Σ-DEFINABILITY IN HEREDITARILY FINITE SUPERSTRUCTURES AND PAIRS OF MODELS

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Abstract

We consider the problem of being Σ-definable for an uncountable model of a c-simple theory in hereditarily finite superstructures over models of another c-simple theory. A necessary condition is specified in terms of decidable models and the concept of relative indiscernibility introduced in the paper. A criterion is stated for the uncountable model of a c-simple theory to be Σ-definable in superstructures over dense linear orders, and over infinite models of the empty signature. We prove the existence of a c-simple theory (of an infinite signature) every uncountable model of which is not Σ-definable in superstructures over dense linear orders. Also, a criterion is given for a pair of models to be recursively saturated.

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