Abstract
This chapter discusses the method of alternating chains. A common construction occurring in contributions to the theory of models is the development of an alternating chain of structures, i.e. of a sequence of structures lying alternately in each of two given classes of structures. Although not always originally formulated in these terms, it has turned out that a substantial number of the applications of these chains can be formulated in terms of separability (or interpolability) tests. The word “hierarchy” is usually used informally in mathematical discussions, so it no doubt suggests more or less structure to different people. By a family of sets we mean a function to a class of sets. “Hierarchy” is most often used in the literature in reference to a special kind of family of sets —namely a family of sets (or classes) of sets. The inclusion relation on the classes of sets induces a partial order on the domain (or index set) of the hierarchy, and in most cases this is a partial well order if not indeed a well order.
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