## Introduction to Mathematical Logic, Fourth EditionThe 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: 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. |

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### Contents

Preface | |

The prepositional calculus 11 | |

Quantification theory 50 | |

Formal number theory 154 | |

Axiomatic set theory 225 | |

Computability 305 | |

Recursively enumerable sets 333 | |

Appendix Secondorder logic 368 | |

Answers to selected exercises 383 | |

Bibliography 412 | |

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### Common terms and phrases

applied Assume axiom of choice axiom schema axiomatic cardinal numbers Church's thesis closed wf computation consistent contains contradicting Corollary deduction theorem defined definition denote denumerable element equinumerous example Exercise expression false finite number first-order theory following wfs formulas free variables function letters Godel number Hence individual constants inductive hypothesis infinite language Lemma logically equivalent logically valid mathematical modus ponens natural numbers non-empty normal algorithm normal model obtain occurrences ordinal partial recursive function positive integers predicate calculus predicate letter prenex normal form primitive recursive Proof Let proper axioms Proposition provable Prove quantifiers real numbers recursive or recursive recursively undecidable relation replace rule A4 satisfies second-order second-order logic sentence sequence set theory Show standard interpretation statement form statement letters subset symbols tape description tautology theory with equality transfinite induction true truth table truth values Turing machine well-ordering xn,y