Intuitionistic Set Theory

Author: John L. Bell
Publisher:
ISBN: 1848901402
Release Date: 2014-02-28
Genre: Mathematics

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.

Foundations of Set Theory

Author: A.A. Fraenkel
Publisher: Elsevier
ISBN: 0080887058
Release Date: 1973-12-01
Genre: Computers

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.

Algebraic Set Theory

Author: Andri Joyal
Publisher: Cambridge University Press
ISBN: 0521558301
Release Date: 1995-09-14
Genre: Mathematics

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.

Logic Colloquium 2004

Author: Alessandro Andretta
Publisher: Cambridge University Press
ISBN: 9780521884242
Release Date: 2008
Genre: Computers

A collection of surveys, tutorials, and research papers from the 2004 Logic Colloquium.

From Sets and Types to Topology and Analysis

Author: Laura Crosilla
Publisher: Oxford University Press on Demand
ISBN: 9780198566519
Release Date: 2005-10-06
Genre: Mathematics

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.

Bulgarian Studies in the Philosophy of Science

Author: D. Ginev
Publisher: Springer Science & Business Media
ISBN: 1402014961
Release Date: 2003-08-31
Genre: Philosophy

This volume attempts to provide a new articulation of issues surrounding scientific realism, scientific rationality, the epistemology of non-classical physics, the type of revolutionary changes in the development of science, the naturalization of epistemology within frameworks of cognitive science and structural linguistics, models of the information technology revolution, and reconstructions of early modern logical systems.

Constructivism in Mathematics

Author: A.S. Troelstra
Publisher: Elsevier
ISBN: 9780080955100
Release Date: 2014-06-28
Genre: Mathematics

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.

Intuitionistic Fuzzy Sets

Author: Krassimir T. Atanassov
Publisher: Physica
ISBN: 9783790818703
Release Date: 2013-03-20
Genre: Mathematics

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.

Harvey Friedman s Research on the Foundations of Mathematics

Author: L.A. Harrington
Publisher: Elsevier
ISBN: 0080960405
Release Date: 1985-11-01
Genre: Mathematics

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.

Logic Mathematics Philosophy Vintage Enthusiasms

Author: David DeVidi
Publisher: Springer Science & Business Media
ISBN: 9400702140
Release Date: 2011-03-23
Genre: Philosophy

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.

Mathematics in Philosophy

Author: Charles Parsons
Publisher: Cornell University Press
ISBN: 0801489814
Release Date: 2005
Genre: Mathematics

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.

Emerging Research on Applied Fuzzy Sets and Intuitionistic Fuzzy Matrices

Author: Adak, Amal Kumar
Publisher: IGI Global
ISBN: 9781522509158
Release Date: 2016-11-17
Genre: Computers

The use of fuzzy logic has become prominent in a variety of fields and applications. By implementing these logic sets, problems and uncertainties are more effectively resolved. Emerging Research on Applied Fuzzy Sets and Intuitionistic Fuzzy Matrices is a pivotal reference source for the latest scholarly perspectives on the interdisciplinary use of fuzzy logic theory, focusing on the application of sets and matrices. Highlighting theoretical framework and empirical research findings, this book is ideally designed for academics, practitioners, upper-level students, and professionals interested in an innovative overview of fuzzy logic sets and matrices.

Logicism Intuitionism and Formalism

Author: Sten Lindström
Publisher: Springer Science & Business Media
ISBN: 9781402089268
Release Date: 2008-11-25
Genre: Mathematics

This anthology reviews the programmes in the foundations of mathematics from the classical period and assesses their possible relevance for contemporary philosophy of mathematics. A special section is concerned with constructive mathematics.