### The primal framework. I

by Baldwin and Shelah. [BlSh:330]

Annals Pure and Applied Logic, 1990

This the first of a series of articles dealing with abstract
classification theory. The apparatus to assign systems of cardinal
invariants to models of a first order theory (or determine its
impossibility) is developed in [Sh:a]. 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
[Sh:88] and for universal classes [Sh:300] led to the conclusion
that an axiomatic approach provided the best setting for developing
a theory of wider application. In the first chapter we describe the
axioms on which the remainder of the article depends and give some
examples and context to justify this level of generality. 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 II. 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 III provides sufficient conditions
for prime models over such trees to exist.

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