Celebratio Mathematica — van den Dries

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L. van den Dries and A. H. Lewenberg: “\( T \)-convexity and tame extensions,” J. Symbolic Logic 60 : 1 (March 1995), pp. 74–102. MR 1324502 Zbl 0856.03028 article

Let \( T \) be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of \( T \) and show that the residue field of such a convex hull has a natural expansion to a model of \( T \). We give a quantifier elimination relative to \( T \) for the theory of pairs \( (\mathscr{R},V) \) where \( \mathscr{R} \models T \) and \( V \neq \mathscr{R} \) is the convex hull of an elementary substructure of \( \mathscr{R} \). We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to \( T \) for the theory of pairs with \( \mathscr{R} \) a model of \( T \) and a proper elementary substructure that is Dedekind complete in \( \mathscr{R} \). We deduce that the theory of such “tame” pairs is complete.