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Table of contents:
The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in mathematical logic. This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. The text contains numerous exercises and an appendix furnishes answers to many of them. Introduction to Mathematical Logic includes: propositional logic first-order logic first-order number theory and the incompleteness and undecidability theorems of G/del, Rosser, Church, and Tarski; and axiomatic set theory theory of computability. The study of mathematical logic, axiomatic set theory, and computability theory provides an understanding of the fundamental assumptions and proof techniques that form basis of mathematics. Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence. Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields.
Contents:
The Propositional CalculusPropositional Connectives
Truth TablesTautologiesAdequate Sets of ConnectivesAn Axiom System for the Propositional CalculusIndependence: Many-Valued LogicsOther AxiomatizationsQuantification TheoryQuantifiersFirst-Order Languages and Their InterpretationsFirst-Order TheoriesProperties of First-Order TheoriesAdditional Metatheorems and Derived RulesRule CCompleteness TheoremsFirst-Order Theories with EqualityDefinitions of New Function Letters and Individual ConstantsPrenex Normal FormsIsomorphism of Interpretations
Categoricity of TheoriesGeneralized First-Order Theories
Completeness and DecidabilityElementary Equivalence
Elementary ExtensionsUltrapowers
Non-Standard AnalysisSemantic TreesQuantification Theory Allowing Empty DomainsFormal Number TheoryAn Axiom SystemNumber-Theoretic Functions and RelationsPrimitive Recursive and Recursive FunctionsArithmatization
G/del NumbersThe Fixed Point Theorem
G/del's Incompleteness TheoremRecursive Undecidability
Church's TheoremAxiomatic Set TheoryAn Axiom SystemOrdinal NumbersEquinumerousity
Finite and Denumerable Sets.Hartog's Theorem
Initial Ordinals
Ordinal ArithmeticThe Axiom of Choice
The Axiom of RegularityOther Axiomatizations of Set TheoryComputabilityAlgorithms
Turing MachinesDiagramsPartial Recursive Functions
Unsolvable Problems.The Kleene-Mosotovski Hierarchy
Recursively Enumerable SetsOther notions of ComputabilityDecision Problems
Brief Description:
Covering the basic topics of a solid first course in mathematical logic, this book includes an appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. It is useful for lecturers and researchers in mathematics, philosophy, and related fields.
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