Intuitionistic Set Theory

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Author: John L. Bell

Publisher: N.A

ISBN: 9781848901407

Category: Mathematics

Page: 132

View: 4269

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.

Intuitionistic Fuzzy Sets

Theory and Applications

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Author: Krassimir T. Atanassov

Publisher: Physica

ISBN: 3790818704

Category: Mathematics

Page: 324

View: 9969

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.

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: 8970

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.

Studies in the History of Mathematical Logic

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Author: Stanislaw Surma

Publisher: N.A

ISBN: 9781938421266

Category: Mathematics

Page: 280

View: 921

This volume contains seventeen essays in the history of modern mathematical logic. The first nine are concerned with the completeness of various logical calculi. The second five essays deal with the completeness of classical first-order predicate logic. One essay deals with the history of Cantor's definition of set, another with the set-theoretical reduction of the concept of relation, and a final essay is devoted to a survey of various meanings of the concept of completeness of formalized deductive theories. The essays were first presented in the national conferences of the Thematic Group for the History of Logic organized by the Department of Logic of the Polish Academy of Sciences in 1966-1971. The Advanced Reasoning Forum is pleased to make available this exact reprint of the original volume first published in 1973 by the Polish Academy of Sciences and the Department of Logic of Jagiellonian University, edited by Stanislaw J. Surma. * * * * The essays are: 1. Emil Post's doctoral dissertation (Stanislaw J. Surma); 2. A historical survey of the significant methods of proving post's theorem about the completeness of the classical propositional calculus (Stanislaw J. Surma); 3. A survey of the results and methods of investigations of the equivalential propositional calculus (Stanislaw J. Surma); 4. A uniform method of proof of the completeness theorem for the equivalential propositional calculus and for some of its extensions (Stanislaw J. Surma); 5. Kolmogorov and Glivenko's papers about intuitionistic logic (Jacek K. Kabzinski); 6. Jaskowski's matrix criterion for the intuitionistic propositional calculus (Stanislaw J. Surma); 7. Axiomatization of the implicational Godel's matrices by Kalmar's method (Andrzej Wronski); 8. A contribution to the history of the investigations into the intermediate propositional calculi (Andrzej Wronski); 9. On Ackermann's rigorous implication (Jan Wolenski); 10. Kurt Godel's doctoral dissertation (Jan Zygmunt); 11. A survey of the methods of proof of the Godel-Malcev's completeness theorem (Jan Zygmunt); 12. The concept of the Lindenbaum algebra: its genesis (Stanislaw J. Surma); 13. On the old and new methods of interpreting quantifiers (Andrzej Wronski); 14. L. Rieger's logical achievement (Wladyslaw Szczech); 15. The development of Cantor's definition of set (Jerzy Perzanowski); 16. On the origins of the set-theoretical concept of relation (Piotr Kossowski); 17. A survey of various concepts of completeness of the deductive theories (Stanislaw J. Surma)."

Toposes and Local Set Theories

An Introduction

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Author: John L. Bell

Publisher: Courier Corporation

ISBN: 0486462862

Category: Mathematics

Page: 267

View: 370

This text introduces topos theory, a development in category theory that unites important but seemingly diverse notions from algebraic geometry, set theory, and intuitionistic logic. Topics include local set theories, fundamental properties of toposes, sheaves, local-valued sets, and natural and real numbers in local set theories. 1988 edition.

A Short Introduction to Intuitionistic Logic

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Author: Grigori Mints

Publisher: Springer Science & Business Media

ISBN: 0306469758

Category: Mathematics

Page: 131

View: 4941

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for making this book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999.

Intuitionism

An Introduction

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Author: Arend Heyting

Publisher: Elsevier

ISBN: 0444534067

Category: Electronic books

Page: 147

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Descriptive Set Theory

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Author: Yiannis N. Moschovakis

Publisher: American Mathematical Soc.

ISBN: 0821848135

Category: Mathematics

Page: 502

View: 9424

Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern ``effective'' theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics. The book includes all the necessary background from (advanced) set theory, logic and recursion theory.

Algebraic Set Theory

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Author: Andri Joyal,Ieke Moerdijk

Publisher: Cambridge University Press

ISBN: 9780521558303

Category: Mathematics

Page: 123

View: 8868

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.

Introduction to the Theory of Sets

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Author: Joseph Breuer

Publisher: Courier Corporation

ISBN: 0486154874

Category: Mathematics

Page: 128

View: 739

This undergraduate text develops its subject through observations of the physical world, covering finite sets, cardinal numbers, infinite cardinals, and ordinals. Includes exercises with answers. 1958 edition.

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: 942

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.

Introduction to Higher-Order Categorical Logic

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Author: J. Lambek,P. J. Scott

Publisher: Cambridge University Press

ISBN: 9780521356534

Category: Mathematics

Page: 304

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Part I indicates that typed-calculi are a formulation of higher-order logic, and cartesian closed categories are essentially the same. Part II demonstrates that another formulation of higher-order logic is closely related to topos theory.

Set Theory, Arithmetic, and Foundations of Mathematics

Theorems, Philosophies

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Author: Juliette Kennedy,Roman Kossak

Publisher: Cambridge University Press

ISBN: 1139504819

Category: Mathematics

Page: N.A

View: 3520

This collection of papers from various areas of mathematical logic showcases the remarkable breadth and richness of the field. Leading authors reveal how contemporary technical results touch upon foundational questions about the nature of mathematics. Highlights of the volume include: a history of Tennenbaum's theorem in arithmetic; a number of papers on Tennenbaum phenomena in weak arithmetics as well as on other aspects of arithmetics, such as interpretability; the transcript of Gödel's previously unpublished 1972–1975 conversations with Sue Toledo, along with an appreciation of the same by Curtis Franks; Hugh Woodin's paper arguing against the generic multiverse view; Anne Troelstra's history of intuitionism through 1991; and Aki Kanamori's history of the Suslin problem in set theory. The book provides a historical and philosophical treatment of particular theorems in arithmetic and set theory, and is ideal for researchers and graduate students in mathematical logic and philosophy of mathematics.

Leo Esakia on Duality in Modal and Intuitionistic Logics

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Author: Guram Bezhanishvili

Publisher: Springer

ISBN: 940178860X

Category: Philosophy

Page: 334

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This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues.

Principles of Harmonic Analysis

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Author: Anton Deitmar,Siegfried Echterhoff

Publisher: Springer

ISBN: 3319057928

Category: Mathematics

Page: 332

View: 848

This book offers a complete and streamlined treatment of the central principles of abelian harmonic analysis: Pontryagin duality, the Plancherel theorem and the Poisson summation formula, as well as their respective generalizations to non-abelian groups, including the Selberg trace formula. The principles are then applied to spectral analysis of Heisenberg manifolds and Riemann surfaces. This new edition contains a new chapter on p-adic and adelic groups, as well as a complementary section on direct and projective limits. Many of the supporting proofs have been revised and refined. The book is an excellent resource for graduate students who wish to learn and understand harmonic analysis and for researchers seeking to apply it.

Set theory

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Author: Charles C. Pinter

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 216

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Mathematical Logic

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Author: Stephen Cole Kleene

Publisher: Courier Corporation

ISBN: 0486317072

Category: Mathematics

Page: 416

View: 8752

Contents include an elementary but thorough overview of mathematical logic of 1st order; formal number theory; surveys of the work by Church, Turing, and others, including Gödel's completeness theorem, Gentzen's theorem, more.

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: 1114

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.