## Models and GamesThis gentle introduction to logic and model theory is based on a systematic use of three important games in logic: the semantic game; the Ehrenfeucht–Fraïssé game; and the model existence game. The third game has not been isolated in the literature before but it underlies the concepts of Beth tableaux and consistency properties. Jouko Väänänen shows that these games are closely related and in turn govern the three interrelated concepts of logic: truth, elementary equivalence and proof. All three methods are developed not only for first order logic but also for infinitary logic and generalized quantifiers. Along the way, the author also proves completeness theorems for many logics, including the cofinality quantifier logic of Shelah, a fully compact extension of first order logic. With over 500 exercises this book is ideal for graduate courses, covering the basic material as well as more advanced applications. |

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### Contents

1 | |

Games | 14 |

Graphs | 35 |

Models | 53 |

FirstOrder Logic | 79 |

Infinitary Logic | 139 |

Model Theory of Infinitary Logic | 176 |

1 | 183 |

Stronger Infinitary Logics | 228 |

Exercises | 278 |

Exercises | 343 |

353 | |

362 | |

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### Common terms and phrases

a„_i assume atomic automorphisms Axiom of Choice back-and-forth sequence back-and-forth set bijection bijection closed cofinality Compactness Theorem consistent constant symbols countable models countable set countable vocabulary Craig Interpolation Definition denote EFn(G,G Ehrenfeucht–Fraıss´e Game element equivalence classes equivalence relation Example Exercise Figure first-order logic following are equivalent function symbols G and G Hyttinen induction hypothesis infinitary logic infinite L-formula L-sentences L-structure Lemma linear order MEG(T Model Existence Game model theory moves natural number partial isomorphism PC-class player I plays player II player II chooses player II wins po-set predicate symbol Proof Let Proof Suppose properties Proposition prove quantifier Q quantifier rank regular cardinal satisfies Scott tree sentence of quantifier Show that player Skolem function strategy of player subset Suppose Q ultraproduct uncountable V¨a¨an¨anen vertex well-order winning strategy