Abstract
We define ranks and degrees for families of theories, similar to Morley rank and degree, as well as Cantor–Bendixson rank and degree, and the notion of totally transcendental family of theories. Bounds for \(e\)-spectra with respect to ranks and degrees are found. It is shown that the ranks and the degrees are preserved under \(E\)-closures and values for the ranks and the degrees are characterized. Criteria for totally transcendental families in terms of cardinality of \(E\)-closure and of the \(e\)-spectrum value, for a countable language, are proved.
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Funding
The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0002) and the Committee of Science in Education and the Science Ministry of the Republic of Kazakhstan (grant no. AP08855544).
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(Submitted by I. Sh. Kalimullin)
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Sudoplatov, S.V. Ranks for Families of Theories and Their Spectra. Lobachevskii J Math 42, 2959–2968 (2021). https://doi.org/10.1134/S1995080221120313
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DOI: https://doi.org/10.1134/S1995080221120313