Ranks for families of theories and their spectra

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.