The number of the models of complete Horn theories of finite depth
Let T be an elementary theory. The spectrum of T is the cardinal function \(J_ T\) that associates with each cardinal \(\omega_{\alpha}\) the number of nonisomorphic T-models of cardinality \(\omega_{\alpha}\). \(J_ T\) is called a maximal spectrum if \(J_ T(\omega_{\alpha})=2^{\omega_{\alpha}}\) for \(\alpha\geq 1\). Developing ideas of S. Shelah, \textit{J. Saffe} [Ann. Pure Appl. Logic 24, 231-261 (1983; Zbl 0518.03010)] had given a complete description of the spectra of \(\omega\)-stable theories. The results of \textit{J. Saffe, E. A. Palyutin} and \textit{S. S. Starchenko} [Algebra Logika 24, No.3, 278-326 (1985; Zbl 0597.03017)] reduced the problem of describing the spectra for complete (countable) Horn theories to the problem of describing the nonmaximal spectra for complete Horn theories of finite depth. In the present paper, this last problem is solved.
scientific article; zbMATH DE number 4039869