Mathematical Logic

Description

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. Preface Contents Part A Chapter I Introduction I.1 An Example from Group Theory I.2 An Example from the Theory of Equivalence Relations I.3 A Preliminary Analysis I.4 Preview Chapter II Syntax of First-Order Languages II.1 Alphabets II.2 The Alphabet of a First-Order Language II.3 Terms and Formulas in First-Order Languages II.4 Induction in the Calculi of Terms and of Formulas II.5 Free Variables and Sentences Chapter III Semantics of First-Order Languages III.1 Structures and Interpretations III.2 Standardization of Connectives III.3 The Satisfaction Relation III.4 The Consequence Relation III.5 Two Lemmas on the Satisfaction Relation III.6 Some Simple Formalizations III.7 Some Remarks on Formalizability III.8 Substitution Chapter IV A Sequent Calculus IV.1 Sequent Rules IV.2 Structural Rules and Connective Rules IV.3 Derivable Connective Rules IV.4 Quantifier and Equality Rules IV.5 Further Derivable Rules IV.6 Summary and Example IV.7 Consistency Chapter V The Completeness Theorem V.1 Henkin’s Theorem V.2 Satisfiability of Consistent Sets of Formulas (the Countable Case) V.3 Satisfiability of Consistent Sets of Formulas (the General Case) V.4 The Completeness Theorem Chapter VI The Löwenheim–Skolem Theorem and the Compactness Theorem VI.1 The Löwenheim–Skolem Theorem VI.2 The Compactness Theorem VI.3 Elementary Classes VI.4 Elementarily Equivalent Structures Chapter VII The Scope of First-Order Logic VII.1 The Notion of Formal Proof VII.2 Mathematics Within the Framework of First-Order Logic VII.3 The Zermelo–Fraenkel Axioms for Set Theory VII.4 Set Theory as a Basis for Mathematics Chapter VIII Syntactic Interpretations and Normal Forms VIII.1 Term-Reduced Formulas and Relational Symbol Sets VIII.2 Syntactic Interpretations VIII.3 Extensions by Definitions VIII.4 Normal Forms Part B Chapter IX Extensions of First-Order Logic IX.1 Second-Order Logic IX.2 The System Lω_1ω IX.3 The System L_Q Chapter X Computability and Its Limitations X.1 Decidability and Enumerability X.2 Register Machines X.3 The Halting Problem for Register Machines X.4 The Undecidability of First-Order Logic X.5 Trakhtenbrot’s Theorem and the Incompleteness of Second-Order Logic X.6 Theories and Decidability X.7 Self-Referential Statements and Gödel’s Incompleteness Theorems X.8 Decidability of Presburger Arithmetic X.9 Decidability of Weak Monadic Successor Arithmetic Chapter XI Free Models and Logic Programming XI.1 Herbrand’s Theorem XI.2 Free Models and Universal Horn Formulas XI.3 Herbrand Structures XI.4 Propositional Logic XI.5 Propositional Resolution XI.6 First-Order Resolution (without Unification) XI.7 Logic Programming Chapter XII An Algebraic Characterization of Elementary Equivalence XII.1 Finite and Partial Isomorphisms XII.2 Fraïssé’s Theorem XII.3 Proof of Fraïssé’s Theorem XII.4 Ehrenfeucht Games Chapter XIII Lindström’s Theorems XIII.1 Logical Systems XIII.2 Compact Regular Logical Systems XIII.3 Lindström’s First Theorem XIII.4 Lindström’s Second Theorem References List of Symbols Subject Index