## A Shorter Model TheoryThis is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory. Besides standard results such as the compactness and omitting types theorems, it also describes various links with algebra, including the Skolem-Tarski method of quantifier elimination, model completeness, automorphism groups and omega-categoricity, ultraproducts, O-minimality and structures of finite Morley rank. The material on back-and-forth equivalences, interpretations and zero-one laws can serve as an introduction to applications of model theory in computer science. Each chapter finishes with a brief commentary on the literature and suggestions for further reading. This book will benefit graduate students with an interest in model theory. |

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### Contents

Naming of parts | 1 |

Structures | 2 |

Homomorphisms and substructures | 5 |

Terms and atomic formulas | 10 |

Parameters and diagrams | 15 |

Canonical models | 17 |

Further reading | 20 |

Classifying structures | 21 |

Indiscernibles | 152 |

Further reading | 156 |

The countable case | 158 |

Omitting types | 165 |

Countable categoricity | 171 |

ωcategorical structures by Fraďssés method | 175 |

Further reading | 181 |

The existential case | 182 |

Definable subsets | 22 |

Definable classes of structures | 30 |

Some notions from logic | 37 |

Maps and the formulas they preserve | 43 |

Translations | 51 |

Quantifier elimination | 59 |

Further reading | 68 |

Structures that look alike | 69 |

Backandforth equivalence | 73 |

Games for elementary equivalence | 82 |

Further reading | 91 |

Interpretations | 93 |

Automorphisms | 94 |

Relativisation | 101 |

Interpreting one structure in another | 107 |

Imaginary elements | 113 |

Further reading | 122 |

The firstorder case compactness | 124 |

Types | 130 |

Elementary amalgamation | 134 |

Amalgamation and preservation | 141 |

Expanding the language | 147 |

Existentially closed structures | 183 |

Constructing ec structures | 188 |

Modelcompleteness | 195 |

Quantifier elimination revisited | 201 |

Further reading | 208 |

Saturation | 210 |

The great and the good | 211 |

Big models exist | 220 |

Syntactic characterisations | 225 |

Onecardinal and twocardinal theorems | 233 |

Ultraproducts and ultrapowers | 237 |

Further reading | 248 |

Structure and categoricity | 250 |

EhrenfeuchtMostowski models | 251 |

Minimal sets | 257 |

Totally transcendental structures | 264 |

Stability | 273 |

Morleys theorem | 286 |

Further reading | 296 |

298 | |

300 | |

### Common terms and phrases

3i formula A-big A-homogeneous A-saturated abelian groups algebraically closed field atomic formula atomic sentences Aut(A automorphism axiomatisable axioms back-and-forth equivalent chain choose closed term compactness theorem complete theory complete type constants contains Corollary countable first-order countable model definition dom(A e.c. model elementarily equivalent elementary embedding elementary extension elementary substructure equivalence classes equivalence relation equivalent modulo example Exercises for section finite set finite subset first-order formula first-order language first-order theory following are equivalent formula p(x function symbol hence homomorphism induction infinite cardinal isomorphism linear ordering minimal set model theory model-complete Morley rank n-ary n-tuple natural numbers non-empty ordinal parameters player positive integer prove quantifier elimination quantifier-free formula realised recursive sequence of elements set of elements set of formulas Show signature Skolem strongly minimal subgroup Suppose Th(A tuple tuple of elements ty(a ty(x ultrafilter ultraproducts V2 theory vector space write