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.

Formulas and properties

“…Following [2] we define the rank RS(•) for families T ⊆ T Σ , similar to Morley rank for a fixed theory, and a hierarchy with respect to these ranks in the following way.…”

“…Definition [2]. For the empty family T we put the rank RS(T ) = −1, for finite nonempty families T we put RS(T ) = 0, and for infinite families T -RS(T ) ≥ 1.…”

“…By the definition a family T is e-minimal iff RS(T ) = 1 and ds(T ) = 1 [2], and iff T has unique accumulation point [5].…”

“…In this section we introduce a measure of complexity for sentences satisfying a property using the RS-rank for families of theories [2,3,4].…”

“…If P is infinite then T belongs to the E-closure Cl E (P ) [2,6] of P as unique element of α-th Cantor-Bendixson derivative of Cl E (P ), i.e., an element of Cl E (P ) having Cantor-Bendixson rank CB(Cl E (P )) = α [2].…”