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The spectrum problem. I: \(\aleph_{\epsilon}\)-saturated models, the main gap. (English) Zbl 0532.03013
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 M. Makkai and L. Harrington, and in forthcoming books by D. Lascar and by J. Baldwin.
Reviewer: A.Pillay

MSC:
03C45 Classification theory, stability and related concepts in model theory
03C50 Models with special properties (saturated, rigid, etc.)
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