Intuitionistic Set Theory

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Author: John L. Bell

Publisher: N.A

ISBN: 9781848901407

Category: Mathematics

Page: 132

View: 9787

While intuitionistic (or constructive) set theory IST has received a certain attention from mathematical logicians, so far as I am aware no book providing a systematic introduction to the subject has yet been published. This may be the case in part because, as a form of higher-order intuitionistic logic - the internal logic of a topos - IST has been chiefly developed in a tops-theoretic context. In particular, proofs of relative consistency with IST for mathematical assertions have been (implicitly) formulated in topos- or sheaf-theoretic terms, rather than in the framework of Heyting-algebra-valued models, the natural extension to IST of the well-known Boolean-valued models for classical set theory. In this book I offer a brief but systematic introduction to IST which develops the subject up to and including the use of Heyting-algebra-valued models in relative consistency proofs. I believe that IST, presented as it is in the familiar language of set theory, will appeal particularly to those logicians, mathematicians and philosophers who are unacquainted with the methods of topos theory.

From Sets and Types to Topology and Analysis

Towards Practicable Foundations for Constructive Mathematics

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Author: Laura Crosilla,Peter Schuster

Publisher: Oxford University Press on Demand

ISBN: 0198566514

Category: Mathematics

Page: 350

View: 8630

Bridging the foundations and practice of constructive mathematics, this text focusses on the contrast between the theoretical developments - which have been most useful for computer science - and more specific efforts on constructive analysis, algebra and topology.

Foundations of Set Theory

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Author: A.A. Fraenkel,Y. Bar-Hillel,A. Levy

Publisher: Elsevier

ISBN: 9780080887050

Category: Computers

Page: 412

View: 6664

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians.

Decision and Game Theory in Management With Intuitionistic Fuzzy Sets

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Author: Deng-Feng Li

Publisher: Springer

ISBN: 3642407129

Category: Computers

Page: 444

View: 9773

The focus of this book is on establishing theories and methods of both decision and game analysis in management using intuitionistic fuzzy sets. It proposes a series of innovative theories, models and methods such as the representation theorem and extension principle of intuitionistic fuzzy sets, ranking methods of intuitionistic fuzzy numbers, non-linear and linear programming methods for intuitionistic fuzzy multi-attribute decision making and (interval-valued) intuitionistic fuzzy matrix games. These theories and methods form the theory system of intuitionistic fuzzy decision making and games, which is not only remarkably different from those of the traditional, Bayes and/or fuzzy decision theory but can also provide an effective and efficient tool for solving complex management problems. Since there is a certain degree of inherent hesitancy in real-life management, which cannot always be described by the traditional mathematical methods and/or fuzzy set theory, this book offers an effective approach to using the intuitionistic fuzzy set expressed with membership and non-membership functions. This book is addressed to all those involved in theoretical research and practical applications from a variety of fields/disciplines: decision science, game theory, management science, fuzzy sets, operational research, applied mathematics, systems engineering, industrial engineering, economics, etc.

Intuitionistic Fuzzy Measures

Theory and Applications

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Author: Adrian I. Ban

Publisher: Nova Publishers

ISBN: 9781594549113

Category: Mathematics

Page: 264

View: 3658

This book is the outcome of about eight years of work performed by the author largely in the field of intuitionistic fuzzy set theory and more in depth on intuitionistic fuzzy measures presented from a point of view characteristic for pure mathematics. The purpose of the book is to present a continuation of studies conducted focusing mainly on measures that evaluate intuitionistic fuzzy sets by real values and crisp sets by intuitionistic fuzzy values.

Mathematics in Philosophy

Selected Essays

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Author: Charles Parsons

Publisher: Cornell University Press

ISBN: 9780801489815

Category: Mathematics

Page: 365

View: 919

This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics. A common point of view, that mathematical thought is central to our thought in general, underlies the essays. In his introduction, Parsons articulates that point of view and relates it to past and recent discussions of the foundations of mathematics.Mathematics in Philosophy is divided into three parts. Ontology—the question of the nature and extent of existence assumptions in mathematics—is the subject of Part One and recurs elsewhere. Part Two consists of essays on two important historical figures, Kant and Frege, and one contemporary, W. V. Quine. Part Three contains essays on the three interrelated notions of set, class, and truth.

Harvey Friedman's Research on the Foundations of Mathematics

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Author: L.A. Harrington,M.D. Morley,A. Šcedrov,S.G. Simpson

Publisher: Elsevier

ISBN: 9780080960401

Category: Mathematics

Page: 407

View: 4241

This volume discusses various aspects of Harvey Friedman's research in the foundations of mathematics over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers.

Algebraic Set Theory

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Author: Andri Joyal,Ieke Moerdijk

Publisher: Cambridge University Press

ISBN: 9780521558303

Category: Mathematics

Page: 123

View: 2182

This book offers a new algebraic approach to set theory. The authors introduce a particular kind of algebra, the Zermelo-Fraenkel algebras, which arise from the familiar axioms of Zermelo-Fraenkel set theory. Furthermore, the authors explicitly construct these algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory. In particular it provides a uniform description of various constructions of the cumulative hierarchy of sets in forcing models, sheaf models and realizability models. Graduate students and researchers in mathematical logic, category theory and computer science should find this book of great interest, and it should be accessible to anyone with a background in categorical logic.

Foundations of Constructive Mathematics

Metamathematical Studies

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Author: M.J. Beeson

Publisher: Springer Science & Business Media

ISBN: 3642689523

Category: Mathematics

Page: 466

View: 4591

This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec tions with philosophy and computer science. Namely, the creation and study of "formal systems for constructive mathematics". The general organization of the book is described in the" User's Manual" which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, "formal systems for constructive mathematics". "Con structive mathematics" refers to mathematics in which, when you prove that l a thing exists (having certain desired properties) you show how to find it. Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind.

On Intuitionistic Fuzzy Sets Theory

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Author: Krassimir T. Atanassov

Publisher: Springer

ISBN: 3642291279

Category: Computers

Page: 324

View: 7209

This book aims to be a comprehensive and accurate survey of state-of-art research on intuitionistic fuzzy sets theory and could be considered a continuation and extension of the author ́s previous book on Intuitionistic Fuzzy Sets, published by Springer in 1999 (Atanassov, Krassimir T., Intuitionistic Fuzzy Sets, Studies in Fuzziness and soft computing, ISBN 978-3-7908-1228-2, 1999). Since the aforementioned book has appeared, the research activity of the author within the area of intuitionistic fuzzy sets has been expanding into many directions. The results of the author ́s most recent work covering the past 12 years as well as the newest general ideas and open problems in this field have been therefore collected in this new book.

Constructivism in Mathematics

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Author: A.S. Troelstra,D. van Dalen

Publisher: Elsevier

ISBN: 008095510X

Category: Mathematics

Page: 129

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Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras. The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The text then examines proof theory of intuitionistic logic, theory of types and constructive set theory, and choice sequences. The book elaborates on semantical completeness, sheaves, sites, and higher-order logic, and applications of sheaf models. Topics include a derived rule of local continuity, axiom of countable choice, forcing over sites, sheaf models for higher-order logic, and complete Heyting algebras. The publication is a valuable reference for mathematicians and researchers interested in mathematics and logic.

Logic, Mathematics, Philosophy, Vintage Enthusiasms

Essays in Honour of John L. Bell

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Author: David DeVidi,Michael Hallett,Peter Clark

Publisher: Springer Science & Business Media

ISBN: 9789400702141

Category: Philosophy

Page: 486

View: 4391

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic (William Lawvere, Peter Aczel, Graham Priest, Giovanni Sambin); analytical philosophy (Michael Dummett, William Demopoulos), philosophy of science (Michael Redhead, Frank Arntzenius), philosophy of mathematics (Michael Hallett, John Mayberry, Daniel Isaacson) and decision theory and foundations of economics (Ken Bimore). Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic.

Intuitionistic Fuzzy Sets

Theory and Applications

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Author: Krassimir T. Atanassov

Publisher: Physica

ISBN: 3790818704

Category: Mathematics

Page: 324

View: 6128

In the beginning of 1983, I came across A. Kaufmann's book "Introduction to the theory of fuzzy sets" (Academic Press, New York, 1975). This was my first acquaintance with the fuzzy set theory. Then I tried to introduce a new component (which determines the degree of non-membership) in the definition of these sets and to study the properties of the new objects so defined. I defined ordinary operations as "n", "U", "+" and "." over the new sets, but I had began to look more seriously at them since April 1983, when I defined operators analogous to the modal operators of "necessity" and "possibility". The late George Gargov (7 April 1947 - 9 November 1996) is the "god father" of the sets I introduced - in fact, he has invented the name "intu itionistic fuzzy", motivated by the fact that the law of the excluded middle does not hold for them. Presently, intuitionistic fuzzy sets are an object of intensive research by scholars and scientists from over ten countries. This book is the first attempt for a more comprehensive and complete report on the intuitionistic fuzzy set theory and its more relevant applications in a variety of diverse fields. In this sense, it has also a referential character.

Computational Intelligence in Medical Informatics

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Author: Arpad Kelemen,Ajith Abraham,Yulan Liang

Publisher: Springer Science & Business Media

ISBN: 3540757678

Category: Medical

Page: 379

View: 8134

Medical Informatics (MI) is an emerging interdisciplinary science. This book deals with the application of computational intelligence in MI. Addressing the various issues of medical informatics using different computational intelligence approaches is the novelty of this edited volume. This volume comprises of 15 chapters selected on the basis of fundamental ideas/concepts including an introductory chapter giving the fundamental definitions and some important research challenges.

Varieties of Logic

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Author: Stewart Shapiro

Publisher: OUP Oxford

ISBN: 0191053864

Category: Philosophy

Page: 240

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Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. In Varieties of Logic, Stewart Shapiro develops several ways in which one can be a pluralist or relativist about logic. One of these is an extended argument that words and phrases like 'valid' and 'logical consequence' are polysemous or, perhaps better, are cluster concepts. The notions can be sharpened in various ways. This explains away the 'debates' in the literature between inferentialists and advocates of a truth-conditional, model-theoretic approach, and between those who advocate higher-order logic and those who insist that logic is first-order. A significant kind of pluralism flows from an orientation toward mathematics that emerged toward the end of the nineteenth century, and continues to dominate the field today. The theme is that consistency is the only legitimate criterion for a theory. Logical pluralism arises when one considers a number of interesting and important mathematical theories that invoke a non-classical logic, and are rendered inconsistent, and trivial, if classical logic is imposed. So validity is relative to a theory or structure. The perspective raises a host of important questions about meaning. The most significant of these concern the semantic content of logical terminology, words like 'or', 'not', and 'for all', as they occur in rigorous mathematical deduction. Does the intuitionistic 'not', for example, have the same meaning as its classical counterpart? Shapiro examines the major arguments on the issue, on both sides, and finds them all wanting. He then articulates and defends a thesis that the question of meaning-shift is itself context-sensitive and, indeed, interest-relative. He relates the issue to some prominent considerations concerning open texture, vagueness, and verbal disputes. Logic is ubiquitous. Whenever there is deductive reasoning, there is logic. So there are questions about logical pluralism that are analogous to standard questions about global relativism. The most pressing of these concerns foundational studies, wherein one compares theories, sometimes with different logics, and where one figures out what follows from what in a given logic. Shapiro shows that the issues are not problematic, and that is usually easy to keep track of the logic being used and the one mentioned.

Studies in Soviet Thought

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Author: J.M. Bochenski,J.E. Blakeley

Publisher: Springer Science & Business Media

ISBN: 9401032661

Category: Social Science

Page: 150

View: 8160

Early in 1958 a number of research projects on Soviet philosophy were started at the Institute of East-European Studies at the University of Fribourg (Switzerland) under the direction of the undersigned. At present, they are all completed or nearly so and their results are described in 18 different volumes. In spite of the fact that all of them have been already published or soon will be (mostly in German), it has been thought worthwhile to present their main conclusions in the form of short English reports. This book contains in the main these reports. Only the two programmatic papers (my own and that of Dr. Buchholz) and the study of Dr. Dahm are not direct results of the above-mentioned projects. But it will be clear to everyone that they, too, are closely connected with the subject envisaged and are written from a similar point of view. It will, perhaps, be convenient to briefly formulate this standpoint. All the writings included here are concerned with recent (i. e. mainly post Stalinist) developments in Soviet philosophy, where "Soviet" is taken in the wide meaning of the word, covering also Marxism-Leninism in Communist countries other than the Soviet Union. All the authors started with the assumption that there are interesting aspects to these more recent developments. There was also a common assumption that only specialized work on first-hand sources can be of relevance in this field.