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Elementary Mathematical Logic | |

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The propositional calculus | |

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Linguistic considerations: formulas | |

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Model theory: truth tables, validity | |

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Model theory: the substitution rule, a collection of valid formulas | |

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Model theory: implication and equivalence | |

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Model theory: chains of equivalences | |

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Model theory: duality | |

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Model theory: valid consequence | |

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Model theory: condensed truth tables | |

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Proof theory: provability and deducibility | |

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Proof theory: the deduction theorem | |

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Proof theory: consistency, introduction and elimination rules | |

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Proof theory: completeness | |

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Proof theory: use of derived rules | |

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Applications to ordinary language: analysis of arguments | |

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Applications to ordinary language: incompletely stated arguments | |

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The predicate calculus | |

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Linguistic considerations: formulas, free and bound occurrences of variables | |

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Model theory: domains, validity | |

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Model theory: basic results on validity | |

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Model theory: further results on validity | |

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Model theory: valid consequence | |

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Proof theory: provability and deducibility | |

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Proof theory: the deduction theorem | |

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Proof theory: consistency, introduction and elimination rules | |

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Proof theory: replacement, chains of equivalences | |

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Proof theory: alterations of quantifiers, prenex form | |

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Applications to ordinary language: sets, Aristotelian categorical forms | |

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Applications to ordinary language: more on translating words into symbols | |

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The predicate calculus with equality | |

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Functions, terms | |

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Equality | |

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Equality vs. equivalence, extensionality | |

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Descriptions | |

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Mathematical Logic and the Foundations of Mathematics | |

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The foundations of mathematics | |

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Countable sets | |

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Cantor's diagonal method | |

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Abstract sets | |

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The paradoxes | |

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Axiomatic thinking vs. intuitive thinking in mathematics | |

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Formal systems, metamathematics | |

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Formal number theory | |

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Some other formal systems | |

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Computability and decidability | |

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Decision and computation procedures | |

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Turing machines, Church's thesis | |

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Church's theorem (via Turing machines) | |

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Applications to formal number theory: undecidability (Church) and incompleteness (Godel's theorem) | |

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Applications to formal number theory: consistency proofs (Godel's second theorem) | |

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Application to the predicate calculus (Church, Turing) | |

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Degrees of unsolvability (Post), hierarchies (Kleene, Mostowski) | |

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Undecidability and incompleteness using only simple consistency (Rosser) | |

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The predicate calculus (additional topics) | |

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Godel's completeness theorem: introduction | |

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Godel's completeness theorem: the basic discovery | |

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Godel's completeness theorem with a Gentzen-type formal system, the Lowenheim-Skolem theorem | |

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Godel's completeness theorem (with a Hilbert-type formal system) | |

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Godel's completeness theorem, and the Lowenheim-Skolem theorem, in the predicate calculus with equality | |

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Skolem's paradox and nonstandard models of arithmetic | |

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Gentzen's theorem | |

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Permutability, Herbrand's theorem | |

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Craig's interpolation theorem | |

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Beth's theorem on definability, Robinson's consistency theorem | |

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Bibliography | |

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Theorem and lemma numbers: pages | |

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List of postulates | |

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Symbols and notations | |

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Index | |