Families of elementary theories

The study of families of elementary theories provides information about the behavior and interrelationships of theories within families, the possibility of generation, and their complexity. This complexity is expressed by rank characteristics both for families and for their elements within families. The paper introduces and describes a hierarchy of families of theories and their rank characteristics, including the dynamics of ranks. We consider regular families based on a family of pre-elements, the theories of a given signature, and a stepwise process that defines the hierarchy we are looking for. An ordinal-valued set-theoretic rank is used to reflect the steps of this process. The RS rank and associated ranks for regular families with respect to propositional definable subfamilies are introduced, and the known RS rank for families of primes is generalized as well as their degree. On the basis of separability of sets of prefamilies, connections and dynamics for these ranks and degrees are described. Graphs and neighborhood families indicative of ranks are introduced and characterized. It is shown that decompositions of neighborhood families and rank relations, for discrete decompositions, specify additivity and the possibility to reduce complexity measures for families to simpler subfamilies.