Elementary approximation theory
We consider elementary approximation theory [see \textit{H. Rasiowa} and \textit{A. Skowron}, Proc. Int. Spring Sch., Wendisch-Rietz/GDR 1985, Math. Res. 31, 123-139 (1986; Zbl 0642.68040)]. Our aim is to investigate some aspects concerning finite models of this theory. This motivates us to consider properties of finite models of a given first-order theory \(T\). Let \(\text{Th((Mod }T)_{\text{fin}})\) \((\text{Th}((\text{Mod }T)_{\inf})\), respectively) be the set of all sentences valid in the class of all finite (infinite, respectively) models of \(T\). The following natural questions arise: 1. What are the necessary and sufficient conditions for the equality \(T=\text{Th}((\text{Mod }T)_{\text{fin}})\)? 2. For which theories \(T\) is the theory \(\text{Th}((\text{Mod }T)_{\text{fin}})\) decidable? This paper is organized into three parts. In Section 1 we give some general facts concerning the first question. It also contains simple examples of finitely axiomatizable first-order theories \(T\) such that \(T=\text{Th}((\text{Mod }T)_{\text{fin}})\). In Section 2 we present some elementary approximation theories. It also contains results concerning questions 1 and 2 for these theories. In Section 3 we consider some approximation problems. In particular a notion of an approximating translation is formulated.
finitely axiomatizable first-order theories
scientific article; zbMATH DE number 92878