Mathematical foundations Books

Book SynopsisThe goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.Table of Contents- Part I Preliminaries. - Ultrafilters. - Nonstandard Analysis. - Hyperfinite Generators of Ultrafilters. - Many Stars: Iterated Nonstandard Extensions. - LoebMeasure. - Part II Ramsey Theory. - Ramsey’s Theorem. - The Theorems of van der Waerden and Hales-Jewett. - From Hindman to Gowers. - Partition Regularity of Equations. - Part III Combinatorial Number Theory. - Densities and Structural Properties. - Working in the Remote Realm. - Jin’s Sumset Theorem. - Sumset Configurations in Sets of Positive Density. - Near Arithmetic Progressions in Sparse Sets. - The Interval Measure Property. - Part IV Other Topics. - Triangle Removal and Szemerédi Regularity. - Approximate Groups. - Foundations of Nonstandard Analysis.

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Book SynopsisThis undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics.Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise.This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts.From reviews of the first edition:Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. … The book can be used as a text for traditional transition or structure courses … but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA ReviewsThe style of writing is careful, but joyously enthusiastic…. The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background … and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATHTable of ContentsPreface to Instructors.- Preface to Students.- Acknowledgments.- I What's Mathematics.- 1 Let's Play a Game!.- 2 What's the Name of the Game?.- 3 How to Make a Statement.- 4 What's True in Mathematics?.- A Ten Famous Conjectures.-B Ten Famous Theorems.- II The Foundations of Mathematics.- 5 Let's Be Logical!.- 6 Setting Examples.- 7 Quantifier Mechanics.- 8 Let's Be Functional!.- C The Foundations of Set Theory.- III How to Prove It.- 9 Universal Proofs.- 10 The Domino Theory.- 11 More Domino Games.- 12 Existential Proofs.- D Ten Famous Problems.- IV Advanced Math for Beginners.- 13 Mathematical Structures.- 14 Working in the Fields (and Other Structures).- 15 Group Work.- 16 Good Relations.- 17 Order, Please!.- 18 Now That's the Limit!.- 19 Sizing It Up.- 20 Infinite Delights.- 21 Number Systems Systematically.- 22 Games Are Valuable!.- E Graphic Content.- F All Games Considered.- G A Top Forty List of Math Theorems.

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Book SynopsisThis undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics.Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise.This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts.From reviews of the first edition:Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. … The book can be used as a text for traditional transition or structure courses … but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA ReviewsThe style of writing is careful, but joyously enthusiastic…. The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background … and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATHTable of ContentsPreface to Instructors.- Preface to Students.- Acknowledgments.- I What's Mathematics.- 1 Let's Play a Game!.- 2 What's the Name of the Game?.- 3 How to Make a Statement.- 4 What's True in Mathematics?.- A Ten Famous Conjectures.-B Ten Famous Theorems.- II The Foundations of Mathematics.- 5 Let's Be Logical!.- 6 Setting Examples.- 7 Quantifier Mechanics.- 8 Let's Be Functional!.- C The Foundations of Set Theory.- III How to Prove It.- 9 Universal Proofs.- 10 The Domino Theory.- 11 More Domino Games.- 12 Existential Proofs.- D Ten Famous Problems.- IV Advanced Math for Beginners.- 13 Mathematical Structures.- 14 Working in the Fields (and Other Structures).- 15 Group Work.- 16 Good Relations.- 17 Order, Please!.- 18 Now That's the Limit!.- 19 Sizing It Up.- 20 Infinite Delights.- 21 Number Systems Systematically.- 22 Games Are Valuable!.- E Graphic Content.- F All Games Considered.- G A Top Forty List of Math Theorems.

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Book SynopsisThis textbook introduces enumerative combinatorics through the framework of formal languages and bijections. By starting with elementary operations on words and languages, the authors paint an insightful, unified picture for readers entering the field. Numerous concrete examples and illustrative metaphors motivate the theory throughout, while the overall approach illuminates the important connections between discrete mathematics and theoretical computer science. Beginning with the basics of formal languages, the first chapter quickly establishes a common setting for modeling and counting classical combinatorial objects and constructing bijective proofs. From here, topics are modular and offer substantial flexibility when designing a course. Chapters on generating functions and partitions build further fundamental tools for enumeration and include applications such as a combinatorial proof of the Lagrange inversion formula. Connections to linear algebra emerge in chapters studying Cayley trees, determinantal formulas, and the combinatorics that lie behind the classical Cayley–Hamilton theorem. The remaining chapters range across the Inclusion-Exclusion Principle, graph theory and coloring, exponential structures, matching and distinct representatives, with each topic opening many doors to further study. Generous exercise sets complement all chapters, and miscellaneous sections explore additional applications. Lessons in Enumerative Combinatorics captures the authors' distinctive style and flair for introducing newcomers to combinatorics. The conversational yet rigorous presentation suits students in mathematics and computer science at the graduate, or advanced undergraduate level. Knowledge of single-variable calculus and the basics of discrete mathematics is assumed; familiarity with linear algebra will enhance the study of certain chapters.Trade Review“The wide variety of slightly unusual topics makes the book an excellent resource for the instructor who wants to craft a combinatorics course that contains a diverse collection of attractive results … . The attentive student will certainly come away from a course based on this book with a solid understanding of the combinatorial way of thinking. … the book is an excellent resource for anyone teaching a class in combinatorics.” (Timothy Y. Chow, Mathematical Reviews, March, 2023)“A whole book whose backbone is enumeration by codifying the objects to be enumerated as words. … They do this in a skillfully structured fashion which makes the connections natural and unforced. … One of the remarkable features of this book is the care the authors have taken to make it reader-friendly and accessible to a wide range of students following a graduate mathematics course or an honours undergraduate course in mathematics and computer science.” (Josef Lauri, zbMATH 1478.05001, 2022)Table of Contents1. Basic Combinatorial Structures.- 2. Partitions and Generating Functions.- 3. Planar Trees and the Lagrange Inversion Formula.- 4. Cayley Trees.- 5. The Cayley–Hamilton Theorem.- 6. Exponential Structures and Polynomial Operators.- 7. The Inclusion-Exclusion Principle.- 8. Graphs, Chromatic Polynomials and Acyclic Orientations.- 9. Matching and Distinct Representatives.

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Book SynopsisThis open access book is the first ever collection of Karl Popper's writings on deductive logic.Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science ("falsificationism") and his social and political philosophy ("open society") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics.This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work.Table of Contents Part I: Articles.- Chapter 1. Introduction to Popper’s Articles on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 2. Are Contradictions Embracing? (1943) (Karl R. Popper).- Chapter 3. Logic without Assumptions (1947) (Karl R. Popper).- Chapter 4. New Foundations for Logic (1947) (Karl R. Popper).- Chapter 5. Functional Logic without Axioms or Primitive Rules of Inference (1947)(Karl R. Popper).- Chapter 6. On the Theory of Deduction, Part I. Derivation and its Generalizations (1948) (Karl R. Popper).- Chapter 7. On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation (1948) (Karl R. Popper).- Chapter 8. The Trivialization of Mathematical Logic (1949) (Karl R. Popper).- Chapter 9. A Note on Tarski’s Definition of Truth (1955) (Karl R. Popper).-Chapter 10. On a Proposed Solution of the Paradox of the Liar (1955) (Karl R. Popper).- Chapter 11. On Subjunctive Conditionals with Impossible Antecedents (1959) (Karl R. Popper).- Chapter 12. Lejewski’s Axiomatization of My Theory of Deducibility (1974) (Karl R. Popper).- Chapter 13. Reviews of Popper’s Articles on Logic (Wilhelm Ackermann et.al).- Part II: Manuscripts.- Chapter 14. Introduction to Popper’s Manuscripts on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 15. On Systems of Rules of Inference (Karl R. Popper and Paul Bernays).- Chapter 16. A General Theory of Inference (Karl R. Popper).- Chapter 17. On the Logic of Negation (Karl R. Popper).- Chapter 18. A Note on the Classical Conditional (Karl R. Popper).- Part III: Correspondence.- Chapter 19. Introduction to Popper’s Correspondence on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 20. Popper’s Correspondence with Paul Bernays (Karl R. Popper and Paul Bernays).- Chapter 21. Popper’s Correspondence with Luitzen Egbertus Jan Brouwer (Karl R. Popper and Luitzen E. J. Brouwer).- Chapter 22. Popper’s Correspondence with Rudolf Carnap (Karl R. Popper and Rudolf Carnap).- Chapter 23. Popper’s Correspondence with Alonzo Church (Karl R. Popper and Alonzo Church).- Chapter 24. Popper’s Correspondence with Kalman Joseph Cohen (Karl R. Popper and Kalman J. Cohen).- Chapter 25. Popper’s Correspondence with Henry George Forder (Karl R. Popper and Henry George Forder).- Chapter 26. Popper’s Correspondence with Harold Jeffreys (Karl R. Popper and Harold Jeffreys).- Chapter 27. Popper’s Correspondence with Stephen Cole Kleene (Karl R. Popper and Stephen C. Kleene).- Chapter 28. Popper’s Correspondence with William Calvert Kneale (Karl R. Popper and William C. Kneale).- Chapter 29. Popper’s Correspondence with Willard Van Orman Quine (Karl R. Popper and Willard V. O. Quine).- Chapter 30. Popper’s Correspondence with Heinrich Scholz (Karl R. Popper and Heinrich Scholz).- Chapter 31. Popper’s Correspondence with Peter Schroeder-Heister (Karl R. Popper and Peter Schroeder-Heister).- Concordances.- Bibliography.- Index.

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Book SynopsisChapter 1. Introduction.- Chapter 2. Cartesian cubical sets.- Chapter 3. The cofibration weak factorization system.- Chapter 4. The fibration weak factorization system.- Chapter 5. The weak equivalences.- Chapter 6. The Frobenius condition.- Chapter 7. A universal fibration.- Chapter 8. The equivalence extension property.- Chapter 9. The fibration extension property.

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Book SynopsisMathematical Foundations.- Basics of Quantum Mechanics.- One-Dimensional Problems.- Angular Momentum.- Quantum Problems in Higher Dimensions.- Identical Particles in Quantum Mechanics.- Approximation Methods for Time-Independent Problems.- Time-Dependent Quantum Mechanics.- Electromagnetic Radiation.- Scattering Theory.- Relativistic Quantum Mechanics.- Appendix Numerical Methods for Simple Problems.

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Book SynopsisThis exploration of a selection of fundamental topics and general purpose tools provides a roadmap to undergraduate students who yearn for a deeper dive into many of the concepts and ideas they have been encountering in their classes whether their motivation is pure curiosity or preparation for graduate studies. The topics intersect a wide range of areas encompassing both pure and applied mathematics. The emphasis and style of the book are motivated by the goal of developing self-reliance and independent mathematical thought. Mathematics requires both intuition and common sense as well as rigorous, formal argumentation. This book attempts to showcase both, simultaneously encouraging readers to develop their own insights and understanding and the adoption of proof writing skills. The most satisfying proofs/arguments are fully rigorous and completely intuitive at the same time.

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Book SynopsisThis is a comprehensive book on the life and works of Leon Henkin (1921–2006), an extraordinary scientist and excellent teacher whose writings became influential right from the beginning of his career with his doctoral thesis on “The completeness of formal systems” under the direction of Alonzo Church. Upon the invitation of Alfred Tarski, Henkin joined the Group in Logic and the Methodology of Science in the Department of Mathematics at the University of California Berkeley in 1953. He stayed with the group until his retirement in 1991. This edited volume includes both foundational material and a logic perspective. Algebraic logic, model theory, type theory, completeness theorems, philosophical and foundational studies are among the topics covered, as well as mathematical education. The work discusses Henkin’s intellectual development, his relation to his predecessors and contemporaries and his impact on the recent development of mathematical logic. It offers a valuable reference work for researchers and students in the fields of philosophy, mathematics and computer science.Table of ContentsPart I Biographical Studies.- Leon Henkin.- Lessons from Leon.- Tracing back “Logic in Wonderland” to my work with Leon Henkin.- Henkin and the Suit.- A Fortuitous Year with Leon Henkin.- Leon Henkin and a Life of Service.- Part II Henkin‘s Contribution to XX Century Logic.- Leon Henkin and Cylindric Algebras.- A Bit of History Related to Logic Based on Equality.- Pairing Logical and Pedagogical Foundations for the Theory of Positive Rational Numbers. Henkin‘s unfinished work.- Leon Henkin the Reviewer.- Henkin‘s Theorem in Textbooks.- Henkin on Completeness.- Part III Extensions and Perspectives in Henkin‘s Work.- The Countable Henkin Principle.- Reflections on a Theorem of Henkin.- Henkin‘s Completeness Proof and Glivenko‘s Theorem.- From Classical to Fuzzy Type Theory.- The Henkin Sentence.- April the 19th.- Henkin and Hybrid Logic.- Changing a Semantics: Oportunism or Courage?.- Appendix Curriculum Vitae: Leon Henkin.

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Book SynopsisThis volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.Trade Review“Swedish logician and philosopher Dag Prawitz and his distinguished contributions to philosophical and mathematical logic are the focus of this book. … This is an excellent book, celebrating not only Prawitz’s career, but also a movement in the contrary direction of W. V. O Quine’s views against the so-called (somehow prejudicially) ‘deviant’ logics, and I cannot forbear from congratulating the editor for the distinctive choice of topics and for the general tone of the book.” (Walter Carnielli, Computing Reviews, May, 2015)Table of ContentsPrawitz, proofs, and meaning; Wansing, Heinrich.- A short scientific autobiography; Prawitz, Dag.- Explaining deductive inference; Prawitz, Dag.- Necessity of Thought; Cozzo, Cesare.- On the Motives for Proof Theory; Detlefsen, Michael.- Inferential Semantics; Došen, Kosta.- Cut elimination, substitution and normalization; Dyckhoff, Roy.- Inversion principles and introduction rules; Milne, Peter.- Intuitionistic Existential Instantiation and Epsilon Symbol; Mints, Grigori.- Meaning in Use; Negri, Sara and von Plato, Jan.- Fusing Quantifiers and Connectives: Is Intuitionistic Logic Different?; Pagin, Peter.- On constructive fragments of Classical Logic; Pereira; Luiz Carlos and Haeusler, Edward Hermann.- General-Elimination Harmony and Higher-Level Rules; Read, Stephen.- Hypothesis-discharging rules in atomic bases; Sandqvist, Tor.- Harmony in proof-theoretic semantics: A reductive analysis; Schroeder-Heister, Peter.- First-order Logic without bound variables: Compositional Semantics; Tait, William W.- On Gentzen’s Structural Completeness Proof; Tennant, Neil.- A Notion of C-Justification for Empirical Statements; Usberti, Gabriele.

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Table of ContentsImplicit definability and compactness in infinitary languages.- Some remarks on the model theory of infinitary languages.- Remarks on the theory of geometrical constructions.- Note on admissible ordinals.- An algebraic proof of the barwise compactness theorem.- Formulas with linearly ordered quantifiers.- Some problems in group theory.- Choice of infinitary languages by means of definability criteria; Generalized recursion theory.- Definability, automorphisms, and infinitary languages.- The hanf number for complete sentences.- Quantified algebras.- Normal derivability in classical logic.- A determinate logic.- (?1, ?) properties of unions of models.

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Table of ContentsInductive definitions and subsystems of analysis.- Proof theoretic equivalences between classical and constructive theories for analysis.- Inductive definitions, constructive ordinals, and normal derivations.- The ??+1-Rule.- Ordinal analysis of ID?.- Proof-theoretical analysis of ID? by the method of local predicativity.

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Book SynopsisThis is a softcover reprint of the English translation of 1968 of N. Bourbaki's, Theorie des Ensembles (1970).Table of ContentsI. Description of Formal Mathematics.- § 1. Terms and relations.- 1. Signs and assemblies.- 2. Criteria of substitution.- 3. Formative constructions.- 4. Formative criteria.- § 2. Theorems.- 1. The axioms.- 2. Proofs.- 3. Substitutions in a theory.- 4. Comparison of theories.- § 3. Logical theories.- 1. Axioms.- 2. First consequences.- 3. Methods of proof.- 4. Conjunction.- 5. Equivalence.- § 4. Quantified theories.- 1. Definition of quantifiers.- 2. Axioms of quantified theories.- 3. Properties of quantifiers.- 4. Typical quantifiers.- § 5. Equalitarian theories.- 1. The axioms.- 2. Properties of equality.- 3. Functional relations.- Appendix. Characterization of terms and relations.- 1. Signs and words.- 2. Significant words.- 3. Characterization of significant words.- 4. Application to assemblies in a mathematical theory.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for the Appendix.- II. Theory of Sets.- § 1. Collectivizing relations.- 1. The theory of sets.- 2. Inclusion.- 3. The axiom of extent.- 4. Collectivizing relations.- 5. The axiom of the set of two elements.- 6. The scheme of selection and union.- 7. Complement of a set. The empty set.- § 2. Ordered pairs.- 1. The axiom of the ordered pair.- 2. Product of two sets.- § 3. Correspondences.- 1. Graphs and correspondences.- 2. Inverse of a correspondence.- 3. Composition of two correspondences.- 4. Functions.- 5. Restrictions and extensions of functions.- 6. Definition of a function by means of a term.- 7. Composition of two functions. Inverse function.- 8. Retractions and sections.- 9. Functions of two arguments.- § 4. Union and intersection of a family of sets.- 1. Definition of the union and the intersection of a family of sets.- 2. Properties of union and intersection.- 3. Images of a union and an intersection.- 4. Complements of unions and intersections.- 5. Union and intersection of two sets.- 6. Coverings.- 7. Partitions.- 8. Sum of a family of sets.- § 5. Product of a family of sets.- 1. The axiom of the set of subsets.- 2. Set of mappings of one set into another.- 3. Definitions of the product of a family of sets.- 4. Partial products.- 5. Associativity of products of sets.- 6. Distributivity formulae.- 7. Extension of mappings to products.- § 6. Equivalence relations.- 1. Definition of an equivalence relation.- 2. Equivalence classes; quotient set.- 3. Relations compatible with an equivalence relation.- 4. Saturated subsets.- 5. Mappings compatible with equivalence relations.- 6. Inverse image of an equivalence relation; induced equivalence relation.- 7. Quotients of equivalence relations.- 8. Product of two equivalence relations.- 9. Classes of equivalent objects.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- III. Ordered Sets, Cardinals, Integers.- § 1. Order relations. Ordered sets.- 1. Definition of an order relation.- 2. Preorder relations.- 3. Notation and terminology.- 4. Ordered subsets. Product of ordered sets.- 5. Increasing mappings.- 6. Maximal and minimal elements.- 7. Greatest element and least element.- 8. Upper and lower bounds.- 9. Least upper bound and greatest lower bound.- 10. Directed sets.- 11. Lattices.- 12. Totally ordered sets.- 13. Intervals.- § 2. Well-ordered sets.- 1. Segments of a well-ordered set.- 2. The principle of transfinite induction.- 3. Zermelo’s theorem.- 4. Inductive sets.- 5. Isomorphisms of well-ordered sets.- 6. Lexicographic products.- § 3. Equipotent sets. Cardinals.- 1. The cardinal of a set.- 2. Order relation between cardinals.- 3. Operations on cardinals.- 4. Properties of the cardinals 0 and 1.- 5. Exponentiation of cardinals.- 6. Order relation and operations on cardinals.- § 4. Natural integers. Finite sets.- 1. Definition of integers.- 2. Inequalities between integers.- 3. The principle of induction.- 4. Finite subsets of ordered sets.- 5. Properties of finite character.- § 5. Properties of integers.- 1. Operations on integers and finite sets.- 2. Strict inequalities between integers.- 3. Intervals in sets of integers.- 4. Finite sequences.- 5. Characteristic functions of sets.- 6. Euclidean division.- 7. Expansion to base b.- 8. Combinatorial analysis.- § 6. Infinite sets.- 1. The set of natural integers.- 2. Definition of mappings by induction.- 3. Properties of infinite cardinals.- 4. Countable sets.- 5. Stationary sequences.- § 7. Inverse limits and direct limits.- 1. Inverse limits.- 2. Inverse systems of mappings.- 3. Double inverse limit.- 4. Conditions for an inverse limit to be non-empty.- 5. Direct limits.- 6. Direct systems of mappings.- 7. Double direct limit. Product of direct limits.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- Exercises for § 7.- Historical Note on § 5.- IV. Structures.- § 1. Structures and isomorphisms.- 1. Echelons.- 2. Canonical extensions of mappings.- 3. Transportable relations.- 4. Species of structures.- 5. Isomorphisms and transport of structures.- 6. Deduction of structures.- 7. Equivalent species of structures.- § 2. Morphisms and derived structures.- 1. Morphisms.- 2. Finer structures.- 3. Initial structures.- 4. Examples of initial structures.- 5. Final structures.- 6. Examples of final structures.- § 3. Universal mappings.- 1. Universal sets and mappings.- 2. Existence of universal mappings.- 3. Examples of universal mappings.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Historical Note on Chapters I-IV.- Summary of Results.- § 1. Elements and subsets of a set.- § 2. Functions.- § 3. Products of sets.- § 4. Union, intersection, product of a family of sets.- § 5. Equivalence relations and quotient sets.- § 6. Ordered sets.- § 7. Powers. Countable sets.- § 8. Scales of sets. Structures.- Index of notation.- Index of terminology.- Axioms and schemes of the theory of sets.

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Book SynopsisThis is a thoroughly revised and enlarged second edition that presents the main results of descriptive complexity theory, that is, the connections between axiomatizability of classes of finite structures and their complexity with respect to time and space bounds. The logics that are important in this context include fixed-point logics, transitive closure logics, and also certain infinitary languages; their model theory is studied in full detail. The book is written in such a way that the respective parts on model theory and descriptive complexity theory may be read independently.Table of ContentsPreliminaries.- The Ehrenfeucht-Fraïssé Method.- More on Games.- 0-1 Laws.- Satisfiability in the Finite.- Finite Automata and Logic: A Microcosm of Finite Model Theory.- Descriptive Complexity Theory.- Logics with Fixed-Point Operators.- Logic Programs.- Optimization Problems.- Logics for PTIME.- Quantifiers and Logical Reductions.

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Book SynopsisLe Livre de Théorie des ensembles qui vient en tête 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.Table of ContentsDescription de la mathématique formelle.- Théorie des ensembles.- Ensembles ordonnés, cardinaux, nombres entiers.- Structures.

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Book SynopsisThis monograph covers the recent major advances in various areas of set theory. From the reviews: "One of the classical textbooks and reference books in set theory....The present ‘Third Millennium’ edition...is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master....This well-written book promises to influence the next generation of set theorists, much as its predecessor has done." --MATHEMATICAL REVIEWSTrade ReviewFrom the reviews of the third edition: "Thomas Jech’s text has long been considered a classic survey of the state of the set theory … . As every logician will know, this is a work of extraordinary scholarship, essential for any graduate logician who needs to know where the current boundaries of research are situated. Each chapter ends with a valuable historical survey and there is an extensive bibliography. This will continue to be the bible for set theorists in the new century." (Gerry Leversha, The Mathematical Gazette, March, 2005) "The book does masterly what it is supposed to do. … every mathematician who wishes to refresh his knowledge of set theory will read it with pleasure. … They will also find historical notes, and precise references … . A very comprehensive bibliography, and detailed indexes complete the work. This book fills a serious gap in the literature and there is no doubt that it will become a standard reference … . One can strongly recommend its acquisition for any mathematical library." (Jean-Roger Roisin, Bulletin of the Belgian Mathematical Society, Vol. 11 (3), 2004) "One of the classical textbooks and reference books in set theory is Jech’s Set Theory. … The present ‘Third Millennium’ edition … is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master. … This well-written book promises to influence the next generation of set theorists, much as its predecessor has done over the last quarter of a century." (Eva Coplakova, Mathematical Reviews, 2004 g) "Jech’s book, ‘Set Theory’ has been a standard reference for over 25 years. This ‘Third Millennium Edition’, not only includes all the materials in the first two editions, but also covers recent developments of set theory during the last 25 years. We believe that this new version will become a standard reference on set theory for the next few years." (Guohua Wu, New Zealand Mathematical Society Newsletter, April, 2004) "Jech’s classic monograph has been a standard reference for a generation of set theorists. Though … labeled ‘The Third Millennium Edition’, the present work is in fact a new book. ... Even sections presenting older results have been rewritten and modernized. Exercises have been moved to the end of each section. The bibliography, the section on notation, and the index have been considerably expanded as well. This new edition will certainly become a standard reference on set theory for years to come." (Jörg D. Brendle, Zentralblatt MATH, Vol. 1007, 2003) "Thomas Jech’s Set Theory contains the most comprehensive treatment of the subject in any one volume. The present third edition is a revised and expanded version … . The third edition has three parts. The first, Jech says, every student of set theory should learn, the second every set theorist should master and the third consists of various results reflecting ‘the state of the art of set theory at the turn of the new millennium’. This last part especially contains a lot of new material." (Martin Bunder, The Australian Mathematical Society Gazette, Vol. 30 (2), 2003)Table of ContentsBasic Set Theory.- Axioms of Set Theory.- Ordinal Numbers.- Cardinal Numbers.- Real Numbers.- The Axiom of Choice and Cardinal Arithmetic.- The Axiom of Regularity.- Filters, Ultrafilters and Boolean Algebras.- Stationary Sets.- Combinatorial Set Theory.- Measurable Cardinals.- Borel and Analytic Sets.- Models of Set Theory.- Advanced Set Theory.- Constructible Sets.- Forcing.- Applications of Forcing.- Iterated Forcing and Martin’s Axiom.- Large Cardinals.- Large Cardinals and L.- Iterated Ultrapowers and L[U].- Very Large Cardinals.- Large Cardinals and Forcing.- Saturated Ideals.- The Nonstationary Ideal.- The Singular Cardinal Problem.- Descriptive Set Theory.- The Real Line.- Selected Topics.- Combinatorial Principles in L.- More Applications of Forcing.- More Combinatorial Set Theory.- Complete Boolean Algebras.- Proper Forcing.- More Descriptive Set Theory.- Determinacy.- Supercompact Cardinals and the Real Line.- Inner Models for Large Cardinals.- Forcing and Large Cardinals.- Martin’s Maximum.- More on Stationary Sets.

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Book SynopsisStochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the book.Table of Contents1 Infinitesimal calculus, consistently and accessibly.- 2 Radically elementary probability theory.- 3 Radically elementary stochastic integrals.- 4 The radically elementary Girsanov theorem and the diffusion invariance principle.- 5 Excursion to nancial economics: A radically elementary approach to the fundamental theorems of asset pricing.- 6 Excursion to financial engineering: Volatility invariance in the Black-Scholes model.- 7 A radically elementary theory of Itô diffusions and associated partial differential equations.- 8 Excursion to mathematical physics: A radically elementary definition of Feynman path integrals.- 9 A radically elementary theory of Lévy processes.- 10 Final remarks.

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Book SynopsisTable of ContentsEinleitung: Überblick über den Inhalt des zweiten Halbbandes.- III. Die logischen Grundlagen des statistischen Schließens.- 1. ,Jenseits von Popper und Carnap‘.- 1.a Programm und Abgrenzung vom Projekt einer induktiven Logik.- 1.b Die relative Häufigkeit auf lange Sicht und die Häufigkeitsdefinition der statistischen Wahrscheinlichkeit.- 1.c Der Vorschlag von Braithwaite, die statistische Wahrscheinlichkeit als theoretischen Begriff einzuführen.- 1.d Vorbereitende Betrachtungen zur Testproblematik statistischer Hypothesen.- 1.e Zusammenfassung und Ausblick.- 2. Präludium: Der intuitive Hintergrund.- 3. Die Grundaxiome. Statistische Unabhängigkeit.- 3.a Die Kolmogoroff-Axiome.- 3.b Unabhängigkeit im statistischen Sinn.- 3.c Hypothesen und Oberhypothesen.- 4. Die komparative Stützungslogik.- 4.a Vorbetrachtungen.- 4.b Einige zusätzliche Zwischenbetrachtungen.- 4.c Die Axiome der Stützungslogik.- 5. Die Likelihood-Regel.- 5.a Kombinierte statistische Aussagen.- 5.b Likelihood und Likelihood-Regel.- 6. Die Leistungsfähigkeit der Likelihood-Regel.- 6.a Die Einzelfall-Regel und ihre Begründung.- 6.b Der statistische Stützungsschluß im diskreten Fall und seine Rechtfertigung.- 6.c Übergang zum stetigen Fall.- 6.d Wahrscheinlichkeitsverteilung und Likelihoodfunktion (,Plausibilitätsverteilung‘).- 6.e Denken in Likelihoods und Bayesianismus.- 7. Vorläufiges Postludium: Ergänzende Betrachtungen zu den statistischen Grundbegriffen.- 7.a Der Begriff des statistischen Datums.- 7.b Chance und Häufigkeit auf lange Sicht.- 7.c Versuchstypen.- 8. Zufall, Grundgesamtheit und Stichprobenauswahl.- 9. Die Problematik der statistischen Testtheorie, erläutert am Beispiel zweier konkurrierender Testtheorien.- 9.a Vorbetrachtungen. Ein warnendes historisches Beispiel.- 9.b Macht und Umfang eines Tests. Die Testtheorie von Neyman-Pearson.- 9.c Die Mehrdeutigkeit der Begriffe „Annahme“ und „Verwerfung“ 159 9.d Einige kritische Bemerkungen zu den Begriffen Umfang und Macht 160 9.e Die Likelihood-Testtheorie.- 10. Probleme der Schätzungstheorie.- 10.a Vorbemerkungen.- 10.b Was ist Schätzung? Klassifikation von Schätzungen.- 10.c Einige spezielle Begriffe der statistischen Schätzungstheorie.- 10.d Die Doppeldeutigkeit von „Schätzung“ und die Mehrdeutigkeit von „Güte einer Schätzung“.- 10.e Theoretische Schätzungen und Schätzhandlungen.- 10.f Das Skalendilemma. Zwecke von Schätzungen.- 10.g Schätzungen im engeren und Schätzungen im weiteren Sinn.- 10.h Kritisches zu den Optimalitätsmerkmalen auf lange Sicht, zur Minimax-Theorie und zur Intervallschätzung.- 10.i Ein Präzisierungsversuch des Begriffes der besser gestützten Schätzung.- 10.j Ist die Schätzungstheorie von Savage das Analogon zur Testtheorie von Neyman-Pearson?.- 11. Kritische Betrachtungen zur Likelihood-Stützungs-und-Testtheorie.- 11.a Ist der Likelihood-Test schlechter als nutzlos ?.- 11.b Das Karten-Paradoxon von Kerridge.- 11.c Die logische Struktur des Stützungsbegriffs.- 12. Subjektivismus oder Objektivismus ?.- 12.a Die subjektivistische (personalistische) Kritik: de Finetti und Savage kontra Objektivismus.- 12.b Die Propensity-Interpretation der statistischen Wahrscheinlichkeit: Popper, Giere und Suppes.- 13. Versuch einer Skizze der logischen Struktur des Fiduzial-Argumentes von R. A. Fisher.- Bibliographie.- IV. ,Statistisches Schließen — Statistische Begründung — Statistische Analyse‘statt,Statistische Erklärung‘.- 1. Elf Paradoxien und Dilemmas.- (I) Die Paradoxie der Erklärung des Unwahrscheinlichen.- (II) Das Paradoxon der irrelevanten Gesetzesspezialisierung.- (III) Das Informationsdilemma.- (IV) Das Erklärungs-Bestätigungs-Dilemma.- (V) Das Paradoxon der reinen ex post facto Kausalerklärung.- (VI) Das Verzahnungsparadoxon.- (VII) Das Erklärungs-Begründungs-Dilemma.- (VIII) Das Dilemma der nomologischen Implikation.- (IX) Das ,Weltanschauungsdilemma‘.- (X) Das Argumentationsdilemma.- (XI) Das Gesetzesparadoxon.- 2. Diskussion.- 2.a Problemreduktionen.- 2.b Das Problem der nomologischen Implikation. Statistisches Schließen und statistische Begründungen.- 2.c Verzahnungen von Erklärungs- und Bestätigungsproblemen.- 2.d Die Leibniz-Bedingung. Unbehebbare intuitive Konflikte.- 3. Statistische Begründungen statt statistische Erklärungen. Der statistische Begründungsbegriff als Explikat der Einzelfall-Regel.- 4. Statistische Analysen.- 4.a Kausale Relevanz und Abschirmung.- 4.b Statistische Oberflächenanalyse und statistisch-kausale Tiefenanalyse von Minimalform.- 4.c Statistische Analyse und statistisches Situationsverständnis.- 4.d Was könnte unter „Statistische Erklärung“ verstanden werden?.- Bibliographie.- Anhang I: Indeterminismus vom zweiten Typ.- Anhang II: Das Repräsentationstheorem von B. de Finetti.- 1. Intuitiver Zugang.- 1.a Bernoulli-Wahrscheinlichkeiten und Mischungen von Bernoulli-Wahrscheinlichkeiten.- 1.b Das Problem des Lernens aus der Erfahrung.- 1.c Die Bedeutung des Begriffs der Vertauschbarkeit.- 2. Formale Skizze. Übergang zum kontinuierlichen Fall.- 2.a Vertauschbarkeit und Symmetrie.- 2.b Mischungen und Lernen aus der Erfahrung: Der Riemannsche Fall..- 2.c Mischungen im abstrakten maßtheoretischen Fall. Das Repräsentationstheorem.- 2.d Diskussion.- Bibliographie.- Anhang III: Metrisierung qualitativer Wahrscheinlichkeitsfelder.- 1. Axiomatische Theorien der Metrisierung. Extensive Größen.- 2. Metrisierung von Wahrscheinlichkeitsfeldern.- 2.a Metrisierung klassischer absoluter Wahrscheinlichkeitsfelder im endlichen und abzählbaren Fall.- 2.b Metrisierung quantenmechanischer Wahrscheinlichkeitsfelder.- 2.c Metrisierung qualitativer bedingter Wahrscheinlichkeitsfelder.- Bibliographie.- Autorenregister.- Verzeichnis der Symbole und Abkürzungen.

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Book SynopsisMathematics is often considered as a body of knowledge that is essen­ tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of clearly formulating and correctly proving it. However, the question is not so simple, and P. Weingartner's paper (Language and Coding-Dependency of Results in Logic and Mathe­ matics) deals with some results in logic and mathematics which reveal that certain notions are in general not invariant with respect to different choices of language and of coding processes. Five example are given: 1) The validity of axioms and rules of classical propositional logic depend on the interpretation of sentential variables; 2) The language­ dependency of verisimilitude; 3) The proof of the weak and strong anti­ inductivist theorems in Popper's theory of inductive support is not invariant with respect to limitative criteria put on classical logic; 4) The language-dependency of the concept of provability; 5) The language­ dependency of the existence of ungrounded and paradoxical sentences (in the sense of Kripke). The requirements of logical rigour and consistency are not the only criteria for the acceptance and appreciation of mathematical proposi­ tions and theories.Table of ContentsGeneral Philosophical Perspectives.- Logic, Mathematics, Ontology.- From Certainty to Fallibility in Mathematics?.- Moderate Mathematical Fictionism.- Language and Coding-Dependency of Results in Logic and Mathematics.- What is a Profound Result in Mathematics?.- The Hylemorphic Schema in Mathematics.- Foundational Approaches.- Categorical Foundations of the Protean Character of Mathematics.- Category Theory and Structuralism in Mathematics: Syntactical Considerations.- Reflection in Set Theory. The Bernays-Levy Axiom System.- Structuralism and the Concept of Set.- Aspects of Mathematical Experience.- Logicism Revisited in the Propositional Fragment of Le?niewski’s Ontology.- The Applicability of Mathematics.- The Relation of Mathematics to the Other Sciences.- Mathematics and Physics.- The Mathematical Overdetermination of Physics.- Gödel’s Incompleteness Theorem and Quantum Thermodynamic Limits.- Mathematical Models in Biology.- The Natural Numbers as a Universal Library.- Mathematical Symmetry Principles in the Scientific World View.- Historical Considerations.- Mathematics and Logics. Hungarian Traditions and the Philosophy of Non-Classical Logic.- Umfangslogik, Inhaltslogik, Theorematic Reasoning.

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