Classification theory over a predicate. I
If T is a complete first-order theory and P a unary predicate in its language, the authors study the structre and number of models M of T over the substructure \(M^ P\) of M determined by the universe \(P^ M\). It is shown that, given N, those models of T with \(M^ P=N\) can be coded to some extent by L'-reducts of expansions of N to a certain theory T'', where \(L(T)\subseteq L'\subseteq L(T'')\). The number of such reducts is estimated. This generalizes the context of the Chang-Makkai theorem. It is shown that if for some N of power \(\Lambda\), the number of non-N- isomorphic models M of T, with \(M^ P=N\), \(| M| =\Lambda\) is not too big, then the set of types \(tp(\bar a/P^ M)\) is definable over \(P^ M\) for every model M of T and every \(\bar a\in M\). Assuming the above definability condition and \(T=T''\) some non-structure theorems are proved.
scientific article; zbMATH DE number 3974950