Fuzzy set theory provides us with a framework which is wider than that of classical set theory. Various mathematical structures, whose features emphasize the effects of ordered structure, can be developed on the theory. Fuzzy topology is one such branch, combining ordered structure with topological structure. This branch of mathematics, emerged from the background - processing fuzziness, and locale theory, proposed from the angle of pure mathematics by the great French mathematician Ehresmann, comprise the two most active aspects of topology on lattice, which affect each other.This book is the first monograph to systematically reflect the up-to-date state of fuzzy topology. It emphasizes the so-called "pointed approach" and the effects of stratification structure appearing in fuzzy sets.The monograph can serve as a reference book for mathematicians, researchers, and graduate students working in this branch of mathematics. After an appropriate rearrangements of the chapters and sections, it can also be used as a text for undergraduates.

Many of the modern variational problems in topology arise in different but overlapping fields of scientific study: mechanics, physics and mathematics. In this work, Professor Fomenko offers a concise and clean explanation of some of these problems (both solved and unsolved), using current methods and analytical topology. The author's skillful exposition gives an unusual motivation to the theory expounded, and his work is recommended reading for specialists and nonspecialists alike, involved in the fields of physics and mathematics at both undergraduate and graduate levels.

Model theory is one of the central branches of mathematical logic. The field has evolved rapidly in the last few decades. This book is an introduction to current trends in model theory, and contains a collection of articles authored by top researchers in the field. It is intended as a reference for students as well as senior researchers.

This introductory graduate text covers modern mathematical logic from propositional, first-order, higher-order and infinite logic and Godel's Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory. Based on the author's more than 35 years of teaching experience, the book develops students' intuition by presenting complex ideas in the simplest context for which they make sense. He also provides extensive introductions to set theory, model theory and recursion (computability) theory, which allows this book to be used as a classroom text, for self-study, and as a reference on the state of modern logic.

A compilation of articles about Intensionality in philosophy, logic, linguistics, and mathematics. The articles approach the concept of Intensionality from different perspectives. Some articles address philosophical issues raised by the possible worlds approach to intensionality; others are devoted to technical aspects of modal logic. The volume highlights the particular interdisciplinary nature of intensionality with articles spanning the areas of philosophy, linguistics, mathematics, and computer science.

A compilation of articles about Intensionality in philosophy, logic, linguistics, and mathematics. The articles approach the concept of Intensionality from different perspectives. Some articles address philosophical issues raised by the possible worlds approach to intensionality; others are devoted to technical aspects of modal logic. The volume highlights the particular interdisciplinary nature of intensionality with articles spanning the areas of philosophy, linguistics, mathematics, and computer science.

Originally published in 1973. This book is directed to the student of philosophy whose background in mathematics is very limited. The author strikes a balance between material of a philosophical and a formal kind, and does this in a way that will bring out the intricate connections between the two. On the formal side, he gives particular care to provide the basic tools from set theory and arithmetic that are needed to study systems of logic, setting out completeness results for two, three, and four valued logic, explaining concepts such as freedom and bondage in quantificational logic, describing the intuitionistic conception of the logical operators, and setting out Zermelo's axiom system for set theory. On the philosophical side, he gives particular attention to such topics as the problem of entailment, the import of the Loewenheim-Skolem theorem, the expressive powers of quantificational logic, the ideas underlying intuitionistic logic, the nature of set theory, and the relationship between logic and set theory. There are exercises within the text, set out alongside the theoretical ideas that they involve.

This book presents a study on the foundations of a large class of paraconsistent logics from the point of view of the logics of formal inconsistency. It also presents several systems of non-standard logics with paraconsistent features.

This comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff's classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff's initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics. The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. Contents Historical foreword on the centenary after Felix Hausdorff's classic Set Theory Fundamentals of the theory of functions Fundamentals of the measure theory Historical notes on the Riesz - Radon - Frechet problem of characterization of Radon integrals as linear functionals

Neutrices and External Numbers: A Flexible Number System introduces a new model of orders of magnitude and of error analysis, with particular emphasis on behaviour under algebraic operations. The model is formulated in terms of scalar neutrices and external numbers, in the form of an extension of the nonstandard set of real numbers. Many illustrative examples are given. The book starts with detailed presentation of the algebraic structure of external numbers, then deals with the generalized Dedekind completeness property, applications in analysis, domains of validity of approximations of solutions of differential equations, particularly singular perturbations. Finally, it describes the family of algebraic laws characterizing the practice of calculations with external numbers. Features Presents scalar neutrices and external numbers, a mathematical model of order of magnitude within the real number system. Outlines complete algebraic rules for the neutrices and external numbers Conducts operational analysis of convergence and integration of functions known up to orders of magnitude Formalises a calculus of error propagation, covariant with algebraic operations Presents mathematical models of phenomena incorporating their necessary imprecisions, in particular related to the Sorites paradox

Over the past 20 years, the emergence of clone theory, hyperequational theory, commutator theory and tame congruence theory has led to a growth of universal algebra both in richness and in applications, especially in computer science. Yet most of the classic books on the subject are long out of print and, to date, no other book has integrated these theories with the long-established work that supports them.

Universal Algebra and Applications in Theoretical Computer Science introduces the basic concepts of universal algebra and surveys some of the newer developments in the field. The first half of the book provides a solid grounding in the core material. A leisurely pace, careful exposition, numerous examples, and exercises combine to form an introduction to the subject ideal for beginning graduate students or researchers from other areas. The second half of the book focuses on applications in theoretical computer science and advanced topics, including Mal'cev conditions, tame congruence theory, clones, and commutators.

The impact of the advances in universal algebra on computer science is just beginning to be realized, and the field will undoubtedly continue to grow and mature. Universal Algebra and Applications in Theoretical Computer Science forms an outstanding text and offers a unique opportunity to build the foundation needed for further developments in its theory and in its computer science applications.

This comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff's classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff's initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics.The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. Contents Fundamentals of the theory of classes, sets, and numbers Characterization of all natural models of Neumann - Bernays - Godel and Zermelo - Fraenkel set theories Local theory of sets as a foundation for category theory and its connection with the Zermelo - Fraenkel set theory Compactness theorem for generalized second-order language

Beyond calculus, the world of mathematics grows increasingly abstract and places new and challenging demands on those venturing into that realm. As the focus of calculus instruction has become increasingly computational, it leaves many students ill prepared for more advanced work that requires the ability to understand and construct proofs.

Introductory Concepts for Abstract Mathematics helps readers bridge that gap. It teaches them to work with abstract ideas and develop a facility with definitions, theorems, and proofs. They learn logical principles, and to justify arguments not by what seems right, but by strict adherence to principles of logic and proven mathematical assertions - and they learn to write clearly in the language of mathematics

The author achieves these goals through a methodical treatment of set theory, relations and functions, and number systems, from the natural to the real. He introduces topics not usually addressed at this level, including the remarkable concepts of infinite sets and transfinite cardinal numbers

Introductory Concepts for Abstract Mathematics takes readers into the world beyond calculus and ensures their voyage to that world is successful. It imparts a feeling for the beauty of mathematics and its internal harmony, and inspires an eagerness and increased enthusiasm for moving forward in the study of mathematics.

In this volume, logic starts from the observation that in everyday arguments, as brought forward say by a lawyer, statements are transformed linguistically, connecting them in formal ways irrespective of their contents. Understanding such arguments as deductive situations, or "sequents" in the technical terminology, the transformations between them can be expressed as logical rules. This leads to Gentzen's calculi of derivations, presented first for positive logic and then, depending on the requirements made on the behaviour of negation, for minimal, intuitionist and classical logic. Identifying interdeducible formulas, each of these calculi gives rise to a lattice-like ordered structure. Describing the generation of filters in these structures leads to corresponding modus ponens calculi, and these turn out to be semantically complete because they express the algorithms generating semantical consequences, as obtained in Volume One of these lectures. The operators transforming derivations from one type of calculus into the other are also studied with respect to changes of the lengths of derivations, and operators eliminating defined predicate and function symbols are described expli

For propositional logic it can be decided whether a formula has a deduction from a finite set of other formulas. The present volume begins with a method to decide this for the quantified formulas of those fragments of arithmetic which express the properties of order-plus-successor and or order-plus-addition (Presburger arithmetic); it makes use of an algorithm eliminating quantifiers which, in turn, is also applied to obtain consistency proofs for these fragments. Stronger fragments of arithmetic, also containing multiplication, are sufficiently rich to express a primitive recursive encoding of terms, formulas and deductions, and this leads to Godel's theorem exhibiting statements already undecidable in these fragments. Its central idea, isolated in Tarski's fixpoint lemma, has a certain analogy with Eubulides' antinomy of the Liar, and in a non-technical chapter, accessible to a wider class of readers, this analogy is exploited for an informal discussion of undefinability and incompleteness. The technical tools required to verify the hypotheses on arithmetical representability, on the other hand, are collected in an independent presentation of recursive functions and relations.

Fuzzy set theory - and its underlying fuzzy logic - represents one of the most significant scientific and cultural paradigms to emerge in the last half-century. Its theoretical and technological promise is vast, and we are only beginning to experience its potential. Clustering is the first and most basic application of fuzzy set theory, but forms the basis of many, more sophisticated, intelligent computational models, particularly in pattern recognition, data mining, adaptive and hierarchical clustering, and classifier design.

Fuzzy Sets and their Application to Clustering and Training offers a comprehensive introduction to fuzzy set theory, focusing on the concepts and results needed for training and clustering applications. It provides a unified mathematical framework for fuzzy classification and clustering, a methodology for developing training and classification methods, and a general method for obtaining a variety of fuzzy clustering algorithms.
The authors - top experts from around the world - combine their talents to lay a solid foundation for applications of this powerful tool, from the basic concepts and mathematics through the study of various algorithms, to validity functionals and hierarchical clustering. The result is Fuzzy Sets and their Application to Clustering and Training - an outstanding initiation into the world of fuzzy learning classifiers and fuzzy clustering.

As the amount of information recorded and stored electronically grows ever larger, it becomes increasingly useful, if not essential, to develop better and more efficient ways to summarize and extract information from these large, multivariate data sets. The field of classification does just that-investigates sets of "objects" to see if they can be summarized into a small number of classes comprising similar objects.
Researchers have made great strides in the field over the last twenty years, and classification is no longer perceived as being concerned solely with exploratory analyses. The second edition of Classification incorporates many of the new and powerful methodologies developed since its first edition. Like its predecessor, this edition describes both clustering and graphical methods of representing data, and offers advice on how to decide which methods of analysis best apply to a particular data set. It goes even further, however, by providing critical overviews of recent developments not widely known, including efficient clustering algorithms, cluster validation, consensus classifications, and the classification of symbolic data.
The author has taken an approach accessible to researchers in the wide variety of disciplines that can benefit from classification analysis and methods. He illustrates the methodologies by applying them to data sets-smaller sets given in the text, larger ones available through a Web site.
Large multivariate data sets can be difficult to comprehend-the sheer volume and complexity can prove overwhelming. Classification methods provide efficient, accurate ways to make them less unwieldy and extract more information. Classification, Second Edition offers the ideal vehicle for gaining the background and learning the methodologies-and begin putting these techniques to use.

This is a classic introduction to set theory in three parts. The first part gives a general introduction to set theory, suitable for undergraduates; complete proofs are given and no background in logic is required. Exercises are included, and the more difficult ones are supplied with hints. An appendix to the first part gives a more formal foundation to axiomatic set theory, supplementing the intuitive introduction given in the first part. The final part gives an introduction to modern tools of combinatorial set theory. This part contains enough material for a graduate course of one or two semesters. The subjects discussed include stationary sets, delta systems, partition relations, set mappings, measurable and real-valued measurable cardinals. Two sections give an introduction to modern results on exponentiation of singular cardinals, and certain deeper aspects of the topics are developed in advanced problems.

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.

Classical and Fuzzy Concepts in Mathematical Logic and Applications provides a broad, thorough coverage of the fundamentals of two-valued logic, multivalued logic, and fuzzy logic. Exploring the parallels between classical and fuzzy mathematical logic, the book examines the use of logic in computer science, addresses questions in automatic deduction, and describes efficient computer implementation of proof techniques. Specific issues discussed include: oPropositional and predicate logic oLogic networks oLogic programming oProof of correctness oSemantics oSyntax oCompletenesss oNon-contradiction oTheorems of Herbrand and Kalman The authors consider that the teaching of logic for computer science is biased by the absence of motivations, comments, relevant and convincing examples, graphic aids, and the use of color to distinguish language and metalanguage. Classical and Fuzzy Concepts in Mathematical Logic and Applications discusses how the presence of these facts trigger a stirring, decisive insight into the understanding process. This view shapes this work, reflecting the authors' subjective balance between the scientific and pedagogic components of the textbook. Usually, problems in logic lack relevance, creating a gap between classroom learning and applications to real-life problems. The book includes a variety of application-oriented problems at the end of almost every section, including programming problems in PROLOG III. With the possibility of carrying out proofs with PROLOG III and other software packages, readers will gain a first-hand experience and thus a deeper understanding of the idea of formal proof.

This introduction to mathematical logic takes G del's incompleteness theorem as a starting point. It goes beyond a standard text book and should interest everyone from mathematicians to philosophers and general readers who wish to understand the foundations and limitations of modern mathematics.

Accessible to all students with a sound background in high school mathematics, A Concise Introduction to Pure Mathematics, Fourth Edition presents some of the most fundamental and beautiful ideas in pure mathematics. It covers not only standard material but also many interesting topics not usually encountered at this level, such as the theory of solving cubic equations; Euler's formula for the numbers of corners, edges, and faces of a solid object and the five Platonic solids; the use of prime numbers to encode and decode secret information; the theory of how to compare the sizes of two infinite sets; and the rigorous theory of limits and continuous functions. New to the Fourth Edition Two new chapters that serve as an introduction to abstract algebra via the theory of groups, covering abstract reasoning as well as many examples and applications New material on inequalities, counting methods, the inclusion-exclusion principle, and Euler's phi function Numerous new exercises, with solutions to the odd-numbered ones Through careful explanations and examples, this popular textbook illustrates the power and beauty of basic mathematical concepts in number theory, discrete mathematics, analysis, and abstract algebra. Written in a rigorous yet accessible style, it continues to provide a robust bridge between high school and higher-level mathematics, enabling students to study more advanced courses in abstract algebra and analysis.

Transition to Real Analysis with Proof provides undergraduate students with an introduction to analysis including an introduction to proof. The text combines the topics covered in a transition course to lead into a first course on analysis. This combined approach allows instructors to teach a single course where two were offered. The text opens with an introduction to basic logic and set theory, setting students up to succeed in the study of analysis. Each section is followed by graduated exercises that both guide and challenge students. The author includes examples and illustrations that appeal to the visual side of analysis. The accessible structure of the book makes it an ideal refence for later years of study or professional work. Combines the author's previous works Elements of Advanced Mathematics with Foundations of Analysis Combines logic, set theory and other elements with a one-semester introduction to analysis. Author is a well-known mathematics educator and researcher Targets a trend to combine two courses into one

Bridging the gap between procedural mathematics that emphasizes calculations and conceptual mathematics that focuses on ideas, Mathematics: A Minimal Introduction presents an undergraduate-level introduction to pure mathematics and basic concepts of logic. The author builds logic and mathematics from scratch using essentially no background except natural language. He also carefully avoids circularities that are often encountered in related books and places special emphasis on separating the language of mathematics from metalanguage and eliminating semantics from set theory. The first part of the text focuses on pre-mathematical logic, including syntax, semantics, and inference. The author develops these topics entirely outside the mathematical paradigm. In the second part, the discussion of mathematics starts with axiomatic set theory and ends with advanced topics, such as the geometry of cubics, real and p-adic analysis, and the quadratic reciprocity law. The final part covers mathematical logic and offers a brief introduction to model theory and incompleteness. Taking a formalist approach to the subject, this text shows students how to reconstruct mathematics from language itself. It helps them understand the mathematical discourse needed to advance in the field.

This volume contains the proceedings of the conference Logical Foundations of Mathematics, Computer Science, and Physics-Kurt Godel's Legacy, held in Brno, Czech Republic on the 90th anniversary of his birth. The wide and continuing importance of Godel s work in the logical foundations of mathematics, computer science, and physics is confirmed by the broad range of speakers who participated in making this gathering a scientific event.