An effective version of Wilkie's theorem of the complement and some effective o-minimality results.
Tarski proved that the theory \(\mathbb{R}= (\mathbb{R},+,\cdot,0,1,<)\) is decidable and asked whether the same holds for the theory \(T_{\text{exp}}\), the theory of \(\mathbb{R}\) with the function \(e^x\) added. This problem is still open. Wilkie showed that \(T_{\text{exp}}\) is model complete. It follows that every definable \(X\subseteq\mathbb{R}^n\) (in the language of \(T_{\text{exp}})\) has a finite number of connected components. This implies that \(T_{\text{exp}}\) is o-minimal. Here, the authors prove an effective version of a theorem of Wilkie: Given an expansion of \(\mathbb{R}\) with finitely many \(C^\infty\) functions, if there are uniform and computable upper bounds on the number of connected components of quantifier-free definable sets, then there are uniform and computable bounds for all definable sets. Further on they show: Let \(L\) be a language expanding the language of ordered rings. If an \(L\)-sentence is true in every o-minimal \(L\)-structure expanding the ordered field of real numbers, then it is true in every o-minimal \(L\)-structure expanding any real closed field.
scientific article; zbMATH DE number 2056205