Throughout this book, we shall work inside a big, κ-saturated κ-homogeneous model C of a complete first-order theory T in a language L; we call C the monster model. Every model we consider will have cardinality less than κ and be an elementary substructure of C; every set of parameters will have cardinality less than κ and be a subset of C, and every tuple will have length less than κ and come from C. Since every model of T of cardinality less than κ can be elementarily embedded into C, this is no restriction, but it simplifies the arguments and helps the intuition, as everything takes place inside a fixed “universal domain”.
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