Mathematical logic 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 book gathers together a colorful set of problems on classical Mathematical Logic, selected from over 30 years of teaching. The initial chapters start with problems from supporting fields, like set theory (ultrafilter constructions), full-information game theory (strategies), automata, and recursion theory (decidability, Kleene’s theorems). The work then advances toward propositional logic (compactness and completeness, resolution method), followed by first-order logic, including quantifier elimination and the Ehrenfeucht– Fraïssé game; ultraproducts; and examples for axiomatizability and non-axiomatizability. The Arithmetic part covers Robinson’s theory, Peano’s axiom system, and Gödel’s incompleteness theorems. Finally, the book touches universal graphs, tournaments, and the zero-one law in Mathematical Logic. Instructors teaching Mathematical Logic, as well as students who want to understand its concepts and methods, can greatly benefit from this work. The style and topics have been specially chosen so that readers interested in the mathematical content and methodology could follow the problems and prove the main theorems themselves, including Gödel’s famous completeness and incompleteness theorems. Examples of applications on axiomatizability and decidability of numerous mathematical theories enrich this volume.Table of ContentsChapter 1 - Special Set Systems.- Chapter 2 - Games and Voting.- Chapter 3 - Formal languages and automata.- Chapter 4 - Recursion Theory.- Chapter 5 - Propositional Calculus.- Chapter 6 - First-order logic.- Chapter 7 - Fundamental Theorems.- Chapter 8 - Elementary Equivalence.- Chapter 9 - Ultraproducts.- Chapter 10 - Arithmetic.- Chapter 11 - Selected Applications.- Chapter 12 - Solutions.

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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 SynopsisMathematical Logic: An Introduction is a textbook that uses mathematical tools to investigate mathematics itself. In particular, the concepts of proof and truth are examined. The book presents the fundamental topics in mathematical logic and presents clear and complete proofs throughout the text. Such proofs are used to develop the language of propositional logic and the language of first-order logic, including the notion of a formal deduction. The text also covers Tarski’s definition of truth and the computability concept. It also provides coherent proofs of Godel’s completeness and incompleteness theorems. Moreover, the text was written with the student in mind and thus, it provides an accessible introduction to mathematical logic. In particular, the text explicitly shows the reader how to prove the basic theorems and presents detailed proofs throughout the book. Most undergraduate books on mathematical logic are written for a reader who is well-versed in logical notation and mathematical proof. This textbook is written to attract a wider audience, including students who are not yet experts in the art of mathematical proof.

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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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Book SynopsisThis textbook offers an introduction to the philosophy of science. It helps undergraduate students from the natural, the human and social sciences to gain an understanding of what science is, how it has developed, what its core traits are, how to distinguish between science and pseudo-science and to discover what a scientific attitude is. It argues against the common assumption that there is fundamental difference between natural and human science, with natural science being concerned with testing hypotheses and discovering natural laws, and the aim of human and some social sciences being to understand the meanings of individual and social group actions. Instead examines the similarities between the sciences and shows how the testing of hypotheses and doing interpretation/hermeneutics are similar activities. The book makes clear that lessons from natural scientists are relevant to students and scholars within the social and human sciences, and vice versa. It teaches its readers how to effectively demarcate between science and pseudo-science and sets criteria for true scientific thinking. Divided into three parts, the book first examines the question What is Science? It describes the evolution of science, defines knowledge, and explains the use of and need for hypotheses and hypothesis testing. The second half of part I deals with scientific data and observation, qualitative data and methods, and ends with a discussion of theories on the development of science. Part II offers philosophical reflections on four of the most important concepts in science: causes, explanations, laws and models. Part III presents discussions on philosophy of mind, the relation between mind and body, value-free and value-related science, and reflections on actual trends in science.Trade Review“Target audience comprises ‘students of engineering, physics, biology, social science, medicine and nursing’ … this book allows readers from a global audience to grasp the ‘flavour’ of the long and rich Germanic and Scandinavian tradition.” (Agustín Adúriz-Bravo, Science and Education, Vol. 28, 2019)“The aim of Lars-Göran Johansson’s textbook Philosophy of Science for Scientists is … to provide an introduction to the philosophy of science for students in all fields of science. … the book is supposed to be suitable for an undergraduate level course in the philosophy of science for philosophy students.” (Amanda Thorell, Theoria, Vol. 83, 2017)“Lars-Göran Johansson’s recent book Philosophy of science for scientists is the only textbook in the philosophy of science that is addressed specifically to an audience consisting of scientists. … In its breadth of treated topics, the book can serve as a basic text to many different courses in the philosophy of science.” (Maarten Franssen, Metascience, Vol. 26, 2017)“Philosophy of Science for Scientists by Lars-Goran Johansson: a lovely textbook for undergraduates. It is a highly readable introduction to how one can view the practice of science. … this is an excellent introduction to understanding science in a general sense. Students and practitioners will find it worthwhile to read and discuss.” (David S. Mazel, MAA Reviews, maa.org, November, 2016)“This is an excellent book that can serve as a very appropriate textbook for the first course in Philosophy of Science. … it is a very well written book and is an enjoyable reading. … It is well written by a great authority in the field and I strongly recommend it to you if you are interested in to understand what science is and why science is important for knowledge and our understanding of reality.” (Philosophy, Religion and Science Book Reviews, Bookinspections.wordpress.com, July, 2016)Table of ContentsPreface and overview of the bookPart 1: What is science?.- 1. The Evolution of Science.- 2. Knowledge.- 3. Hypotheses and Hypothesis Testing.- 4. On Scientific Data.- 5. Qualitative Data and Methods.- 6. Theories about the Development of Science.- Part 2. Philosophical reflections on four core concepts in science: causes, explanations, laws and models.- 7. On Causes and Correlations.- 8. Explanations.- 9. Explanation in the Humanities and Social Sciences.- 10. Scientific Laws.- 11. Theories, Models and Reality.- Part 3. Some auxiliaries.- 12. The Mind-Body Problem.- 13. Science and Values.- 14. Some trends in science.- Appendix.- Logical Forms.- Index.

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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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