An old friend revisited: countable models of \(\omega\)-stable theories (Q998143)

Borel complexity is a tool for analyzing the complexity of the class of countable models -- beyond the number of isomorphism types -- of \(\omega\)-stable theories. In this paper, the author proves that the set of models on \(\omega\) of an \(\omega\)-stable theory is Borel complete if either the theory has ENI-DOP or it is ENI-NDOP but is ENI-deep. Here, `NDOP' is for the negation of the dimensional order property, and a type is called ENI if it is strongly regular, based on a finite set, stationary and nonisolated. The author gives a natural and algebraic characterization of ENI-NDOP, also. Examples and explanations he presents help understanding.