The primal framework. I (Q1813741)

This is the first of a series of articles dealing with abstract classification theory. The apparatus to design systems of cardinal invariants to models of a first-order theory (or determine its impossibility) is developed in ``Classification theory and the number of nonisomorphic models'' (1978; Zbl 0388.03009) by the second author. It is natural to try to extend this theory to classes of models which are described in other ways. Work on the classification theory for nonelementary classes [the second author, Lect. Notes Math. 1292, 419-497 (1988; Zbl 0639.03034)] and for universal classes [the second author, ibid., 264-419 (1988; Zbl 0637.03028)] led to the conclusion that an axiomatic approach provided the best setting for developing a theory of wider application. This approach is reminiscent of the early work of Fraissé and Jónson on the existence of homogeneous-universal models. As this will be a long project it seems appropriate to report our progress as we go along. In large part this series of articles will parallel the development in the last cited paper by the second author. A survey of that paper, which could serve as an introduction to this one, is given by the first author [ibid., 1-23 (1988; Zbl 0659.03010)]. In the first chapter of this article we describe the axioms on which the remainder of the article depends and give some examples and context to justify this level of generality. As is detailed later the principal goal of this series is indicated by its title. The study of universal classes takes as a primitive the notion of closing a subset under functions to obtain a model. We replace that concept by the notion of a prime model. We begin the detailed discussion of this idea in Chapter 2. One of the important contributions of classification theory is the recognition that large models can often be analyzed by means of a family of small models indexed by a tree of height at most \(\omega\). More precisely, the analyzed model is prime over such a tree. Chapter 3 provides sufficient conditions for prime models over such trees to exist. The discussion of properties of a class which guarantee that each model in the class is prime over such a tree will appear later in the series. We introduce in Chapters 1 and 2 a number of principles which we loosely refer to as axioms. At the beginning of Chapter 3 we define the notion of an adequate class --- a class which satisfies those axioms that we assume in the mainline of the study. This notion of an adequate class will be embellished by further axioms in later papers of this series. Our use of the word axiom in this content is somewhat inexact; postulate might be better. In exploring an unknown area we list certain principles which appear to make important distinctions. In our definition of an adequate class we collect a family of these principles that is sufficient to establish a coherent collection of results.