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• Algebra of sets
• Axioms of set theory, relations, funcilons
• Natual numbers, finite and infinite sets
• Generalized union, intersection and cartesian product
• Theory of cardinal numbers
• Linearly ordered sets
• Well-ordered sets
• Alephs and related topics
• Inaccessible cardinals : the continuum hypothesis
• Introduction to the theory of analytic and projective sets.

• 1. Historical Roots
• 2. The Notion of Set
• 3. The Zermelo-Fraenkel Axioms
• 4. Immediate Consequences
• 5. Number Systems within Set Theory
• 6. Infinities
• 7. The Axiom of Choice
• 8. Topics in Higher Set Theory
• 9. Metamathematics of Set Theory
• 10. Large Cardinals and Determinacy
• 11. Concluding Philosophical Remarks.
• (source: Nielsen Book Data)

Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms and undecided questions, and philosophical issues raised by technical developments.
(source: Nielsen Book Data)

A new approach to the standard axioms of set theory, relating the theory to the philosophy of science and metametaphysics.

• Prima parte
• Storia
• Fondamenti della matematica
• Seconda parte
• La teoria
• Applicazioni.

Gli insegnanti si trovano in difficoltà a proposito dello spazio e dell'enfasi da dare agli argomenti di teoria degli insiemi, nella propria preparazione e nel proprio lavoro, perché all'università non è stata loro fornita una conoscenza adeguata. Si può tranquillamente affermare, sulla base di molta esperienza, che il matematico medio, anche chi fa ricerca, non sa cosa sia la teoria degli insiemi. Due pregiudizi si frappongono a una buona conoscenza della teoria: uno, di tipo minimalista, è la sua identificazione con una non meglio precisata "insiemistica", un linguaggio austero fin troppo impegnativo ove lo si voglia imporre prematuramente; l'altro è di tipo massimalista e consiste nel supposto, ed effettivo legame con le questioni più sottili dei fondamenti della matematica. Ma la teoria ha un contenuto matematico importante, e con molti risvolti di interesse didattico. Si può dire in una parola che è lo studio dell'infinito, il che comporta anche per complemento che sia uno studio del finito. Attraverso gli insiemi numerabili ed effettivamente generati si stabilisce anche un collegamento con la più concreta teoria della calcolabilità. Il libro è solo una guida, non un manuale: sono indicati gli argomenti di maggior rilievo; sono offerti commenti sui risultati più significativi; sono segnalati anche temi da non approfondire, pur conoscendone l'esistenza; sono presentate con dettagli formali poche dimostrazioni, tipiche dello stile della materia; sono proposti, come istruzioni per l'uso, alcuni esercizi che potrebbero essere presentarti anche a studenti delle scuole secondarie.

• Éléments et parties d'un cnsernble
• Fonctions
• Produit de plusieurs ensembles
• Réunion, intersection, produit d'une famille d'ensembles
• Relations d'équivalence : ensemble quotient
• Ensembles ordonnés
• Puissances. Ensembles dénombrables
• Échelles d'ensembles et structures.

Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Le Livre de Théorie des ensembles qui vient en tÃate du traité présente les fondements axiomatiques de la théorie des ensembles. Il comprend les chapitres: 1. Description de la mathématique formelle; 1. Théorie des ensembles; 2. Ensembles ordonnés. Cardinaux. 3. nombres entiers; 4. Structures. Il contient également un fascicule de résultats et une note historique. Ce volume est une réimpression de l édition de 1.

• Part I. Elementary Theory: 1. Preliminaries
• 2. The constructible universe
• 3. 1-Trees in L
• 4. +-Trees in L and the fine structure theory
• 5. The story of 0#
• Part II. Advanced theory: 6. The fine structure theory
• 7. Trees and large cardinals in L
• 8. Morasses and the cardinal transfer theorem
• 9. Silver machines
• Remarks and historical notes
• Bibliography
• Glossary of notation
• Index.
• (source: Nielsen Book Data)

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the sixth publication in the Perspectives in Logic series, Keith J. Devlin gives a comprehensive account of the theory of constructible sets at an advanced level. The book provides complete coverage of the theory itself, rather than the many and diverse applications of constructibility theory, although applications are used to motivate and illustrate the theory. The book is divided into two parts: Part I (Elementary Theory) deals with the classical definition of the L -hierarchy of constructible sets and may be used as the basis of a graduate course on constructibility theory. and Part II (Advanced Theory) deals with the J -hierarchy and the Jensen 'fine-structure theory'.
(source: Nielsen Book Data) Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the sixth publication in the Perspectives in Logic series, Keith J. Devlin gives a comprehensive account of the theory of constructible sets at an advanced level. The book provides complete coverage of the theory itself, rather than the many and diverse applications of constructibility theory, although applications are used to motivate and illustrate the theory. The book is divided into two parts: Part I (Elementary Theory) deals with the classical definition of the Lα-hierarchy of constructible sets and may be used as the basis of a graduate course on constructibility theory. and Part II (Advanced Theory) deals with the Jα-hierarchy and the Jensen 'fine-structure theory'.

• Sets, relations and functions
• Counting
• Recurrence Relations
• Logic
• Algebraic structure
• Lattices
• Boolean algebra
• Graphs
• Finite state automata
• Language and grammars.

• 1. Axiom CPAcube and its consequences: properties (A)-(E)
• 2. Games and axiom CPAgame/cube
• 3. Prisms and axioms CPAgame/prism and CPAprism
• 4. CPAprism and coverings with smooth functions
• 5. Applications of CPAgame/prism
• 6. CPA and properties (F*) and (G)
• 7. CPA in the Sacks model.
• (source: Nielsen Book Data)

Here the authors formulate and explore a new axiom of set theory, CPA, the Covering Property Axiom. CPA is consistent with the usual ZFC axioms, indeed it is true in the iterated Sacks model and actually captures the combinatorial core of this model. A plethora of results known to be true in the Sacks model easily follow from CPA. Replacing iterated forcing arguments with deductions from CPA simplifies proofs, provides deeper insight, and leads to new results. One may say that CPA is similar in nature to Martin's axiom, as both capture the essence of the models of ZFC in which they hold. The exposition is self contained and there are natural applications to real analysis and topology. Researchers who use set theory in their work will find much of interest in this book.
(source: Nielsen Book Data)

• Introduction
• 1. Good extender sequences
• 2. Fine structure
• 3. Squashed mice
• 4. Ultrapowers
• 5. Iteration trees
• 6. Uniqueness of wellfounded branches
• 7. The comparison process
• 8. Solidarity and condensation
• 9. Uniqueness of the next extender
• 10. Closure under initial segment
• 11. The construction
• 12. Iterability
• References
• Index of definitions
• Index.
• (source: Nielsen Book Data)

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the third publication in the Lecture Notes in Logic series, Mitchell and Steel construct an inner model with a Woodin cardinal and develop its fine structure theory. This work builds upon the existing theory of a model of the form L[E], where E is a coherent sequence of extenders, and relies upon the fine structure theory of L[E] models with strong cardinals, and the theory of iteration trees and 'backgrounded' L[E] models with Woodin cardinals. This work is what results when fine structure meets iteration trees.
(source: Nielsen Book Data)

• The Basic Classical Notions. kappa-Suslin and lambda-Borel. Basic Notions of the Effective Theory. Structure Theory for Pointclasses. The Constructible Universe. The Playful Universe. The Recursion Theorem. Metamathematics. References. Index.
• (source: Nielsen Book Data)

Now available in paperback, this monograph is a self-contained exposition of the main results and methods of descriptive set theory. It develops all the necessary background material from logic and recursion theory, and treats both classical descriptive set theory and the effective theory developed by logicians.
(source: Nielsen Book Data)

• Introduction
• 1. Plural Quantifications
• 1. Singular Versus Plural Quantifications
• 2. Implication 2.a The Implication Argument 2.b On the Three Theses on Implications 2.c Implication Versus Metaphysical Necessity
• 3. Paraphrase
• 2. The Logic of Plurals
• 1. Language 1.a Elementary Notation *1.b. The Term Connective "and" 1.c. Predicates 1.c.i. Singular, Plural, and Neutral Predicates 1.c.ii. Neutral Expansions and Singular Reducts 1.c.iii. Logical Predicates 1.d. Quantifiers and Variables 1.d.i. Singular and Plural Quantifiers 1.d.ii. Paraphrasing Plural Quantifiers 1.d.iii. Singularizable Plural Quantifications 1.e. The Canonical Notation for the Logic of Plurals 1.e.i. Terms 1.e.ii. Predicates and Simple Sentences 1.e.iii. Quantifiers and Complex Sentences
• 2. Logic 2.a. The Logic of Plurals: Partial Axiomatization 2.a.i. System A 2.a.ii. System B 2.a.iii. System C 2.a.iv. System D *2.b. The Logic of Plurals: Model Theory Appendix: Partial Axiomatization of the Logic of Plurals
• 3. Is Two a Property?
• 1. Why Property Two? 1.a. What Is It to Be a Property? 1.b. Irregularities of Numerical Facts 1.c. Why Not the Set Analysis?
• 2. A Theory of Plural Properties 2.a. The Predicative Part 2.b. The Subject Part 2.c. Instantiation 2.d. Various Kinds of Plural Properties
• 3. Two As an Intrinsic Plural Property
• 4. Concluding Remarks
• 4. What Numbers Should Be
• 1. Are Numbers Objects?
• 2. Plural Properties As Components of Numerical Facts
• 3. Analysis of Numerical Facts
• 4. Numbers Are Properties
• 5. How Sets Are Determined by Their Members
• 1. The Hierarchy of Sets and the Determination of Sets by Their Members
• 2. Reference to Set-like Objects
• 3. Plural Reference to Mundane Objects
• 4. Exclusive Reference to Mundane Objects
• 5. Concluding Remarks References Index.
• (source: Nielsen Book Data)

First Published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.
(source: Nielsen Book Data)

• 1. Iterated Forcing James E. Baumgartner
• 2. The Yorkshireman's guide to proper forcing Keith J. Devlin
• 3. The singular cardinals problem
• independence results Sharon Shelah
• 4. Trees, norms and scales David Guaspari
• 5. On the regularity of ultrafilters Karel Prikry
• 6. Morasses in combinatorial set theory Akihiro Kanamori
• 7. A short course on gap-one morasses with a review of the fine structure of L Lee Stanley.
• (source: Nielsen Book Data)

This book comprises five expository articles and two research papers on topics of current interest in set theory and the foundations of mathematics. Articles by Baumgartner and Devlin introduce the reader to proper forcing. This is a development by Saharon Shelah of Cohen's method which has led to solutions of problems that resisted attack by forcing methods as originally developed in the 1960s. The article by Guaspari is an introduction to descriptive set theory, a subject that has developed dramatically in the last few years. Articles by Kanamori and Stanley discuss one of the most difficult concepts in contemporary set theory, that of the morass, first created by Ronald Jensen in 1971 to solve the gap-two conjecture in model theory, assuming Goedel's axiom of constructibility. The papers by Prikry and Shelah complete the volume by giving the reader the flavour of contemporary research in set theory. This book will be of interest to graduate students and research workers in set theory and mathematical logic.
(source: Nielsen Book Data)

• Introduction.- Fraenkel-Mostowski Set Theory: A Framework for Finitely Supported Mathematics.- Algebraic Structures in Finitely Supported Mathematics.- Extended Fraenkel-Mostowski Set Theory.- Process Calculi in Finitely Supported Mathematics.- References.
• (source: Nielsen Book Data)

In this book the authors present an alternative set theory dealing with a more relaxed notion of infiniteness, called finitely supported mathematics (FSM). It has strong connections to the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel (ZF) set theory with atoms and to the theory of (generalized) nominal sets. More exactly, FSM is ZF mathematics rephrased in terms of finitely supported structures, where the set of atoms is infinite (not necessarily countable as for nominal sets). In FSM, 'sets' are replaced either by 'invariant sets' (sets endowed with some group actions satisfying a finite support requirement) or by 'finitely supported sets' (finitely supported elements in the powerset of an invariant set). It is a theory of 'invariant algebraic structures' in which infinite algebraic structures are characterized by using their finite supports. After explaining the motivation for using invariant sets in the experimental sciences as well as the connections with the nominal approach, admissible sets and Gandy machines (Chapter 1), the authors present in Chapter 2 the basics of invariant sets and show that the principles of constructing FSM have historical roots both in the definition of Tarski 'logical notions' and in the Erlangen Program of Klein for the classification of various geometries according to invariants under suitable groups of transformations. Furthermore, the consistency of various choice principles is analyzed in FSM. Chapter 3 examines whether it is possible to obtain valid results by replacing the notion of infinite sets with the notion of invariant sets in the classical ZF results. The authors present techniques for reformulating ZF properties of algebraic structures in FSM. In Chapter 4 they generalize FM set theory by providing a new set of axioms inspired by the theory of amorphous sets, and so defining the extended Fraenkel-Mostowski (EFM) set theory. In Chapter 5 they define FSM semantics for certain process calculi (e.g., fusion calculus), and emphasize the links to the nominal techniques used in computer science. They demonstrate a complete equivalence between the new FSM semantics (defined by using binding operators instead of side conditions for presenting the transition rules) and the known semantics of these process calculi. The book is useful for researchers and graduate students in computer science and mathematics, particularly those engaged with logic and set theory.
(source: Nielsen Book Data)

• The concept of reverse clustering.- Reverse clustering: the essence and the interpretations.- Case studies: an introduction.- The road traffic data.- The chemicals in the natural environment.- Administrative units, Part I.- Administrative units, Part II.- Academic examples.- Summary and conclusions.
• (source: Nielsen Book Data)

This book presents a new perspective on and a new approach to a wide spectrum of situations, related to data analysis, actually, a kind of a new paradigm. Namely, for a given data set and its partition, whose origins may be of any kind, the authors try to reconstruct this partition on the basis of the data set given, using very broadly conceived clustering procedure. The main advantages of this new paradigm concern the substantive aspects of the particular cases considered, mainly in view of the variety of interpretations, which can be assumed in the framework of the paradigm. Due to the novel problem formulation and the flexibility in the interpretations of this problem and its components, the domains, which are encompassed (or at least affected) by the potential use of the paradigm, include cluster analysis, classification, outlier detection, feature selection, and even factor analysis as well as geometry of the data set. The book is useful for all those who look for new, nonconventional approaches to their data analysis problems.
(source: Nielsen Book Data)

• Preliminaries
• Idea of foundations for mathematics
• Simple arithmetic
• Basic set theory
• Semantics, ontology, and logic
• Principal axioms and definitions of set theory
• Cantorian set theory
• Cantorian finitism
• Axiomatic method
• Axiomatic set theory
• Euclidean set theory
• Euclidean finitism
• Euclidean theory of cardinality
• Euclidean theory of simply infinite systems
• Euclidean set theory from the cantorian standpoint
• Envoi.

This 2001 book presents a unified approach to the foundations of mathematics in the theory of sets, covering both conventional and finitary (constructive) mathematics. It is based on a philosophical, historical and mathematical analysis of the relation between the concepts of 'natural number' and 'set'. This leads to an investigation of the logic of quantification over the universe of sets and a discussion of its role in second order logic, as well as in the analysis of proof by induction and definition by recursion. The subject matter of the book falls on the borderline between philosophy and mathematics, and should appeal to both philosophers and mathematicians with an interest in the foundations of mathematics.
(source: Nielsen Book Data)

• Theoretical Contributions to Rough Set Theory.- Rough Sets on Fuzzy Approximation Spaces and Intuitionistic Fuzzy Approximation Spaces.- Categorical Innovations for Rough Sets.- Granular Structures and Approximations in Rough Sets and Knowledge Spaces.- On Approximation of Classifications, Rough Equalities and Rough Equivalences.- Rough Set Data Mining Activities.- Rough Clustering with Partial Supervision.- A Generic Scheme for Generating Prediction Rules Using Rough Sets.- Rough Web Caching.- Software Defect Classification: A Comparative Study of Rough-Neuro-fuzzy Hybrid Approaches with Linear and Non-linear SVMs.- Rough Hybrid Models to Classification and Attribute Reduction.- Rough Sets and Evolutionary Computation to Solve the Feature Selection Problem.- Nature Inspired Population-Based Heuristics for Rough Set Reduction.- Developing a Knowledge-Based System Using Rough Set Theory and Genetic Algorithms for Substation Fault Diagnosis.
• (source: Nielsen Book Data)

Along the years, rough set theory has earned a well-deserved reputation as a sound methodology for dealing with imperfect knowledge in a simple though mathematically sound way. This edited volume aims at continue stressing the benefits of applying rough sets in many real-life situations while still keeping an eye on topological aspects of the theory as well as strengthening its linkage with other soft computing paradigms. The volume comprises 11 chapters and is organized into three parts. Part 1 deals with theoretical contributions while Parts 2 and 3 focus on several real world data mining applications. Chapters authored by pioneers were selected on the basis of fundamental ideas/concepts rather than the thoroughness of techniques deployed. Academics, scientists as well as engineers working in the rough set, computational intelligence, soft computing and data mining research area will find the comprehensive coverage of this book invaluable.
(source: Nielsen Book Data)

• Ensembles Discovered Predictive Learning and Decision Trees Model Complexity, Model Selection and Regularization Importance Sampling and the Classic Ensemble Methods Rule Ensembles and Interpretation Statistics Ensemble Complexity.
• (source: Nielsen Book Data)

Ensemble methods have been called the most influential development in Data Mining and Machine Learning in the past decade. They combine multiple models into one usually more accurate than the best of its components. Ensembles can provide a critical boost to industrial challenges - from investment timing to drug discovery, and fraud detection to recommendation systems - where predictive accuracy is more vital than model interpretability. Ensembles are useful with all modeling algorithms, but this book focuses on decision trees to explain them most clearly. After describing trees and their strengths and weaknesses, the authors provide an overview of regularization - today understood to be a key reason for the superior performance of modern ensembling algorithms. The book continues with a clear description of two recent developments: Importance Sampling (IS) and Rule Ensembles (RE). IS reveals classic ensemble methods - bagging, random forests, and boosting - to be special cases of a single algorithm, thereby showing how to improve their accuracy and speed. REs are linear rule models derived from decision tree ensembles. They are the most interpretable version of ensembles, which is essential to applications such as credit scoring and fault diagnosis. Lastly, the authors explain the paradox of how ensembles achieve greater accuracy on new data despite their (apparently much greater) complexity. This book is aimed at novice and advanced analytic researchers and practitioners -- especially in Engineering, Statistics, and Computer Science. Those with little exposure to ensembles will learn why and how to employ this breakthrough method, and advanced practitioners will gain insight into building even more powerful models. Throughout, snippets of code in R are provided to illustrate the algorithms described and to encourage the reader to try the techniques. The authors are industry experts in data mining and machine learning who are also adjunct professors and popular speakers. Although early pioneers in discovering and using ensembles, they here distill and clarify the recent groundbreaking work of leading academics (such as Jerome Friedman) to bring the benefits of ensembles to practitioners.
(source: Nielsen Book Data)

• Beginnings of Communication Technology in the 20th Century
• Logical Tolerance, Ensembles Flous and Probabilistic Metrics
• General Systems Theory and Cybernetics
• From Circuit Theory to System Theory
• Fuzzy Sets and Fuzzy Systems
• Fuzzifications
• The Fuzzification of Medical Diagnostics
• Conculsion.

In 1965 Lotfi Zadeh, a professor of electrical engineering at the University of California in Berkeley, published the first of his papers on his new Fuzzy Set Theory. Since the 1980s this mathematical theory of "unsharp amounts" has been applied in many different fields with great success. The word "fuzzy" has also become very well-known among non-scientists thanks to extensive advertising campaigns for fuzzy-controlled household appliances and to their prominent presence in the media, first in Japan and then in other countries. On the other hand, the story of how Fuzzy Set Theory and its earliest applications originated remains largely unknown. In this book, the history of Fuzzy Set Theory and the ways it was first used are incorporated into the history of 20th century science and technology. Influences from philosophy, system theory and cybernetics stemming from the earliest part of the 20th century are considered alongside those of communication and control theory from mid-century. Today, Fuzzy Set Theory is the core discipline of "soft computing, " and provides new impetus for research in the field of Artificial Intelligence