The spectrum problem. I: \(\aleph_{\epsilon}\)-saturated models, the main gap
type: Item
instance of: scholarly article
title: The spectrum problem. I: \(\aleph_{\epsilon}\)-saturated models, the main gap
zbMATH Open document ID: 0532.03013
DOI: 10.1007/BF02761237
author: S. Shelah
published in: Israel Journal of Mathematics
publication date: 1982-01-01T00:00:00Z
review text: 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}.
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C50
zbMATH DE Number: 3843730
zbMATH Keywords: superstable theory
zbMATH Keywords: orthogonality
zbMATH Keywords: saturated models
zbMATH Keywords: DOP
zbMATH Keywords: dimensional order property
zbMATH Keywords: depth
zbMATH Keywords: stable theories
reviewed by: Anand Pillay
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3843730