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