## A Course on Mathematical LogicThis book provides a distinctive, well-motivated introduction to mathematical logic. It starts with the definition of first order languages, proceeds through propositional logic, completeness theorems, and finally the two Incompleteness Theorems of Godel. |

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### Contents

1 | |

12 Terms of a Language | 4 |

13 Formulas of a Language | 6 |

14 FirstOrder Theories | 10 |

Semantics of FirstOrder Languages | 15 |

21 Structures of FirstOrder Languages | 16 |

22 Truth in a Structure | 17 |

23 Model of a Theory | 19 |

Completeness Theorem and Model Theory | 65 |

52 Interpretations in a Theory | 70 |

53 Extension by Definitions | 72 |

54 Compactness Theorem and Applications | 74 |

55 Complete Theories | 77 |

56 Applications in Algebra | 79 |

Recursive Functions and Arithmetization of Theories | 82 |

61 Recursive Functions and Recursive Predicates | 84 |

24 Embeddings and Isomorphisms | 20 |

Propositional Logic | 29 |

31 Syntax of Propositional Logic | 30 |

33 Compactness Theorem for Propositional Logic | 33 |

34 Proof in Propositional Logic | 37 |

35 Metatheorems in Propositional Logic | 38 |

36 Post Tautology Theorem | 42 |

Proof and Metatheorems in FirstOrder Logic | 45 |

42 Metatheorems in FirstOrder Logic | 46 |

43 Some Metatheorems in Arithmetic | 59 |

44 Consistency and Completeness | 62 |

62 Semirecursive Predicates | 93 |

63 Arithmetization of Theories | 96 |

64 Decidable Theories | 103 |

Incompleteness Theorems and Recursion Theory | 107 |

72 First Incompleteness Theorem | 115 |

73 Arithmetical Sets | 116 |

74 Recursive Extensions of Peano Arithemetic | 125 |

75 Second Incompleteness Theorem | 131 |

References | 135 |

137 | |

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### Common terms and phrases

abelian groups algebraically closed fields algorithm assume atomic formula axiomatized binary relation called cardinality closed formula closure properties complete conservative extension consistent constant symbol countable denote detachment rule distinct variables elementary equivalent Example Exercise extension by definitions finite sequence finitely satisfiable first-order language First-Order Logic first-order theory function f Gödel number group theory Hence incompleteness theorem induction hypothesis interpretation introduced isomorphic k-ary Lemma Metatheorems n-ary function symbol n-ary relation symbol natural number nonempty set nonlogical axioms nonlogical symbols open formula ordered fields Peano arithmetic pointclass Proof propositional logic prove the result quantifiers R-formula recursion theory recursive extension recursive functions recursive predicates recursive substitutions representable rules of inference semirecursive predicate sentence set of formulas set theory statement structure subformulas subset substitution rule Suppose symbol f syntactical tautological consequence tautology theorem true validity theorem variable-free terms