An exposition of Shelah's 'main gap': counting uncountable models of \(\omega\)-stable and superstable theories
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.
scientific article; zbMATH DE number 3946101