation Functions Reprinted from: Axioms 2017, 6, 25, doi: 10.3390/axioms6030025 ................... 39 Paolo Bevilacqua, Gianni Bosi and Magal`ı Zuanon Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness Reprinted from: Axioms 2017, 6, 29, doi: 10.3390/axioms6040029 ................... 49 Juan-Jos´eMi˜nana and Oscar Valero On Indistinguishability Operators, Fuzzy Metrics and Modular Metrics Reprinted from: Axioms 2017, 6, 34, doi: 10.3390/axioms6040034 ................... 59 Teresa Gonz´alez-Artega, Rocio de Andr´es Calle and Luis Mart´ınez Managing Interacting Criteria: Application to Environmental Evaluation Practices Reprinted from: Axioms 2018, 7, 4, doi: 10.3390/axioms7010004 ................... 77 Young Bae Jun, Seok-Zun Song and Seon Jeong Kim Cubic Interval-Valued Intuitionistic Fuzzy Sets and TheirApplication in BCK/BCI-Algebras Reprinted from: Axioms 2018, 7, 7, doi: 10.3390/axioms7010007 ...................101 Christian Servin, Gerardo D. Muela and Vladik Kreinovich Fuzzy Analogues of Sets and Functions Can Be UniquelyDetermined from the Corresponding Ordered Category: A Theorem Reprinted from: Axioms 2018, 7, 8, doi: 10.3390/axioms7010008 ...................118 Krzysztof Piasecki Revision of the Kosinski’s´ Theory of Ordered Fuzzy Numbers Reprinted from: Axioms 2018, 7, 16, doi: 10.3390/axioms7010016 ...................125 Juan C. Candeal An Abstract Result on Projective Aggregation Functions Reprinted from: Axioms 2018, 7, 17, doi: 10.3390/axioms7010017 ...................141 Sundas Shahzadi and Muhammad Akram Graphs in an Intuitionistic Fuzzy Soft Environment Reprinted from: Axioms 2018, 7, 20, doi: 10.3390/axioms7020020 ...................149 v Marta Cardin Quantiles in Abstract Convex Structures Reprinted from: Axioms 2018, 7, 35, doi: 10.3390/axioms7020035 ...................165 vi About the Special Issue Editors Esteban Indur´ain, Dr., was born in Pamplona, Spain in 1960. He got his degree in Mathematical Sciences from the University of Zaragoza, Spain in 1982 and his doctorate (Ph.D.) in Mathematical Sciences, also from the University of Zaragoza, in 1985. He is currently full professor of Mathematical Analysis at the Public University of Navarre, Pamplona (Spain), since 2000. Dr. Indurain´ has published ca. 85 papers in well-ranked journals. His impact factor is h = 14 in “Web of Science” and ”Scopus”. He has been the main researcher in several projects. Dr. Indurain´ has co-advised seven Ph.D. theses to-date. One of them was awarded a prize for the best Ph.D. thesis in its ambit and period. Humberto Bustince, Dr., received his Bs.C. degree on Physics from the Salamanca University, Spain, in 1983 and his Ph.D. degree in Mathematics from the Public University of Navarra, Pamplona, Spain, in 1994. He has been a teacher at the Public University of Navarra since 1991, and he is currently a Full Professor with the Department of Automatics and Computation. He served as subdirector of the Technical School for Industrial Engineering and Telecommunications from 01/01/2003 to 30/10/2008, and he was involved in the implantation of Computer Science courses at the Public University of Navarra. He is currently involved in teaching artificial intelligence for students of computer sciences. Dr. Bustince has authored more than 120 journal papers (Web of Knowledge) and more than 100 contributions to international conferences. He has also been the co-author of four books on fuzzy theory and extensions of fuzzy sets. His main research interests are interval-valued fuzzy sets (interval type-2 fuzzy sets), Atanassov’s intuitionistic fuzzy sets, aggregation functions, implication operators, inclusion measures, image processing, decision making, and approximate reasoning. Javier Fernandez, Dr., graduated in Mathematics from the University of Zaragoza (1999) and got his doctorate (Ph.D.) in Mathematics from the University of the Basque Country in 2003. He has been a postdoctoral researcher at the CNRS (France) in 2003–2004 and 2005–2007, in the framework of the European Hyperbolic and Kynetic Equations (HYKE) research project. He has been a lecturer in the Public University of Navarre since 2008. He has co-advised two Ph.D. thesis. Dr. Fernandez is a member of the research Group on Artificial Intelligence and Approximate Reasoning (GIARA) of the Public University of Navarre, whose leader is Dr. Humberto Bustince. His main research interests are fuzzy theory, extensions of fuzzy theory, aggregation functions, computing, differential equations, and unique continuation, as well as image processing. He has authored more than 30 papers in JCR journals, most of them in Q1. Some of the most relevant ones are on linear orders for intervals. vii Preface to ”New Trends in Fuzzy Set Theory and Related Items” Fuzzy set theory is one of the main trend topics in Mathematics and Computer Sciences nowadays. Its wide, multidisciplinary applications, as well as the recent developments in its theoretical bases, provide an excellent set of frameworks for researchers interested in different scientific disciplines. Owing to these reasons, we have addressed the task of issuing this Special Issue, having in mind the possibility of collecting new contributions devoted to this topic, from many different points of view. In this sense, this book includes works authored by authors working from very different perspectives, but all of them are in the first line of research of their areas. The articles (twelve thereof) that have been included in this Special Issue cover a large set of the newest developments in this framework of research. Thus, always bearing in mind the notion of a fuzzy set, we can find ideas and new results on orderings, logic, aggregation of functions, functional equations, operators, metric, etc. We have also included some works devoted to applications in, e.g., social choice. Of course, the field of fuzzy sets theory is so broad nowadays that it is not possible to cover all of them, but we consider that the book provides a rather complete view of the state-of-the-art in the field. EstebanIndur´ain,HumbertoBustince,JavierFernandez SpecialIssueEditors ix axioms Editorial Introduction to Special Issue: New Trends in Fuzzy Set Theory and Related Items Humberto Bustince 1 ID , Javier Fernandez 1 ID and Esteban Induráin 2,* ID 1 Institute of Smart Cities (ISC), Departamento de Estadística, Matemáticas e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain; [email protected] (H.B.); [email protected] (J.F.) 2 Institute for Advanced Materials (InaMat), Departamento de Estadística, Matemáticas e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain * Correspondence: [email protected]; Tel.: +34-948-169-551 Received: 10 May 2018; Accepted: 1 June 2018; Published: 5 June 2018 Abstract: We focus on the articles recently published in the special issue of Axioms devoted
Load more

Recommended publications
  • 1 Elementary Set Theory
    1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection).
    [Show full text]
  • Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T
    University of Texas at El Paso DigitalCommons@UTEP Departmental Technical Reports (CS) Department of Computer Science 8-1-2008 Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T. Nguyen Vladik Kreinovich University of Texas at El Paso, [email protected] Follow this and additional works at: http://digitalcommons.utep.edu/cs_techrep Part of the Computer Engineering Commons Comments: Technical Report: UTEP-CS-08-27a Published in Proceedings of the 9th International Conference on Intelligent Technologies InTech'08, Samui, Thailand, October 7-9, 2008, pp. 47-55. Recommended Citation Nguyen, Hung T. and Kreinovich, Vladik, "Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms" (2008). Departmental Technical Reports (CS). Paper 94. http://digitalcommons.utep.edu/cs_techrep/94 This Article is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for inclusion in Departmental Technical Reports (CS) by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T. Nguyen Vladik Kreinovich Department of Mathematical Sciences Department of Computer Science New Mexico State University University of Texas at El Paso Las Cruces, NM 88003, USA El Paso, TX 79968, USA [email protected] [email protected] Abstract—Subsethood A ⊆ B and set equality A = B are Thus, for two fuzzy sets A and B, it is reasonable to define among the basic notions of set theory. For traditional (“crisp”) degree of subsethood and degree of similarity.
    [Show full text]
  • Mathematics 144 Set Theory Fall 2012 Version
    MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction
    [Show full text]
  • A Taste of Set Theory for Philosophers
    Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics.
    [Show full text]
  • Data Envelopment Analysis with Fuzzy Parameters: an Interactive Approach
    International Journal of Operations Research and Information Systems, 2(3), 39-53, July-September 2011 39 Data Envelopment Analysis with Fuzzy Parameters: An Interactive Approach Adel Hatami-Marbini, Universite Catholique de Louvain, Belgium Saber Saati, Islamic Azad University, Iran Madjid Tavana, La Salle University, USA ABSTRACT Data envelopment analysis (DEA) is a methodology for measuring the relative efficiencies of a set of deci- sion making units (DMUs) that use multiple inputs to produce multiple outputs. In the conventional DEA, all the data assume the form of specific numerical values. However, the observed values of the input and output data in real-life problems are sometimes imprecise or vague. Previous methods have not considered the preferences of the decision makers (DMs) in the evaluation process. This paper proposes an interactive evaluation process for measuring the relative efficiencies of a set of DMUs in fuzzy DEA with consideration of the DMs’ preferences. The authors construct a linear programming (LP) model with fuzzy parameters and calculate the fuzzy efficiency of the DMUs for different α levels. Then, the DM identifies his or her most preferred fuzzy goal for each DMU under consideration. A modified Yager index is used to develop a ranking order of the DMUs. This study allows the DMs to use their preferences or value judgments when evaluating the performance of the DMUs. Keywords: Data Envelopment Analysis, Efficiency Evaluation, Fuzzy Mathematical Programming, Interactive Solution, Preference Modeling INTRODUCTION efficiency. A DMU is considered efficient when no other DMU can produce more outputs using The changing economic conditions have chal- an equal or lesser amount of inputs.
    [Show full text]
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • Surviving Set Theory: a Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory
    Surviving Set Theory: A Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Angela N. Ripley, M.M. Graduate Program in Music The Ohio State University 2015 Dissertation Committee: David Clampitt, Advisor Anna Gawboy Johanna Devaney Copyright by Angela N. Ripley 2015 Abstract Undergraduate music students often experience a high learning curve when they first encounter pitch-class set theory, an analytical system very different from those they have studied previously. Students sometimes find the abstractions of integer notation and the mathematical orientation of set theory foreign or even frightening (Kleppinger 2010), and the dissonance of the atonal repertoire studied often engenders their resistance (Root 2010). Pedagogical games can help mitigate student resistance and trepidation. Table games like Bingo (Gillespie 2000) and Poker (Gingerich 1991) have been adapted to suit college-level classes in music theory. Familiar television shows provide another source of pedagogical games; for example, Berry (2008; 2015) adapts the show Survivor to frame a unit on theory fundamentals. However, none of these pedagogical games engage pitch- class set theory during a multi-week unit of study. In my dissertation, I adapt the show Survivor to frame a four-week unit on pitch- class set theory (introducing topics ranging from pitch-class sets to twelve-tone rows) during a sophomore-level theory course. As on the show, students of different achievement levels work together in small groups, or “tribes,” to complete worksheets called “challenges”; however, in an important modification to the structure of the show, no students are voted out of their tribes.
    [Show full text]
  • Fuzzy Integer Linear Programming with Fuzzy Decision Variables
    Applied Mathematical Sciences, Vol. 4, 2010, no. 70, 3493 - 3502 Fuzzy Integer Linear Programming with Fuzzy Decision Variables C. Sudhagar1 Department of Mathematics and Applied Sciences Middle East College of Information Technology, Muscat, Oman [email protected] K. Ganesan Department of Mathematics, SRM University, Chennai, India gansan [email protected] Abstract In this paper a new method for dealing with Fuzzy Integer Linear Programming Problems (FILPP) has been proposed. FILPP with fuzzy variables model was taken for solution. This solution method is based on the fuzzy ranking method. The proposed method can serve deci- sion makers by providing the reasonable range of values for the fuzzy variable, which is comparatively better than the currently available solu- tions. Numerical examples demonstrate the effectiveness and accuracy of the proposed method. Mathematics Subject Classification: 65K05, 90C10, 90C70, 90C90 Keywords: Fuzzy numbers, Ranking, Fuzzy integer linear programming 1 Introduction Linear Programming Problems(LPP) have an outstanding relevance in the field of Decision making, Artificial intelligence, Control theory, Management sciences, Job placement interventions etc. In many practical applications the available information in the system under consideration are not precise. In such situations, it is more appropriate to use the fuzzy LPP. 1Corresponding author 3494 C. Sudhagar and K. Ganesan The concept of fuzzy linear programming problems was first introduced by Tanaka et al.,[13, 12] After his work, several kinds of fuzzy linear programming problems have appeared in the literature and different methods have been proposed to solve such problems. Numerous methods for comparison of fuzzy numbers have been suggested in the literature. In Campos Verdegay paper[2] linear programming problems with fuzzy constraints and fuzzy coefficients in both matrix and right hand side of the constraint set are considered.
    [Show full text]
  • Basic Concepts of Set Theory, Functions and Relations 1. Basic
    Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6
    [Show full text]
  • Applied Mathematics for Database Professionals
    7451FM.qxd 5/17/07 10:41 AM Page i Applied Mathematics for Database Professionals Lex de Haan and Toon Koppelaars 7451FM.qxd 5/17/07 10:41 AM Page ii Applied Mathematics for Database Professionals Copyright © 2007 by Lex de Haan and Toon Koppelaars All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without the prior written permission of the copyright owner and the publisher. ISBN-13: 978-1-59059-745-3 ISBN-10: 1-59059-745-1 Printed and bound in the United States of America 9 8 7 6 5 4 3 2 1 Trademarked names may appear in this book. Rather than use a trademark symbol with every occurrence of a trademarked name, we use the names only in an editorial fashion and to the benefit of the trademark owner, with no intention of infringement of the trademark. Lead Editor: Jonathan Gennick Technical Reviewers: Chris Date, Cary Millsap Editorial Board: Steve Anglin, Ewan Buckingham, Gary Cornell, Jonathan Gennick, Jason Gilmore, Jonathan Hassell, Chris Mills, Matthew Moodie, Jeffrey Pepper, Ben Renow-Clarke, Dominic Shakeshaft, Matt Wade, Tom Welsh Project Manager: Tracy Brown Collins Copy Edit Manager: Nicole Flores Copy Editor: Susannah Davidson Pfalzer Assistant Production Director: Kari Brooks-Copony Production Editor: Kelly Winquist Compositor: Dina Quan Proofreader: April Eddy Indexer: Brenda Miller Artist: April Milne Cover Designer: Kurt Krames Manufacturing Director: Tom Debolski Distributed to the book trade worldwide by Springer-Verlag New York, Inc., 233 Spring Street, 6th Floor, New York, NY 10013.
    [Show full text]
  • Connes on the Role of Hyperreals in Mathematics
    Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V.
    [Show full text]
  • A Class of Fuzzy Theories*
    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 85, 409-451 (1982) A Class of Fuzzy Theories* ERNEST G. MANES Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003 Submitted by L. Zadeh Contenfs. 0. Introduction. 1. Fuzzy theories. 2. Equality and degree of membership. 3. Distributions as operations. 4. Homomorphisms. 5. Independent joint distributions. 6. The logic of propositions. 7. Superposition. 8. The distributional conditional. 9. Conclusions. References. 0. INTR~DUCTTON At the level of syntax, a flowchart scheme [25, Chap. 41 decomposes into atomic pieces put together by the operations of structured programming [ 11. Our definition of “fuzzy theory” is motivated solely by providing the minimal machinery to interpret loop-free schemes in a fuzzy way. Indeed, a fuzzy theory T = (T, e, (-)“) is defined in Section 1 by the data (A, B, C). For each set X there is given a new set TX of “distributions on X” or “vague specifications (A) of elements of X.” For each set X there is given a distinguished function e, : X-+ TX, “a crisp 03) specification is a special case of a vague one.” For each “fuzzy function” a: X-, TY there is given a distinguished “extension” (Cl a#: TX-+ TY. The data are all subject to three axioms. This definition is motivated by the flowchart scheme l.E. Some fundamental examples are crisp set theory: TX = X, CD) fuzzy set theory: TX = [0, llx, (E) * The research reported in this paper was supported in part by the National Science Foun- dation under Grant MCS76-84477. 409 0022-247X/82/020409-43$02.0010 Copyright 0 1982 by Academic Press, Inc.
    [Show full text]