The spectrum problem. I: \(\aleph_{\epsilon}\)-saturated models, the main gap
In this important paper, Shelah introduces the notions of DOP (dimensional order property) and depth, and shows that for superstable T: \(I^ a_{\aleph_ 0}(\aleph_{\alpha},T) (=\) the number of \(F^ a_{\aleph_ 0}\)-saturated models of T of cardinality \(\aleph_{\alpha})\) is either \(2^{\alpha} (\forall \aleph_{\alpha}\geq \lambda(T)+\aleph_ 1)\) or is \(<\beth_{(2^{| T|})}+(| \alpha |)\) (for T countable, \(<\beth_{\omega_ 1}(| \alpha |))\) for all such \(\aleph_{\alpha}\). The first case holds just if T has the DOP or is deep. This solves the ''Main gap'' for the class of \(F^ a_{\aleph_ 0}\)-saturated models of superstable T. The notion DOP can be defined for arbitrary stable theories, T having NDOP (T not having DOP) meaning that whenever \(M\prec M_ 1,M_ 2\) are \(F^ a_{\kappa(T)}\)-saturated models, \(M_ 1\), \(M_ 2\) independent over M, then the \(F^ a_{\kappa(T)}\)-prime model over \(M_ 1\cup M_ 2\) is \(F^ a_{\kappa(T)}\)-minimal. If T is superstable with NDOP then any \(F^ a_{\aleph_ 0}\)-saturated model of T is \(F^ a_{\aleph_ 0}\)-prime and minimal over a ''nonforking tree'' of ''small'' \(F^ a_{\aleph_ 0}\)-saturated models. If some such tree is not well- founded, then T is said to be deep. Some of the proofs are rather sketchy. It is worth mentioning that there are other expositions of this work and analysis of the concepts involved, notably in ''An exposition of Shelah's 'Main gap''' by \textit{M. Makkai} and \textit{L. Harrington}, and in forthcoming books by \textit{D. Lascar} and by \textit{J. Baldwin}.
scientific article; zbMATH DE number 3843730