This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science.The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function.Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science.The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function.Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

Finite model theory, the model theory of finite structures, has roots in clas­ sical model theory; however, its systematic development was strongly influ­ enced by research and questions of complexity theory and of database theory. Model theory or the theory of models, as it was first named by Tarski in 1954, may be considered as the part of the semantics of formalized languages that is concerned with the interplay between the syntactic structure of an axiom system on the one hand and (algebraic, settheoretic, . . . ) properties of its models on the other hand. As it turned out, first-order language (we mostly speak of first-order logic) became the most prominent language in this respect, the reason being that it obeys some fundamental principles such as the compactness theorem and the completeness theorem. These principles are valuable modeltheoretic tools and, at the same time, reflect the expressive weakness of first-order logic. This weakness is the breeding ground for the freedomwhich modeltheoretic methods rest upon. By compactness, any first-order axiom system either has only finite models of limited cardinality or has infinite models. The first case is trivial because finitely many finite structures can explicitly be described by a first-order sentence. As model theory usually considers all models of an axiom system, modeltheorists were thus led to the second case, that is, to infinite structures. In fact, classical model theory of first-order logic and its generalizations to stronger languages live in the realm of the infinite.

The 1999 Annual Conference of the European Association for Computer Science Logic, CSL’99, was held in Madrid, Spain, on September 20-25, 1999. CSL’99 was the 13th in a series of annual meetings, originally intended as Internat- nal Workshops on Computer Science Logic, and the 8th to be held as the - nual Conference of the EACSL. The conference was organized by the Computer Science Departments (DSIP and DACYA) at Universidad Complutense in M- rid (UCM). The CSL’99 program committee selected 34 of 91 submitted papers for p- sentation at the conference and publication in this proceedings volume. Each submitted paper was refereed by at least two, and in almost all cases, three di erent referees. The second refereeing round, previously required before a - per was accepted for publication in the proceedings, was dropped following a decision taken by the EACSL membership meeting held during CSL’98 (Brno, Czech Republic, August 25, 1998).

This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science.The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function.Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science.The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function.Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

Finite model theory, the model theory of finite structures, has roots in clas­ sical model theory; however, its systematic development was strongly influ­ enced by research and questions of complexity theory and of database theory. Model theory or the theory of models, as it was first named by Tarski in 1954, may be considered as the part of the semantics of formalized languages that is concerned with the interplay between the syntactic structure of an axiom system on the one hand and (algebraic, settheoretic, . . . ) properties of its models on the other hand. As it turned out, first-order language (we mostly speak of first-order logic) became the most prominent language in this respect, the reason being that it obeys some fundamental principles such as the compactness theorem and the completeness theorem. These principles are valuable modeltheoretic tools and, at the same time, reflect the expressive weakness of first-order logic. This weakness is the breeding ground for the freedomwhich modeltheoretic methods rest upon. By compactness, any first-order axiom system either has only finite models of limited cardinality or has infinite models. The first case is trivial because finitely many finite structures can explicitly be described by a first-order sentence. As model theory usually considers all models of an axiom system, modeltheorists were thus led to the second case, that is, to infinite structures. In fact, classical model theory of first-order logic and its generalizations to stronger languages live in the realm of the infinite.

The 1999 Annual Conference of the European Association for Computer Science Logic, CSL’99, was held in Madrid, Spain, on September 20-25, 1999. CSL’99 was the 13th in a series of annual meetings, originally intended as Internat- nal Workshops on Computer Science Logic, and the 8th to be held as the - nual Conference of the EACSL. The conference was organized by the Computer Science Departments (DSIP and DACYA) at Universidad Complutense in M- rid (UCM). The CSL’99 program committee selected 34 of 91 submitted papers for p- sentation at the conference and publication in this proceedings volume. Each submitted paper was refereed by at least two, and in almost all cases, three di erent referees. The second refereeing round, previously required before a - per was accepted for publication in the proceedings, was dropped following a decision taken by the EACSL membership meeting held during CSL’98 (Brno, Czech Republic, August 25, 1998).