This book is an introduction to modern cardinal arithmetic, developed in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice. It splits into three parts. Part one, which is contained in Chapter 1, describes the classical cardinal arithmetic due to Bernstein, Cantor, Hausdorff, Konig, and Tarski. The results were found in the years between 1870 and 1930. Part two, which is Chapter 2, characterizes the development of cardinal arith metic in the seventies, which was led by Galvin, Hajnal, and Silver. The third part, contained in Chapters 3 to 9, presents the fundamental investigations in pcf-theory which has been developed by S. Shelah to answer the questions left open in the seventies. All theorems presented in Chapter 3 and Chapters 5 to 9 are due to Shelah, unless otherwise stated. We are greatly indebted to all those set theorists whose work we have tried to expound. Concerning the literature we owe very much to S. Shelah's book [Sh5] and to the article by M. R. Burke and M. Magidor [BM] which also initiated our students' interest for Shelah's pcf-theory.

Reprint. Paperback. 387 pp. Diophantus of Alexandria, sometimes called "the father of algebra," was an Alexandrian mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost. In studying Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of diophantine equations ("diophantine geometry") and of diophantine approximations remain important areas of mathematical research. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation. Heath's work is one of the standard books in the field.

This volume is a compilation of the best papers presented at the First International Symposium on Fuzzy and Rough Sets (ISFUROS 2006) held in Santa Clara, Cuba. They contain valuable contributions both in the theoretical field and in several application domains such as intelligent control, data analysis, decision making and machine learning, just to name a few. Together, they capture the huge potential of the aforementioned methodologies.

Now in a new edition --the classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient formal logic to give a full development of G del's incompleteness theorems. Part IV considers the significance of the technical work with a discussion of Church's Thesis and readings on the foundations of mathematics. This new edition contains the timeline "Computability and Undecidability" as well as the essay "On mathematics."

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."

This volume is both a tribute to Ulrich Felgner's research in algebra, logic, and set theory and a strong research contribution to these areas. Felgner's former students, friends and collaborators have contributed sixteen papers to this volume that highlight the unity of these three fields in the spirit of Ulrich Felgner's own research. The interested reader will find excellent original research surveys and papers that span the field from set theory without the axiom of choice via model-theoretic algebra to the mathematics of intonation.

Russell's paradox arises when we consider those sets that do not belong to themselves. The collection of such sets cannot constitute a set. Step back a bit. Logical formulas define sets (in a standard model). Formulas, being mathematical objects, can be thought of as sets themselves-mathematics reduces to set theory. Consider those formulas that do not belong to the set they define. The collection of such formulas is not definable by a formula, by the same argument that Russell used. This quickly gives Tarski's result on the undefinability of truth. Variations on the same idea yield the famous results of Godel, Church, Rosser, and Post. This book gives a full presentation of the basic incompleteness and undecidability theorems of mathematical logic in the framework of set theory. Corresponding results for arithmetic follow easily, and are also given. Godel numbering is generally avoided, except when an explicit connection is made between set theory and arithmetic. The book assumes little technical background from the reader. One needs mathematical ability, a general familiarity with formal logic, and an understanding of the completeness theorem, though not its proof. All else is developed and formally proved, from Tarski's Theorem to Godel's Second Incompleteness Theorem. Exercises are scattered throughout.

This is a collection of articles on set theory written by some of the participants in theResearchProgrammeonSetTheoryanditsApplicationsthattookplaceatthe Centre de Recerca Matem' atica (CRM) in Bellaterra (Barcelona). The Programme run from September 2003 to July 2004 and included an international conference on set theory in September 2003, an advanced course on Ramsey methods in ? analysis in January 2004, and a joint CRM-ICREA workshop on the foundations of set theory in June 2004, the latter held in Barcelona. A total of 33 short and long term visitors from 15 countries participated in the Programme. This volume consists of two parts, the ?rst containing survey papers on some of the mainstream areas of set theory, and the second containing original research papers. All of them are authored by visitors who took part in the set theory Programme or by participants in the Programme's activities. The survey papers cover topics as Omega-logic, applications of set theory to lattice theory and Boolean algebras, real-valued measurable cardinals, complexity of sets and relations in continuum theory, weak subsystems of axiomatic set t- ory, de?nable versions of large cardinals, and selection theory for open covers of topological spaces. As for the research papers, they range from topics such as the number of near-coherence classes of ultra?lters, the consistency strength of bounded forcing axioms,P (?) combinatorics,someapplicationsof morasses,subgroupsofAbelian ? Polish groups, adding club subsets of ? with ?nite conditions, the consistency 2 strength of mutual stationarity, and new axioms of set theory.

This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.

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 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.

Fuzzy set and logic theory suggest that all natural language linguistic expressions are imprecise and must be assessed as a matter of degree. But in general membership degree is an imprecise notion which requires that Type 2 membership degrees be considered in most applications related to human decision making schemas. Even if the membership functions are restricted to be Type1, their combinations generate an interval - valued Type 2 membership. This is part of the general result that Classical equivalences breakdown in Fuzzy theory. Thus all classical formulas must be reassessed with an upper and lower expression that are generated by the breakdown of classical formulas.

Key features:

- Ontological grounding
- Epistemological justification
- Measurement of Membership
- Breakdown of equivalences
- FDCF is not equivalent to FCCF
- Fuzzy Beliefs
- Meta-Linguistic axioms
- Ontological grounding
- Epistemological justification
- Measurement of Membership
- Breakdown of equivalences
- FDCF is not equivalent to FCCF
- Fuzzy Beliefs
- Meta-Linguistic axioms

The capabilities of modern technology are rapidly increasing, spurred on to a large extent by the tremendous advances in communications and computing. Automated vehicles and global wireless connections are some examples of these advances. In order to take advantage of such enhanced capabilities, our need to model and manipulate our knowledge of the geophysical world, using compatible representations, is also rapidly increasing. In response to this one fundamental issue of great concern in modern geographical research is how to most effectively capture the physical world around us in systems like geographical information systems (GIS). Making this task even more challenging is the fact that uncertainty plays a pervasive role in the representation, analysis and use of geospatial information. The types of uncertainty that appear in geospatial information systems are not the just simple randomness of observation, as in weather data, but are manifested in many other forms including imprecision, incompleteness and granularization. Describing the uncertainty of the boundaries of deserts and mountains clearly require different tools than those provided by probability theory. The multiplicity of modalities of uncertainty appearing in GIS requires a variety of formalisms to model these uncertainties. In light of this it is natural that fuzzy set theory has become a topic of intensive interest in many areas of geographical research and applications This volume, Fuzzy Modeling with Spatial Information for Geographic Problems, provides many stimulating examples of advances in geographical research based on approaches using fuzzy sets and related technologies.

Game theory has already proved its tremendous potential for con?ict resolution problems in the ?elds of Decision Theory and Economics. In the recent past, there have been attempts to extend the results of crisp game theory to those con?ict resolution problems which are fuzzy in nature e.g. Nishizaki and Sakawa [61] and references cited there in. These developments have lead to the emergence of a new area in the literature called fuzzy games. Another area in the fuzzy decision theory, which has been growing very fast is the area of fuzzy mathematical programming and its applications to various branches of sciences, Engineering and Management. In the crisp scenario, there exists a beautiful relationship between two person zero sum matrix game theory and duality in linear p- gramming. It is therefore natural to ask if something similar holds in the fuzzy scenario as well. This discussion essentially constitutes the core of our presentation. The objective of this book is to present a systematic and focussed study of the application of fuzzy sets to two very basic areas of decision theory, namely Mathematical Programming and Matrix Game Theory.

Problems in Set Theory, Mathematical Logic and the Theory of Algorithms by I. Lavrov & L. Maksimova is an English translation of the fourth edition of the most popular student problem book in mathematical logic in Russian. It covers major classical topics in proof theory and the semantics of propositional and predicate logic as well as set theory and computation theory. Each chapter begins with 1-2 pages of terminology and definitions that make the book self-contained. Solutions are provided. The book is likely to become an essential part of curricula in logic.

This is the first book presenting cardinality theory of fuzzy sets with triangular norms, including its scalar and "fuzzy" streams. This theory constitutes not only a powerful basis but also a useful tool for modelling and processing vague and imprecise quantitative information. The multiple application areas of the theory encompass computer science, soft computing, computing with words, and decision-making. Starting with a presentation of the fundamentals of triangular norms and fuzzy set theory, the book offers a self-contained, concise and systematic exposition of cardinalities of fuzzy sets that includes many examples.

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.

After the pioneering works by Robbins {1944, 1945) and Choquet (1955), the notation of a set-valued random variable (called a random closed set in literatures) was systematically introduced by Kendall {1974) and Matheron {1975). It is well known that the theory of set-valued random variables is a natural extension of that of general real-valued random variables or random vectors. However, owing to the topological structure of the space of closed sets and special features of set-theoretic operations ( cf. Beer [27]), set-valued random variables have many special properties. This gives new meanings for the classical probability theory. As a result of the development in this area in the past more than 30 years, the theory of set-valued random variables with many applications has become one of new and active branches in probability theory. In practice also, we are often faced with random experiments whose outcomes are not numbers but are expressed in inexact linguistic terms.

Since their inception, fuzzy sets and fuzzy logic became popular. The reason is that the very idea of fuzzy sets and fuzzy logic attacks an old tradition in science, namely bivalent (black-or-white, all-or-none) judg ment and reasoning and the thus resulting approach to formation of scientific theories and models of reality. The idea of fuzzy logic, briefly speaking, is just the opposite of this tradition: instead of full truth and falsity, our judgment and reasoning also involve intermediate truth values. Application of this idea to various fields has become known under the term fuzzy approach (or graded truth approach). Both prac tice (many successful engineering applications) and theory (interesting nontrivial contributions and broad interest of mathematicians, logicians, and engineers) have proven the usefulness of fuzzy approach. One of the most successful areas of fuzzy methods is the application of fuzzy relational modeling. Fuzzy relations represent formal means for modeling of rather nontrivial phenomena (reasoning, decision, control, knowledge extraction, systems analysis and design, etc. ) in the pres ence of a particular kind of indeterminacy called vagueness. Models and methods based on fuzzy relations are often described by logical formulas (or by natural language statements that can be translated into logical formulas). Therefore, in order to approach these models and methods in an appropriate formal way, it is desirable to have a general theory of fuzzy relational systems with basic connections to (formal) language which enables us to describe relationships in these systems."

Lectori salutem! The kind reader opens the book that its authors would have liked to read it themselves, but it was not written yet. Then, their only choice was to write this book, to fill a gap in the mathematicalliterature. The idea of convexity has appeared in the human mind since the antiquity and its fertility has led to a huge diversity of notions and of applications. A student intending a thoroughgoing study of convexity has the sensation of swimming into an ocean. It is due to two reasons: the first one is the great number of properties and applications of the classical convexity and second one is the great number of generalisations for various purposes. As a consequence, a tendency of writing huge books guiding the reader in convexity appeared during the last twenty years (for example, the books of P. M. Gruber and J. M. Willis (1993) and R. J. Webster (1994)). Another last years' tendency is to order, from some point of view, as many convexity notions as possible (for example, the book of I. Singer (1997)). These approaches to the domain of convexity follow the previous point of view of axiomatizing it (A. Ghika (1955), W. Prenowitz (1961), D. Voiculescu (1967), V. W. Bryant and R. J. Webster (1969)). Following this last tendency, our book proposes to the reader two classifications of convexity properties for sets, both of them starting from the internal mechanism of defining them.

Assessing the degree to which two objects, an object and a query, or two concepts are similar or compatible is a fundamental component of human reasoning and consequently is critical in the development of automated diagnosis, classification, information retrieval and decision systems. The assessment of similarity has played an important role in such diverse disciplines such as taxonomy, psychology, and the social sciences. Each discipline has proposed methods for quantifying similarity judgments suitable for its particular applications. This book presents a unified approach to quantifying similarity and compatibility within the framework of fuzzy set theory and examines the primary importance of these concepts in approximate reasoning. Examples of the application of similarity measures in various areas including expert systems, information retrieval, and intelligent database systems are provided.

The book aims at surveying results in the application of fuzzy sets and fuzzy logic to economics and engineering. New results include fuzzy non-linear regression, fully fuzzified linear programming, fuzzy multi-period control, fuzzy network analysis, each using an evolutionary algorithm; fuzzy queuing decision analysis using possibility theory; fuzzy differential equations; fuzzy difference equations; fuzzy partial differential equations; fuzzy eigenvalues based on an evolutionary algorithm; fuzzy hierarchical analysis using an evolutionary algorithm; fuzzy integral equations. Other important topics covered are fuzzy input-output analysis; fuzzy mathematics of finance; fuzzy PERT (project evaluation and review technique). No previous knowledge of fuzzy sets is needed. The mathematical background is assumed to be elementary calculus.

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.

How to draw plausible conclusions from uncertain and conflicting sources of evidence is one of the major intellectual challenges of Artificial Intelligence. It is a prerequisite of the smart technology needed to help humans cope with the information explosion of the modern world. In addition, computational modelling of uncertain reasoning is a key to understanding human rationality. Previous computational accounts of uncertain reasoning have fallen into two camps: purely symbolic and numeric. This book represents a major advance by presenting a unifying framework which unites these opposing camps. The Incidence Calculus can be viewed as both a symbolic and a numeric mechanism. Numeric values are assigned indirectly to evidence via the possible worlds in which that evidence is true. This facilitates purely symbolic reasoning using the possible worlds and numeric reasoning via the probabilities of those possible worlds. Moreover, the indirect assignment solves some difficult technical problems, like the combinat ion of dependent sources of evidcence, which had defeated earlier mechanisms. Weiru Liu generalises the Incidence Calculus and then compares it to a succes sion of earlier computational mechanisms for uncertain reasoning: Dempster-Shafer Theory, Assumption-Based Truth Maintenance, Probabilis tic Logic, Rough Sets, etc. She shows how each of them is represented and interpreted in Incidence Calculus. The consequence is a unified mechanism which includes both symbolic and numeric mechanisms as special cases. It provides a bridge between symbolic and numeric approaches, retaining the advantages of both and overcoming some of their disadvantages."

Decision makers in managerial and public organizations often encounter de cision problems under conflict or competition, because they select strategies independently or by mutual agreement and therefore their payoffs are then affected by the strategies of the other decision makers. Their interests do not always coincide and are at times even completely opposed. Competition or partial cooperation among decision makers should be considered as an essen tial part of the problem when we deal with the decision making problems in organizations which consist of decision makers with conflicting interests. Game theory has been dealing with such problems and its techniques have been used as powerful analytical tools in the resolution process of the decision problems. The publication of the great work by J. von Neumann and O. Morgen stern in 1944 attracted attention of many people and laid the foundation of game theory. We can see remarkable advances in the field of game theory for analysis of economic situations and a number of books in the field have been published in recent years. The aim of game theory is to specify the behavior of each player so as to optimize the interests of the player. It then recommends a set of solutions as strategies so that the actions chosen by each decision maker (player) lead to an outcome most profitable for himself or her self."