Labyrinth of Thought discusses the emergence and development of set theory and the set-theoretic approach to mathematics during the period 1850-1940. Rather than focusing on the pivotal figure of Georg Cantor, it analyzes his work and the emergence of transfinite set theory within the broader context of the rise of modern mathematics. The text has a tripartite structure. Part 1, The Emergence of Sets within Mathematics, surveys the initial motivations for a mathematical notion of a set within several branches of the discipline (geometry, algebra, algebraic number theory, real and complex analysis), emphasizing the role played by Riemann in fostering acceptance of the set-theoretic approach. In Part 2, Entering the Labyrinth, attention turns to the earliest theories of sets, their evolution, and their reception by the mathematical community; prominent are the epoch-making contributions of Cantor and Dedekind, and the complex interactions between them. Part 3, In Search of an Axiom System, studies the four-decade period from the discovery of set-theoretic paradoxes to Godel s independence results, an era during which set theory gradually became assimilated into mainstream mathematics; particular attention is given to the interactions between axiomatic set theory and modern systems of formal logic, especially the interplay between set theory and type theory. A new Epilogue for this second edition offers further reflections on the foundations of set theory, including the "dichotomy conception" and the well-known iterative conception."

A survey of the philosophical implications and practical applications of fuzzy systems

Fuzzy mathematical concepts such as fuzzy sets, fuzzy logic, and similarity relations represent one of the most exciting currents in modern engineering and have great potential in applications ranging from control theory to bioinformatics. Data Engineering guides the reader through a number of concepts interconnected by fuzzy mathematics and discusses these concepts from a systems engineering perspective to showcase the continuing vitality, attractiveness, and applicability of fuzzy mathematics.

The author discusses the fundamental aspects of data analysis, systems modeling, and uncertainty calculi. He avoids a narrow discussion of specialized methodologies and takes a holistic view of the nature and application of fuzzy systems, considering principles, paradigms, and methodologies along the way. This broad coverage includes:

• Fundamentals of modeling, identification, and clustering
• System analysis
• Uncertainty techniques
• Random-set modeling and identification
• Fuzzy inference engines
• Fuzzy classification, control, and mathematics

In the important emerging field of bioinformatics, the book sets out how to encode a natural system in mathematical models, describes methods to identify interrelationships and interactions from data, and thereby helps the practitioner to decide which variables to measure and why.

Data Engineering serves as an up-to-date and informative survey of the theoretical and practical tools for analyzing complex systems. It offers a unique treatment of complex issues that is accessible to students and researchers from a variety of backgrounds.

This book is an introduction to set theory for beginning graduate students who want to get a sound grounding in those aspects of set theory used extensively throughout other areas of mathematics. Topics covered include formal languages and models, the power and limitation of the Axiomatic Method, the Axiom of Choice, including the fascinating Banach-Tarski Paradox, applications of Zorn's Lemma, ordinal arithmetic, including transfinite induction, and cardinal arithmetic. The style of writing, more a dialogue with the reader than that of the Master indoctrinating the pupil, makes this also very suitable for self-study.

The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. It is also viewed as a foundation of mathematics so that "to make a notion precise" simply means "to define it in set theory." This book gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation.

The requirement to reason logically forms the basis of all mathematics, and hence mathematical logic is one of the most fundamental topics that students will study. Assuming no prior knowledge of the topic, this book provides an accessible introduction for advanced undergraduate students.

The breathtakingly rapid pace of change in computing makes it easy to overlook the pioneers who began it all. The Universal Computer: The Road from Leibniz to Turing explores the fascinating lives, ideas, and discoveries of seven remarkable mathematicians. It tells the stories of the unsung heroes of the computer age - the logicians.

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In "Higher Topos Theory," Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.

The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

This book provides an approach to the study of perturbation and discretization effects on the long-time behavior of dynamical and control systems. It analyzes the impact of time and space discretizations on asymptotically stable attracting sets, attractors, asumptotically controllable sets and their respective domains of attractions and reachable sets. Combining robust stability concepts from nonlinear control theory, techniques from optimal control and differential games and methods from nonsmooth analysis, both qualitative and quantitative results are obtained and new algorithms are developed, analyzed and illustrated by examples.

This book is an excellent starting point for any curriculum in fuzzy systems fields such as computer science, mathematics, business/economics and engineering. It covers the basics leading to: fuzzy clustering, fuzzy pattern recognition, fuzzy database, fuzzy image processing, soft computing, fuzzy applications in operations research, fuzzy decision making, fuzzy rule based systems, fuzzy systems modeling, fuzzy mathematics. It is not a book designed for researchers - it is where you really learn the "basics" needed for any of the above-mentioned applications. It includes many figures and problem sets at the end of sections.

This two-volume work bridges the gap between introductory expositions of logic (or set theory) and the research literature. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly lecture style that makes them equally effective for self-study or class use. Volume I includes formal proof techniques, applications of compactness (including nonstandard analysis), computability and its relation to the completeness phenonmenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen.

By its nature, set theory does not depend on any previous mathematical knowl edge. Hence, an individual wanting to read this book can best find out if he is ready to do so by trying to read the first ten or twenty pages of Chapter 1. As a textbook, the book can serve for a course at the junior or senior level. If a course covers only some of the chapters, the author hopes that the student will read the rest himself in the next year or two. Set theory has always been a sub ject which people find pleasant to study at least partly by themselves. Chapters 1-7, or perhaps 1-8, present the core of the subject. (Chapter 8 is a short, easy discussion of the axiom of regularity). Even a hurried course should try to cover most of this core (of which more is said below). Chapter 9 presents the logic needed for a fully axiomatic set th ory and especially for independence or consistency results. Chapter 10 gives von Neumann's proof of the relative consistency of the regularity axiom and three similar related results. Von Neumann's 'inner model' proof is easy to grasp and yet it prepares one for the famous and more difficult work of GOdel and Cohen, which are the main topics of any book or course in set theory at the next level."

The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems.

This book comprises revised full versions of lectures given during the 9th European Summer School in Logic, Languages, and Information, ESSLLI'97, held in Aix-en-Provence, France, in August 1997. The six lectures presented introduce the reader to the state of the art in the area of generalized quantifiers and computation. Besides an introductory survey by the volume editor various aspects of generalized quantifiers are studied in depth.

The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad vanced Study Institute on "Difference sets, sequences and their correlation properties." This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized."

Introduction Model theorists have often joked in recent years that the part of mathemat- ical logic known as "pure model theory" (or stability theory), as opposed to the older and more traditional "model theory applied to algebra" , turns out to have more and more to do with other subjects ofmathematics and to yield gen- uine applications to combinatorial geometry, differential algebra and algebraic geometry. We illustrate this by presenting the very striking application to diophantine geometry due to Ehud Hrushovski: using model theory, he has given the first proof valid in all characteristics of the "Mordell-Lang conjecture for function fields" (The Mordell-Lang conjecture for function fields, Journal AMS 9 (1996), 667-690). More recently he has also given a new (model theoretic) proof of the Manin-Mumford conjecture for semi-abelian varieties over a number field. His proofyields the first effective bound for the cardinality ofthe finite sets involved (The Manin-Mumford conjecture, preprint). There have been previous instances of applications of model theory to alge- bra or number theory, but these appl~cations had in common the feature that their proofs used a lot of algebra (or number theory) but only very basic tools and results from the model theory side: compactness, first-order definability, elementary equivalence...

This book is dedicated to Andrzej Ehrenfeucht on the occasion of his 65th birthday. On personal invitation by the volume editors, 22 internationally well-known scientists from mathematical logics and theoretical computer science participated in this project honoring an excellent scientist with excellent papers centered around his scientific work.
The 22 invited papers are presented in topical sections on model theory, games and logic, graphs and algorithms, pattern matching and learning, combinatorics of words, algebra of languages, formal language theory, and computational molecular biology.

Fuzzy logic has become an important tool for a number of different applications ranging from the control of engineering systems to artificial intelligence. In this concise introduction, the author presents a succinct guide to the basic ideas of fuzzy logic, fuzzy sets, fuzzy relations, and fuzzy reasoning, and shows how they may be applied. The book culminates in a chapter which describes fuzzy logic control: the design of intelligent control systems using fuzzy if-then rules which make use of human knowledge and experience to behave in a manner similar to a human controller. Throughout, the level of mathematical knowledge required is kept basic and the concepts are illustrated with numerous diagrams to aid in comprehension. As a result, all those curious to know more about fuzzy concepts and their real-world application will find this a good place to start.

This text provides a wide-ranging introduction to convex sets and functions, suitable for final-year undergraduates and also graduate students. Demanding only a modest knowledge of analysis and linear algebra, it discusses such diverse topics as number theory, classical extremum problems, combinatorial geometry, linear programming, game theory, polytopes, bodies of constant width, the gamma function, minimax approximation, and the theory of linear, classical, and matrix inequalities.

The purpose of this monograph is to develop a very general approach to the algebra ization of sententiallogics, to show its results on a number of particular logics, and to relate it to other existing approaches, namely to those based on logical matrices and the equational consequence developed by Blok, Czelakowski, Pigozzi and others. The main distinctive feature of our approachlies in the mathematical objects used as models of a sententiallogic: We use abstract logics, while the dassical approaches use logical matrices. Using models with more structure allows us to reflect in them the metalogical properties of the sentential logic. Since an abstract logic can be viewed as a "bundle" or family of matrices, one might think that the new models are essentially equivalent to the old ones; but we believe, after an overall appreciation of the work done in this area, that it is precisely the treatment of an abstract logic as a single object that gives rise to a useful -and beautiful- mathematical theory, able to explain the connections, not only at the logical Ievel but at the metalogical Ievel, between a sentential logic and the particular dass of models we associate with it, namely the dass of its full models. Traditionally logical matrices have been regarded as the most suitable notion of model in the algebraic studies of sentential logics; and indeed this notion gives sev eral completeness theorems and has generated an interesting mathematical theory."

An advanced graduate course. Some knowledge of forcing is assumed, and some elementary Mathematical Logic, e.g. the Lowenheim-Skolem Theorem. A student with one semester of mathematical logic and 1 of set theory should be prepared to read these notes. The first half deals with the general area of Borel hierarchies. What are the possible lengths of a Borel hierarchy in a separable metric space? Lebesgue showed that in an uncountable complete separable metric space the Borel hierarchy has uncountably many distinct levels, but for incomplete spaces the answer is independent. The second half includes Harrington's Theorem - it is consistent to have sets on the second level of the projective hierarchy of arbitrary size less than the continuum and a proof and appl- ications of Louveau's Theorem on hyperprojective parameters.

In recent years, applied mathematics has been used in all novel disciplines of scientific development. Advances in Applied Mathematical Problems summarizes interdisciplinary work within the field of applied mathematics. The topics discussed in the book include: * Similarity Solutions of Spherical Shock Waves in a Self-Gravitating Ideal Gas * Dual Solutions for Finite Element Analysis of Unsteady Hydromagnetic Stagnation Point Flow of Water Nanofluid Generated by Stretching Sheet * Multiparametric modeling of carbon cycle in temperate wetlands for regional climate change analysis using satellite data * An Intelligent Neuro Fuzzy System for Pattern Classification * Fuzzy inventory model with demand, deterioration and inflation: a comparative study through NGTFN and CNTFN * Summability and its application for the stability of the system * Design Of Manufacturing, Control, And Automation Systems * SEIR - Application for Crop through Water and Soil Texture * Advances in radial basis functions * Modeling for Time Period of Natural Frequency for Non-Homogeneous Square Plate With Variable Thickness and Temperature Effect * A Study On Metric Fixed Point Theorems Satisfying Integral Type Contractions * Objective Function - In Radiometric Studies - Application to Agrs Surveys Associated With Radon * Modelling Kernel Function in Black body Radiation Inversion

Die Weltwirtschaft steht an der Schwelle zu einem Zeitalter intelligenter Systeme. Viele Untemehmen werden sich kiinftig im zunehmend globalen Wettbewerb nur dann behaupten konnen, wenn es ihnen gelingt, ihre Produkte und Verfahren mit einem hohen MaB an Intelligenz auszustatten (, Machine Intelligence Quotient'). Gerade die Bundesrepublik Deutschland ist als sogenanntes Hochlohnland und mit ihren sehr begrenzten namrlichen Ressourcen zur Sicherung ihres wirtschaftlichen Wohlstandes auf eine schnelle Umsetzung fortschrittlicher Methoden und Technologien angewiesen. Eine zentrale Rolle bei der Entwicklung intelligenter Produkte und Verfahren spielen neue methodische Ansatze der Informatik. Die gesellschaftlichen Krafte in der Bundesrepublik haben dies erkannt, bis hin zu GewerkschaftsfUhrem, die intelligente Produkte fiir den Weltrnarkt fordem. Der erste Absatz dieses V orwortes starnmt iibrigens von Giinther Einert, dem Minister fUr Wirtschaft, Mittelstand und Technologie des Landes NRW. Neben der Mikro-Elektronik und der Mikro-Systemtechnik inklusive cler Mikro Sensorik als Basistechnologien werden Fuzzy-Logik, Kiinstliche Neuronale Netze und Evolutionare Algorithmen sowie Kombinationen davon als Problemlosungstechniken bei der Entwicklung der geforderten intelligenten Produkte eine wesentliche Rolle spielen. In einem irrt der Minister allerdings: neu sind die methodischen Ansatze, auf die er sich bezieht, durchaus nicht. Neu ist nur ihre erfolgreiche Umsetzung. Wir haben die Thematik der 4. Dortmunder Fuzzy-Tage erweitert und dies mit dem Untertitel "Fachtagung iiber Computational Intelligence" zum Ausdruck gebracht. Wir sind nicht die ersten, die eine Fachtagung tiber Computational Intelligence ankiindigen (das war IEEE), aber wir sind nach meiner Kenntnis die ersten, die eine so1che Fachtagung durchfUhren."

There seems to be no doubt that geometry originates from such practical activ ities as weather observation and terrain survey. But there are different manners, methods, and ways to raise the various experiences to the level of theory so that they finally constitute a science. F. Engels said, "The objective of mathematics is the study of space forms and quantitative relations of the real world. " Dur ing the time of the ancient Greeks, there were two different methods dealing with geometry: one, represented by the Euclid's "Elements," purely pursued the logical relations among geometric entities, excluding completely the quantita tive relations, as to establish the axiom system of geometry. This method has become a model of deduction methods in mathematics. The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a spherical surface and of a parabolic sector. Though these approaches vary in style, have their own features, and reflect different viewpoints in the development of geometry, both have made great contributions to the development of mathematics. The development of geometry in China was all along concerned with quanti tative relations."

The book concerns limit theorems and laws of large numbers for scaled unionsof independent identically distributed random sets. These results generalizewell-known facts from the theory of extreme values. Limiting distributions (called union-stable) are characterized and found explicitly for many examples of random closed sets. The speed of convergence in the limit theorems for unions is estimated by means of the probability metrics method.It includes the evaluation of distances between distributions of random sets constructed similarly to the well-known distances between distributions of random variables. The techniques include regularly varying functions, topological properties of the space of closed sets, Choquet capacities, convex analysis and multivalued functions. Moreover, the concept of regular variation is elaborated for multivalued (set-valued) functions. Applications of the limit theorems to simulation of random sets, statistical tests, polygonal approximations of compacts, limit theorems for pointwise maxima of random functions are considered. Several open problems are mentioned. Addressed primarily to researchers in the theory of random sets, stochastic geometry and extreme value theory, the book will also be of interest to applied mathematicians working on applications of extremal processes and their spatial counterparts. The book is self-contained, and no familiarity with the theory of random sets is assumed.

In the last 25 years, the fuzzy set theory has been applied in many disciplines such as operations research, management science, control theory, artificial intelligence/expert system, etc. In this volume, methods and applications of fuzzy mathematical programming and possibilistic mathematical programming are first systematically and thoroughly reviewed and classified. This state-of-the-art survey provides readers with a capsule look into the existing methods, and their characteristics and applicability to analysis of fuzzy and possibilistic programming problems. To realize practical fuzzy modelling, we present solutions for real-world problems including production/manufacturing, transportation, assignment, game, environmental management, resource allocation, project investment, banking/finance, and agricultural economics. To improve flexibility and robustness of fuzzy mathematical programming techniques, we also present our expert decision-making support system IFLP which considers and solves all possibilities of a specific domain of (fuzzy) linear programming problems. Basic fuzzy set theories, membership functions, fuzzy decisions, operators and fuzzy arithmetic are introduced with simple numerical examples in aneasy-to-read and easy-to-follow manner. An updated bibliographical listing of 60 books, monographs or conference proceedings, and about 300 selected papers, reports or theses is presented in the end of this study.