Trivial pursuit: remarks on the main gap
A trivial remark: the title of this paper does not imply that the authors' pursuit is trivial. Indeed, ``a dependence relation is trivial when a single point which depends on a pair of independent points depends on just one of them''. The authors use this notion in the context of stability theory, and define the triviality of a type via forking. They pursue consequences of this triviality, as well as constructions of various trees consisting of submodels of a given model, in the context of the main gap theorem in the spectrum problem (i.e., approximately, the number of isomorphism types of infinite models of a theory is either always maximal possible, or always rather small). For instance, the last theorem of the paper reads: if the theory T is \(\omega\)-stable, countable, without the dimensional order property [for this, cf., e.g., the paper by \textit{L. Harrington} and \textit{M. Makkai}, Notre Dame J. Formal Logic 26, 139-177 (1985; Zbl 0589.03015)] and of depth \(\geq \omega\), then for \(1\leq \beta <\omega\), the number of isomorphism types of cardinalities \(\leq \aleph_{\beta}\), is \(2^{\aleph_{\beta}}\). Of course, there are many more results about and considerations of the spectrum problem as well as triviality and trees. This paper amalgamates and extends various sections of Baldwin's book: Fundamentals of stability theory (Berlin, 1988). And so, it is very readable and instructive: there are a lot of explanations of what is going on, why the authors do things as they do, what the connections are with other approaches, and so forth. A drawback is the presence of many careless typographical errors, although they are mostly trivial.
scientific article; zbMATH DE number 4029523