234 INDEX Baumgartner example, 141 big type, 97 brimful, 73 brimmed, 73 chain, 27 chain , 76 closed map, 10 coextension, 147 coherence property, 28 coherent chain, 83 collection cardinal, 106 compact galois types, 84 complete sentence, 45 complete sentence, 𝐿𝜔1,𝜔(𝑄), 54 completeness, 159 cotorsion theories, 30 countable closure property, 8 cub, 6 cut-pair, 58 definability of types, 187 determined over, 9 Devlin-Shelah weak diamond, 127 diagram, 50 direct limit, 28 direct union, 28 disjoint amalgamation over limit models, 79 disjoint amalgamation property, 29 dominance lemma, 183 downward closed, 32 EM-set, 50 end extension, 40 end extension sentence, 212 enumeration, 152 excellence, first order, 179 excellence, quasiminimal, 10 excellent, 160 exchange axiom, 3 extendibility sentence, 212 extension lemma, 145 few, 165 filtration, 160 finitary AEC, 184 finite character, AEC, 184 finite diagram, 35 fragment, 5, 211 full, 159 Galois saturated, 69 Galois type, 68, 69 Galois-stable, 71 generalized symmetry, 158 good, 141 good atomic class, 160 good enumeration, 154 good system, 151 group, aqm, 15 Hanf Function, 33 Hanf Number, 33 Hanf Number, AEC, 34 Hanf number, Galois types, 106 Hanf number: categoricity, 117 Hart-Shelah example, 194 hereditary cardinality, 110 homogeneity: ℵ0-over models, 8 homogeneous geometry, 3 homogeneous model theory, 35 homogeneous, sequence, 22 Hrushovski construction, 29 independence, atomic classes, 146 independent family, 125 independent system, 10, 151 inhomogeneity, 21 isomorphic systems, 167 J. Knight example, 138 joint embedding, 29, 64 L¨ owenheim-Skolem number, 28 Lachlan example, 193 limit model, 76, 80, 123 linearization, 152 local galois types, 84 Lopez-Escobar theorem, 40 Marcus Example, 22, 139, 142 Martin’s Axiom, 126 maximal model, 123 maximal triple, 57 metric AEC, 36 minimal type, 97 model homogeneous, 67 monster model, 68 Morley omitting types theorem, 205 Morley’s categoricity theorem, 45 Non-elementary class, 26 nonalgebraic, 94 omitting Galois types, 106 omitting types theorem, 33 omitting types: set theoretic method, 111 order property, 116 order type, 73 partial monomorphism, 7 partial type, 33 pregeometry, 3 Presentation Theorem, 30 primary, 141 prime over, 10, 141 proper for linear orders, 50 proper pair, 57

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