An exposition of Shelah's 'main gap': counting uncountable models of \(\omega\)-stable and superstable theories
type: Item
instance of: scholarly article
title: An exposition of Shelah's 'main gap': counting uncountable models of \(\omega\)-stable and superstable theories
zbMATH Open document ID: 0589.03015
author: Michael Makkai
author: Leo Harrington
published in: Notre Dame Journal of Formal Logic
publication date: 1985-01-01T00:00:00Z
review text: For a countable complete first-order theory T and a cardinal number \(\lambda\) let I(\(\lambda\),T) denote the number of isomorphism types of models of T having the cardinality \(\lambda\). In 1982, using his profound classification of complete theories, S. Shelah almost completely described possible spectrum functions \(\lambda\) \(\mapsto I(\lambda,T)\) for uncountable \(\lambda\). (Formulations of the results are given and commented by \textit{S. Shelah} [Bull. Amer. Math. Soc., New Ser. 12, 227- 232 (1985; Zbl 0578.03016)]; proofs will appear in the forthcoming second edition of his book ''Classification theory and the number of non- isomorphic models'' (1978; Zbl 0388.03009).) In particular, he proved that either \(I(\aleph_{\alpha},T)=2^{\aleph_{\alpha}}\) for every \(\alpha >0\), or \(I(\aleph_{\alpha},T)<\beth_{\omega_ 1}(| \omega +\alpha |)\) for every \(\alpha >0\). There is no complete description of possible spectrum functions yet, but the proof of the result above gives a nearly exhaustive information on the question. In Shelah's opinion, the main point in the spectrum problem is the ''main gap theorem'', establishing the dichotomy above. In the paper under review (circulating as a preprint since 1981) the authors expound earlier partial \textit{S. Shelah}'s results concerning the ''main gap theorem'' [Isr. J. Math. 43, 324-356, 357-364 (1982; Zbl 0532.03013 and Zbl 0532.03014)], in particular, a proof of the ''main gap theorem'' for totally transcendental theories.
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 3946101
zbMATH Keywords: forking
zbMATH Keywords: spectrum functions
zbMATH Keywords: totally transcendental theories
reviewed by: Oleg Belegradek
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093870822
OpenAlex ID: W1994734519
DOI: 10.1305/NDJFL/1093870822
description: scientific article; zbMATH DE number 3946101