Theory of models with generalized atomic formulas

Abstract

IntroductionWe shall prove the following theorem, which gives a necessary and sufficient condition for an elementary class to be characterized by a set of sentences having a prescribed number of alternations of quantifiers. A finite sequence of relational systems is said to be a sandwich of order n if each is an elementary extension of (i ≦ n—2), and each is an extension of (i ≦ n—2). If K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n.